![Preface
Mathematical morphology is a discipline of image analysis that was introduced
in the mid-1960s by two researchers at the École des Mines in Paris: Georges
Matheron [MAT 75] and Jean Serra [SER 82, SER 88c]. Historically, this was the
first consistent nonlinear image analysis theory, which from the very start included
not only theoretical results but also many practical aspects. Its initial objective was to
facilitate studies of mineral deposits via sampling. It was implemented using dedicated
image processing hardware, akin to analog computers in many ways. Mathematical
morphology was endowed from the very beginning with the three pillars which
ensured its success: a solid theoretical foundation, a large body of applications and
an efficient implementation.
Since this heroic era, many developments have been proposed. Indeed, many
unforeseen applications have been developed: in materials science and in the life
sciences, for example. The techniques eventually become popular internationally and
improved to the level where they are now, more than 40 years after their beginning.
Since 1993, a regular and well-attended series of international symposiums dedicated
to the discipline have taken place and many journals have mathematical morphology
tracks and special issues. Mathematical morphology is now part of the basic body of
techniques taught to any student of image processing courses anywhere; most image
processing software packages feature morphology toolboxes and filters, including the
most popular programs such as Photoshop or Matlab. Far from being an academic
pursuit, morphology is used in industry and businesses at many levels, for example:
quality control in industrial production, medical imaging, document processing and
much more.
In spite of this popularity, researchers and practitioners in mathematical morphol-
ogy often find that their operators and functions are not understood as well as they
could be. For instance, many newcomers to the discipline think it only applies to
binary images (images featuring only two levels: pure black and pure white). On the
contrary, mathematical morphology is a complete theory capable of handling the most
xv](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-19-320.jpg)
![xvi Mathematical Morphology
varied image types in a way that is often subtle yet efficient. Morphology can be
used to process certain types of noise in images, but can also be used more generally
in filtering, segmentation, classification, analysis and coding of visual-type data. It
can also be used to process general graphs, surfaces, implicit and explicit volumes,
manifolds and time or spectral series in both deterministic and stochastic contexts.
One of the reasons for this lack of understanding might be the relative lack of
recent and comprehensive books on the topic [DOU 93, DOU 03b, HEI 94a, SOI 03a].
We were therefore very honored when Henri Maître, director of the Image and Signal
collection at Hermès Publishing in France, asked us to propose, compile and edit
contributions from some of the best-known researchers and practitioners in the field in
order to showcase the capabilities of mathematical morphology. Thanks to ISTE and
John Wiley and Sons, we are now pleased to provide this book in English. Its content
has been thoroughly revised and significantly expanded from the French language
version.
The primary goal of this book is to expose the state of the art in mathematical
morphology in a didactic fashion. However, our authors did not limit themselves to
this exercise, but also developed some original and novel content. They took advantage
of this opportunity to reformulate, rework and rethink the themes they work with most
often, in order to make them available to a greater audience in a unique format. We
are also of course very honored by the confidence afforded to us by all our numerous
contributors. We take this opportunity to thank them and applaud their efforts. This
book has taken a very long time to come to fruition, but our authors have been a
pleasure to work with all along. We hope the end result meets their expectation.
Among our authors, we particularly wish to thank Christian Ronse and Jean Serra,
who have both helped us immensely to improved the general quality of the book.
The 20 chapters are divided into 5 parts as follows:
– The first part explains the fundamental aspects of the discipline. Starting with
a general introduction, two more theoretical chapters follow. The first of these is
concerned with mathematical structure, including a modernized formalism which is
the result of several decades of work.
– The second part extends morphology into image analysis, in particular detailing
how estimations, choices and measurements can be made. This is achived through
links with other disciplines such as stereology, geostatistics and fuzzy logic.
– The third part concerns the theory of morphological filtering and segmentation,
insisting on modern connected approaches from both the theoretical and practical
aspects.
– The fourth part exposes some practical aspects of mathematical morphology, in
particular, how to deal with color and multivariate data. Links to discrete geometry
and topology and some algorithmic aspects are included, without which applications
would be impossible.](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-20-320.jpg)
![Chapter 1
Introduction to Mathematical Morphology
In this chapter we endeavor to introduce in a concise way the main aspects
of Mathematical Morphology, as well as what constitutes its field. This question
is difficult, not so much as a technical matter but as a question of starting point.
Historically, mathematical morphology began as a technique to study random sets
with applications to the mining industry. It was rapidly extended to work with two-
dimensional (2D) images in a deterministic framework first with binary images, then
gray-level and later to color and multispectral data and in dimensions > 2. The
framework of mathematical morphology encompasses many various mathematical
disciplines from set theory including lattice theory, random sets, probabilities, measure
theory, topology, discrete and continuous geometry, as well as algorithmic considera-
tions and finally applications.
The main principle of morphological analysis is to extract knowledge from the
response of various transformations which are generally nonlinear.
One difficulty in the way mathematical morphology has been developed and
expanded [MAT 75, SER 82, SER 88c] (see also [HEI 94a, SCH 94, SOI 03a]) is
that its general properties do not fall within the general topics taught at school
and universities (with the exception of relatively advanced graduate-level courses).
Classical mathematics define a function as an operator associating a single point in
a domain with a single value. A contrario, in morphology we associate whole sets
with other whole sets. The consequences of this are important. For instance, if a point
generally has zero measure, this is not generally the case for sets. Consequently, while
a probability of the presence of a point may be zero, this is not the case for a set.
Chapter written by Laurent NAJMAN and Hugues TALBOT.
3
Mathematical Morphology: From Theory to Applications Edited by Laurent Najman and Hugues Talbot
© 2010 ISTE Ltd. Published 2010 by ISTE Ltd.](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-25-320.jpg)
![4 Mathematical Morphology
In addition, we can compare morphology to other image processing disciplines.
For instance, linear operator theory assumes that images are merely a multidimen-
sional signal. We also assume that signals combine themselves additively. The main
mathematical structure is the vector space and basic operators are those that preserve
this structure and commute with basic rules (in this case, addition and multiplication
by a constant). From this point deriving convolution operators is natural; hence it
is also natural to study Fourier or wavelet transforms. It is also natural to study
decomposition by projections on basis vectors. This way is of course extremely
productive and fruitful, but it is not the complete story.
Indeed, very often a 2D image is not only a signal but corresponds to a projection
of a larger 3D ‘reality’ onto a sensor via an optical system of some kind. Two objects
that overlap each other due to the projections do not add their feature but, on the
contrary, create occlusions. The addition is not the most natural operator in this case.
It makes more sense to think in terms of overlapping objects and therefore, in terms
of sets, their union, intersections and so on. With morphology, we characterize what is
seen via geometrical transforms, taking into account shapes, connectivity, orientation,
size, etc. The mathematical structure that is most adapted to this context is not the
vector space, but the generalization of set theory to complete lattices [BIR 95].
1.1. First steps with mathematical morphology: dilations and erosions
In order to be able to define mathematical morphology operators, we need to
introduce the abstract notion of complete lattice. We shall then be able to ‘perform’
morphology on any instance of such a lattice.
1.1.1. The notion of complete lattice
A lattice [BIR 95] (E, ≤) is a set E (the space) endowed with an ordering
relationship ≤ which is reflexive (∀x ∈ E, x ≤ x), anti-symmetric (x ≤ y and
y ≤ x ⇒ x = y) and transitive (x ≤ y and y ≤ z ⇒ x ≤ z). This ordering is such
that for all x and y, we can define both a larger element x ∨ y and a smaller element
x ∧ y. Such a lattice is said to be complete if any subset P of E has a supremum
W
P
and an infimum
V
P that both belong to E. The supremum is formally the smallest of
all elements of E that are greater than all the elements of P. Conversely, the infimum
is the largest element of E that is smaller than all the elements of P. In a lattice,
supremum and infimum play symmetric roles. In particular, if we consider the lattice
P[E] constituted by the collection of all the subsets of set E, two operators ψ and ψ∗
are dual if, for all X, ψ(Xc
= [ψ∗
(X)]) where Xc
= E X is the complement of X
in E.](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-26-320.jpg)
![Introduction to Mathematical Morphology 5
1.1.2. Examples of lattices
Figure 1.1 is an example of a lattice. This instance is simple but informative, as
it corresponds to the lattice of primary additive colors (red, green and blue). Each
element of the lattice is a binary 3-vector, where 0 represents the absence of a primary
color and 1 its presence. The color black is represented by [0, 0, 0] and white by
[1, 1, 1]. Pure red is [1, 0, 0], pure green is [0, 1, 0], and so on. Magenta is represented
by [1, 0, 1]. In this lattice, there does not exist a way to directly compare pure green
and pure blue or magenta and yellow: the order is not total. However, white is greater
(brighter) than all colors and black is smaller (darker). Whatever subset of colors is
chosen, it is always possible to define a supremum by selecting the maximal individual
component among the colors of the set (e.g. the supremum of [1, 0, 0] and [0, 0, 1] is
[1, 0, 1]). This supremum may not be in the subset, but it belongs to the original lattice.
Similarly, the infimum is defined by taking the minimal individual component.
1 1
1
0
0 0
1 0 0 0 1 0 0 0 1
0
1
1 1 0 1 0 1 1
Figure 1.1. An example of a lattice: the lattice of additive primary colors
Another example of a lattice is the set of real numbers R endowed with the
usual order relation. This lattice is not complete since, for instance, the subset of
integer numbers has +∞ as supremum but +∞ is not part of R. In contrast, R =
R ∪ {−∞, +∞} is a complete lattice. Through these examples, we can see that the
notion of complete lattice is not fundamentally difficult.
1.1.2.1. Lattice and order
Many morphological operators preserve the ordering structure. We call such
operators Φ increasing and express it by ∀x, y ∈ (E, ≤), x ≤ y ⇒ Φ(x) ≤ Φ(y).
Others will transform input lattice elements into larger or smaller elements. If we have
an operator Ψ which is such that ∀x ∈ (E, ≤), x ≤ Ψ(x) then the operator is called
extensive: it will enlarge elements. Conversely, if Ψ(x) ≤ x, then the operator is anti-
extensive: it will shrink them. The simplest operators we can introduce on a lattice are
those that commute with the supremum or the infimum. Respectively, these operators
are called abstract dilation and erosion. Under various conditions such operators can
combine some of these properties, as we will see shortly.
While these definitions are straightforward and relatively easily understood after
some period of familiarization, there is a legitimate question as to why morphologists](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-27-320.jpg)
![Introduction to Mathematical Morphology 9
set Xc
using the symmetric structuring element of B, denoted B̌. Let p, q be two
points where p ∈ Bq ⇔ q ∈ B̌p. This amounts to B̌ = {−b|b ∈ B}:
(X ⊕ B)c
= Xc
⊖ B̌, and (X ⊖ B)c
= Xc
⊕ B̌.
We illustrate this property in Figure 1.5 using the erosion as an example.
Figure 1.5. The erosion of the cross of Figure 1.3, using the property that the dilation with the
symmetric structuring element is the dual of this operation
One way to extend the binary operators to the gray-level case is to take the
hypograph SG(F) of a function F:
SG(F) = {(x, t) ∈ E × T |t ≤ F(x)}.
Using this approach, dilating (respectively, eroding) a gray-level image is equiva-
lent to dilating (respectively, eroding) each of its thresholds.
An equivalent approach consists of using the lattice of functions, using the order
structure provided by the order on T . In particular, for two functions F, G ∈ T E
, we
obtain:
F ≤ G ⇐⇒ ∀x ∈ E, F(x) ≤ G(x).
In this way, equations (1.2) and (1.3) translate in the following manner:
δG(F)(x) = (F ⊕ G)(x) = sup
y∈E
{F(y) + G(x − y)} (1.4)
and
εG(F)(x) = (F ⊖ G)(x) = inf
y∈E
{F(y) − G(y − x)}. (1.5)
In these equations, function G is a structuring function. This function may be
arbitrary, for instance sometimes parabolic functions are used in operations [BOO 96]
such as the Euclidean distance transform [MEH 99].](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-31-320.jpg)
![Introduction to Mathematical Morphology 17
a b
(a) (b)
Figure 1.13. The local grid. In the square grid case, we can specify that each point is
connected to its four nearest neighbors as in (a), or its 8 neighbors including the diagonal
pixels as in (b). In the case of the hexagonal grid in (c), each pixel has 6 neighbors
simple closed curve (a closed curve that does not self-intersect) divides the plane into
two distinct regions which are connected within themselves: one is of finite extent
and the other not. In the discrete case, this property is not true by default. The Jordan
problem is illustrated in Figure 1.14.
(a) (b) (c)
Figure 1.14. The discrete Jordan property (not true in the square grid by default): (a) the
non-degenerate, simple path separates the discrete plane into three connected components; (b)
the path does not separate anything at all; and (c) the Jordan property is true (always the case
with the hexagonal grid)
If the grid in Figure 1.14a is 4-connected, the subset of the plane delimited by the
path is not connected. If the grid in Figure 1.14b is 8-connected the path does not
separate the inside of the curve from the outside. In contrast, with the hexagonal grid
it is possible to show that these problems never occur.
In order to solve this problem in a pragmatic way, image analysts often consider
two kinds of connectivity [ROS 73, ROS 75] concurrently: one for the foreground ob-
jects (inside the curves) and one for the background (outside). A more mathematically
meaningful way of solving this problem is to consider a more complete topology for
the discrete grid, e.g. following Khalimski [KHA 90].](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-39-320.jpg)
![Introduction to Mathematical Morphology 19
(a) (b)
Figure 1.15. Area closing using a parameter of 20 square pixels. Small minima in the image
were filled adaptively. Maxima in the image are unaffected (e.g. the small fiber)
opening by a structuring element of area λ that preserves C, for instance the opening
that uses C itself as a structuring element (or any subset of C with area λ).
Knowing that we should preserve all connected components with area at least λ,
we deduce that we can consider (at least conceptually) all possible openings with all
connected structuring elements of area λ. It is easy to show that the supremum of
these openings, i.e. the operator that at each point preserves the maximum of all these
openings, is itself an opening and that it preserves all connected components with area
at least λ. The supremum is therefore the area opening with parameter λ.
It would be theoretically possible to implement the area opening operator by
computing the result of all the possible openings using all connected structuring
elements with area λ. However, this would be very inefficient as the size of the family
of structuring elements increases exponentially with λ. However, the representation of
an opening (or a closing) by such a morphological family is useful from the theoretical
point of view. There exists a theorem by Matheron [MAT 75] that demonstrates the
existence of a morphological decomposition for all openings and closings. For more
details, see section 2.4.3.
From the practical point of view, it is useful to remember that a combination by a
supremum of openings is itself an opening. Respectively, a composition by an infimum
of closings is also a closing. These filters are called algebraic openings (respectively,
closings).
In Chapter 12, we study how to implement some algebraic filters in practice.
As an illustration, Figure 1.16 depicts an application of various algebraic filters
to the denoising of thin objects. We used a closing by infimum combination of
closings using various structuring elements families, either line segments or adaptive
paths [HEI 05]. The objective here is to preserve the object of interest while filtering](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-41-320.jpg)
![Introduction to Mathematical Morphology 23
What makes the morphological approach different from and complementary to
many other approaches is the fact that morphological operators do not seek to preserve
information present in an image. Indeed, since the basic operators of morphology
are not invertible, we expect a reduction in information content after each operator
application. The key to success with morphology is to realize this, and to use this
defining characteristic to our advantage. We can achieve this by selectively destroying
the undesirable content of the image: noise, background irregularities, etc. while
preserving the desired content for as long as possible. Figures 1.15 and 1.16 are direct
illustrations of this philosophy.
It is sometimes necessary to destroy undesirable content in an image, but not
practical to do so. A complementary tactic is to effectively erase the desirable portion
of an image, but to restore it through a difference with the original image. This gives
rise to the idea of residues.
Simply put, residues are transforms that involve combinations of morphological
operators with the differences (or subtractions). Top-hat transforms, morphological
gradients [RIV 93] and other similar transforms that we present in the next section are
all examples of residues.
Residues are generally well behaved in morphology, precisely because the basic
properties of morphological operators are in our favour. For instance, because
openings are anti-extensive, the difference between the original image and any
opening derived from that image will always be positive.
1.3.1. Gradients
The gradient of an image is basically its first derivative. For a 2D or generally nD
image (n > 1), the gradient operator produces an n-vector at each point, where each
component corresponds to the slope along the n principal directions of the grid in the
discrete space. With morphology, we most often only consider the magnitude of the
gradient at each point, which is a scalar irrespective of the dimension of the image.
Gradients show the amount of local variation in the image. Zones of high gradient
typically correspond to object contours or texture.
Using erosions and dilations, we can define the morphological gradient as follows:
Grad(F) = δB(F) − εB(F). (1.17)
There are two other definitions:
Grad+
(F) = δB(F) − F
Grad−
(F) = F − εB(F). (1.18)](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-45-320.jpg)
![Introduction to Mathematical Morphology 25
(a) (b)
Figure 1.19. Top-hat on an image of cells: (a) original image (small bright spots are vesicles
in the cells, made fluorescent through the use of a bio-molecular marker); (b) the result of a
thresholding of the white top-hat overlaid over the original image
(a) (b) (c)
Figure 1.20. The distance transform of a set X is the application that from each point of X
associates its distance to the complementary set. In this illustration we consider the 8-distance
but this is by no means a rule: (a) set; (b) distance transform; (c) level sets
1.4.1. Maximal balls and skeletons
A maximal ball with radius R is the set of points located at a distance less than or
equal to R from a central point p. It is obvious this definition depends on the distance
used. For instance, using the 8-distance in 2D, the ball of radius 3 is a 7 × 7 square.
When the Euclidean distance is used, the ball is a discrete disk.
A maximal ball B relative to a set E is a ball such that there does not exist a ball
B′
such that B ⊂ B′
⊂ E (see Figure 1.21). This apparently simple notion is useful
to define some interesting residues, in particular the skeleton. This notion has been
known since the 1930s [BOU 32, DUR 30, DUR 31], but was popularized in image
processing contexts in the 1960s [BLU 61, CAL 68] under the name of medial axis of
E. The medial axis is defined as the collection of all centers of maximal balls of E.
In Euclidean space, the medial axis is called the skeleton. The skeleton of a set O
that is connected, open, non-empty and bounded has many interesting properties from](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-47-320.jpg)
![26 Mathematical Morphology
Figure 1.21. Two balls included in a binary set. The ball centered at point M is maximal
because no other ball from the same family (here a family of disks) can contain it and
simultaneously be included in the binary set. The ball centered in N is not maximal
the geometrical and topological point of view: it is connected, homotopic to O and
negligible from the point of view of the Lebesgue measure (although it may be dense
everywhere in O) [RIV 87].
In the discrete case, the centers of maximal balls in O are well defined but are
not necessarily located on the grid; the medial axis may therefore not be connected.
We then define informally the discrete skeleton S(O) as a connected set, included in
O, homotopic to O and as thin as possible. This notion is disjoint from the notion of
medial axis, but it is possible to constrain the discrete skeleton to contain the medial
axis. In practice, both discrete skeletons and medial axes are often noisy. To be able to
use them in practice, we must be able to filter them.
In Chapter 10, these notions are more precisely defined and algorithms are given
to compute them. An example of filtered medial axis and filtered skeleton are given in
Figure 1.22.
(a) (b) (c)
Figure 1.22. An example of a skeleton: (a) a binary image; (b) its filtered medial axis; and (c)
a filtered skeleton of (a) that contains the medial axis](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-48-320.jpg)
![Introduction to Mathematical Morphology 27
1.4.2. Granulometric curves
Mathematical morphology, even within itself, is capable of providing information
on the size of objects in images in several different ways. One of these methods
relies on the notion of granulometries, which is directly derived from the notion of
morphological filter (see section 1.2.6 and Chapter 10).
The granulometric curve of an image is a representation of the distribution of
sizes in an image. This is based on the observation that intermediate residues of a
granulometric family γn, indexed by n, are characteristic of the size of objects in
images. More formally, the granulometric curve is the function Gf (λ) defined on the
interval of λ, such that:
GI(λ) =
X
I −
X
γλ(I) λ ∈ [0, 1, 2, . . ., R] (1.19)
where I is the input image,
P
I is the sum (or integral) of all the pixels in the image
and R is the value of λ for which no further change occurs due to γR (since the image
has become constant). For finite discrete images, the value R always exists. If we use
closings instead of openings, the operands on either side of the subtraction sign in
equation (1.19) are exchanged (since all closings are extensive, and so ϕ(I) ≥ I).
Granulometric curves by openings make it possible to estimate the size of peaks
in images, while granulometries by closings measure the size of troughs.
It is possible to build a granulometric curve using both openings and closings.
In order to obtain a single continuous curve, convention states that the parameter of
the closings is given in abscissa from the origin towards the negative value and the
parameter of the openings is given from the origin towards the positive values. The
same sign convention is used for both openings and closings; such a curve is depicted
in Figure 1.23.
In this example, we illustrate the fact that the granulometric curve records the
volume of image (i.e. the area of features times their gray level) that is erased beyond
a certain size, both for openings and closings.
1.4.2.1. Applications
The granulometric curve summarizes the distribution of size of objects in an
image, without necessitating a segmentation step. Consequently, the notion of object
is not well defined in this context. We can only talk about volumes of gray levels. In
addition, since the granulometric curve is one-dimensional, the information content is
necessarily reduced from that of the whole image.
The interpretation of this content is not always easy. It can sometimes be
interpreted in terms of texture energy, as in the example given in Figure 1.24](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-49-320.jpg)
![28 Mathematical Morphology
(a)
-10
-5
0
5
10
15
20
25
30
35
-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80
Arbitrary
area
units
Closing size Opening size
Granulometry curve
(b)
Figure 1.23. A granulometric curve by openings and closings of an artificial image, using a
family of Euclidean disks as structuring elements (indexed by their radii). The increasing sizes
of openings go from the origin towards +∞ and the increasing size of closings go from the
origin to −∞: (a) image and (b) granulometric curve
in the context of a study involving the aging of steam pipes used in electricity
production. Many applications use granulometries to estimate size-related parameters
for subsequent procedures; see [COM 07].
1.4.2.2. Granulometries by erosions and dilations
It is also possible to produce granulometries by using only erosion or dilations.
There is a strong link between these and skeletons [MAT 92].
1.4.3. Median set and morphological interpolation
Another application of distances worth mentioning, also related to skeletons,
is their capacity for computing a median set used as an interpolation algorithm.
In the literature, median sets appeared in the work of Casas [CAS 96] and Meyer
[MEY 96]. The equation of the underlying operation and its basic properties were
given by Serra in [SER 98b]. Iwanowski has successfully developed it for various
morphings on still images and video sequences, in black and white and in color
[IWA 00a]. More recently, Vidal et al. used a recursive technique for improving the
interpolations [VID 07].
Recall that the Hausdorff distance is the maximum distance of a set from the
nearest point in the other set. It measures how far two subsets of a metric space are
from each other. Informally, two sets are close in the Hausdorff distance if every point
of either set is close to some point of the other set.](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-50-320.jpg)
![Introduction to Mathematical Morphology 29
(a) (b)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100
rate
of
loss
radius
(c)
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
rate
of
loss
radius
(d)
Figure 1.24. Application of granulometries: granulometric curves of surface microscopy
images of pig iron steam pipes, in the case of a young pipe and an old pipe. Age deteriorates
grain boundaries, which results in a larger number of small grains compared with younger
samples: (a) young sample; (b) old sample; (c) young sample curve; (d) old sample curve
Consider an ordered pair of closed sets {X, Y } with X ⊆ Y and such that
their Hausdorff distance is finite. Their median element is the closed set M(X, Y ),
composed of X and Y and whose boundary points are equidistant from X and the
complement Y c
to Y . In other words, the boundary of M is nothing but the skeleton
by zone of influence (also known as the generalized Voronoï diagram) between X and
Y c
.
The set M can depend on a parameter α which weights the relative importances
of X and Y in the interpolation. The analytic expression of the weighted median set
Mα(X, Y ) is obtained from its two primitives X, Y by taking the union [SER 98b]:
Mα(X, Y ) = ∪λ {(X ⊕ αλB) ∩ (Y ⊖ (1 − α)λB)} . (1.20)
In the general case, for two sets A and B with non-empty intersection (i.e.
A ∩ B 6= ∅), we set X = A ∩ B and Y = A ∪ B so that X ⊂ Y and apply
equation (1.20). Figure 1.25, whose steps are described in more detail in [IWA 00b],
depicts the progressive passage from a bird to a plane as α varies from 0 to 1. As the
map defined by equation (1.20) is increasing for both X and Y , it extends directly to
digital numerical functions by simply replacing union and intersection by supremum
and infimum, respectively.](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-51-320.jpg)
![Introduction to Mathematical Morphology 31
bodies meet. Although both visions are equivalent in the continuous domain under
sufficient regularity assumptions [NAJ 94b], they are not compatible in all discrete
frameworks and, notably, not compatible in the pixel framework [COU 07c, NAJ 05].
Although the previous explanation may not appear to be very formal, the literature
on the topic of watershed properties and algorithms is abundant; see [BEU 79b,
COU 05, MEY 94b, ROE 01, VIN 91c]. The formalization of the various concepts
derived from the watershed in the discrete case, as well as the mathematical properties
of the objects so obtained, are presented in Chapter 3.
The watershed line transform forms the basis of a powerful and flexible segmen-
tation methodology introduced at the Centre de Morphologie Mathematique in the
1970s [BEU 79b] and further developed in the 1990s [MEY 90b, VIN 91c]. This
methodology was later unified using hierarchical approaches [BEU 94, NAJ 96].
The general idea is that we first need to produce internal markers of the objects
under study. These are binary sets which can be labeled (i.e. given a distinct gray
level for each connected component), that are included in the objects sought. The
shape of these markers is unimportant; only their position and their extent matter.
In a similar manner, we seek markers that are external to the objects, i.e. totally
included in the background. A function which exhibits high values near contours and
low values in a near-constant area of the image is used. Usually some regularized
version of the gradient operator can be employed. This function is then reconstructed
using the geodesic reconstruction operator of section 1.2.2 by imposing all markers
(both internal and external) as minima in this function, and by eliminating the original
minima present in the function. A single watershed line is then present. This separates
internal and external markers, and tends to place itself on the contour of objects to be
segmented.
Many chapters of the Applications part of this book (Part V, notably Chapters 14
and 17) use one of the many variations of the watershed. It is therefore useful to
illustrate the above procedure on a simple example due to Gratin [GRA 93]. Here we
seek to segment a 2D magnetic resonance image (MRI) of an egg. On this image
(Figure 1.27), markers for the exterior of the egg, the white and the yolk are set
manually, but it is of course possible to obtain these through an automated procedure.
Contrary to expectations and despite the simple nature of the problem, a simple
thresholding does not yield good results due to the high level of noise. In contrast,
the watershed segmentation procedure result is almost perfect.
The general methodology for morphological segmentation is developed further in
Chapter 9. It relies on defining some criterion that induces a hierarchy of segmenta-
tions, i.e. a nested sequence of connected partitions. Any hierarchy of segmentations
is equivalent to a specific watershed referred to as a saliency map [NAJ 96] or](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-53-320.jpg)
![32 Mathematical Morphology
(a) (b)
(c) (d)
Figure 1.27. MRI of an egg: (a) original image (courtesy of N. Roberts, University of
Liverpool); (b) the gradient of this image (seen as a 3D terrain); (c) manually set markers;
and (d) result of the segmentation
ultrametric watershed [NAJ 09a, NAJ 09b]. Filtering such a watershed amounts to
transforming the hierarchy into another watershed.
Figure 1.28 illustrates the principle on an image of uranium oxide. We want
to extract the cells but, unfortunately, a brute-force watershed application gives an
oversegmented image. Instead of trying to find some markers, we can filter the image
to remove the background noise. Here the chosen filter depends on a depth criterion
(see Chapters 7 and 9). Rather than setting a fixed level of noise reduction for the
filtering, it is better to compute the whole hierarchy of segmentations that can be
obtained by varying the parameter.
The resulting hierarchy is represented as a saliency map in Figure 1.28b. Any
threshold of Figure 1.28b gives a segmentation. The more a contour is present in
the hierarchy of segmentations, the more visible it is. It can be seen that there is
a large difference between the noise contours and the ‘true’ contours; choosing the
correct level of thresholding is therefore easy (Figure 1.28c). It is even possible to](https://image.slidesharecdn.com/2153899-250605120412-bae68fd0/85/Mathematical-Morphology-From-Theory-To-Applications-Laurent-Najman-54-320.jpg)
Mathematical Morphology From Theory To Applications Laurent Najman
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- 6. To our families who have put up with us as we were writing this book, for nearly three years now. Deepest love and thanks to Laurence, Annick Zoé, Ilan, Sophie and Shaï.
- 7. Mathematical Morphology From Theory to Applications Edited by Laurent Najman Hugues Talbot
- 8. First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from two volumes Morphologie mathématique 1 & 2 published 2008 and 2010 in France by Hermes Science/Lavoisier © LAVOISIER 2008, 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2010 The rights of Laurent Najman and Hugues Talbot to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Mathematical morphology / edited by Laurent Najman, Hugues Talbot. p. cm. “Adapted and updated from two volumes Morphologie mathématique 1, 2 published 2008 and 2010 in France by Hermes Science/Lavoisier” Includes bibliographical references and index. ISBN 978-1-84821-215-2 1. Image analysis. 2. Image processing--Mathematics. I. Najman, Laurent. II. Talbot, Hugues. TA1637.M35963 2010 621.36'70151--dc22 2010020106 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-215-2 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.
- 9. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv PART I. FOUNDATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 1. Introduction to Mathematical Morphology . . . . . . . . . . . . 3 Laurent NAJMAN, Hugues TALBOT 1.1. First steps with mathematical morphology: dilations and erosions . . . 4 1.1.1. The notion of complete lattice . . . . . . . . . . . . . . . . . . . . 4 1.1.2. Examples of lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3. Elementary operators . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.4. Hit-or-miss transforms . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2. Morphological filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1. Openings and closings using structuring elements . . . . . . . . . 12 1.2.2. Geodesy and reconstruction . . . . . . . . . . . . . . . . . . . . . . 13 1.2.3. Connected filtering and levelings . . . . . . . . . . . . . . . . . . . 18 1.2.4. Area openings and closings . . . . . . . . . . . . . . . . . . . . . . 18 1.2.5. Algebraic filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.6. Granulometric families . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.7. Alternating sequential filters . . . . . . . . . . . . . . . . . . . . . 21 1.3. Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.1. Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.2. Top-hat transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4. Distance transform, skeletons and granulometric curves . . . . . . . . . 24 1.4.1. Maximal balls and skeletons . . . . . . . . . . . . . . . . . . . . . 25 1.4.2. Granulometric curves . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4.3. Median set and morphological interpolation . . . . . . . . . . . . . 28 1.5. Hierarchies and the watershed transform . . . . . . . . . . . . . . . . . . 30 1.6. Some concluding thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . 33 v
- 10. vi Mathematical Morphology Chapter 2. Algebraic Foundations of Morphology . . . . . . . . . . . . . . . 35 Christian RONSE, Jean SERRA 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2. Complete lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.1. Partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.2. Complete lattices and isomorphisms . . . . . . . . . . . . . . . . . 37 2.2.3. Remarkable elements and families . . . . . . . . . . . . . . . . . . 39 2.2.4. Distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.5. Boolean lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3. Examples of lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.1. Lattices of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.2. Lattices of numerical functions . . . . . . . . . . . . . . . . . . . . 44 2.3.3. Lattice of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3.4. Lattice of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.5. Monotone convergence and continuity . . . . . . . . . . . . . . . . 50 2.4. Closings and openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.4.1. Moore families and closings . . . . . . . . . . . . . . . . . . . . . 51 2.4.2. Openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.4.3. Generation of closings and openings . . . . . . . . . . . . . . . . . 54 2.5. Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.5.1. Adjunctions, dilations and erosions . . . . . . . . . . . . . . . . . 57 2.5.2. Set-theoretical case . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.5.3. Case of numerical functions . . . . . . . . . . . . . . . . . . . . . . 62 2.6. Connections and connective segmentation . . . . . . . . . . . . . . . . . 64 2.6.1. Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.6.2. Connective segmentation . . . . . . . . . . . . . . . . . . . . . . . 67 2.6.3. Examples of connective segmentations . . . . . . . . . . . . . . . 69 2.6.4. Partial connections and compound segmentations . . . . . . . . . 72 2.7. Morphological filtering and hierarchies . . . . . . . . . . . . . . . . . . 75 2.7.1. The lattice of filters . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.7.2. Connected filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.7.3. Hierarchies and Matheron semigroups . . . . . . . . . . . . . . . . 79 Chapter 3. Watersheds in Discrete Spaces . . . . . . . . . . . . . . . . . . . . 81 Gilles BERTRAND, Michel COUPRIE, Jean COUSTY, Laurent NAJMAN 3.1. Watersheds on the vertices of a graph . . . . . . . . . . . . . . . . . . . 82 3.1.1. Extensions and watersheds . . . . . . . . . . . . . . . . . . . . . . 83 3.1.2. W-thinnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.1.3. Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.1.4. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.2. Watershed cuts: watershed on the edges of a graph . . . . . . . . . . . . 90 3.2.1. Edge-weighted graphs . . . . . . . . . . . . . . . . . . . . . . . . . 90
- 11. Contents vii 3.2.2. Watershed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.2.3. Minimum spanning forests and watershed optimality . . . . . . . 94 3.2.4. Optimal thinnings . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.2.5. Watershed cuts and topological watersheds . . . . . . . . . . . . . 99 3.2.6. Application example . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.3. Watersheds in complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.3.1. Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.3.2. Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.3.3. Cuts in complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.3.4. Watersheds in complexes . . . . . . . . . . . . . . . . . . . . . . . 106 PART II. EVALUATING AND DECIDING . . . . . . . . . . . . . . . . . . . . . 109 Chapter 4. An Introduction to Measurement Theory for Image Analysis . 111 Hugues TALBOT, Jean SERRA, Laurent NAJMAN 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2. General requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.3. Convex ring and Minkowski functionals . . . . . . . . . . . . . . . . . . 113 4.3.1. The Euler–Poincaré characteristic . . . . . . . . . . . . . . . . . . 115 4.3.2. Euler–Poincaré characteristics in discrete space . . . . . . . . . . 116 4.4. Stereology and Minkowski functionals . . . . . . . . . . . . . . . . . . . 119 4.4.1. Generation of the Minkowski functionals . . . . . . . . . . . . . . 119 4.5. Change in scale and stationarity . . . . . . . . . . . . . . . . . . . . . . . 121 4.6. Individual objects and granulometries . . . . . . . . . . . . . . . . . . . 122 4.6.1. Unbiased counting estimates . . . . . . . . . . . . . . . . . . . . . 123 4.6.2. Number and measure granulometries . . . . . . . . . . . . . . . . 124 4.6.3. Linear granulometries . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.7. Gray-level extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.7.1. Area and volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.7.2. Gradient and perimeter . . . . . . . . . . . . . . . . . . . . . . . . 129 4.7.3. Numerical Euler–Poincaré characteristic . . . . . . . . . . . . . . 129 4.7.4. A counter-example: the length of a curve . . . . . . . . . . . . . . 130 4.8. As a conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Chapter 5. Stochastic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Christian LANTUÉJOUL 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.2. Random transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2.1. Estimating an integral . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2.2. Individual particle analysis . . . . . . . . . . . . . . . . . . . . . . 136 5.3. Random image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.3.1. Statistical characterization . . . . . . . . . . . . . . . . . . . . . . . 138 5.3.2. Integral range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
- 12. viii Mathematical Morphology 5.3.3. Specific parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.3.4. Synthesizing textures . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.3.5. Gaussian random function . . . . . . . . . . . . . . . . . . . . . . . 147 5.3.6. Boolean model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Chapter 6. Fuzzy Sets and Mathematical Morphology . . . . . . . . . . . . 155 Isabelle BLOCH 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.2. Background to fuzzy sets . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.2.1. Fuzzy sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2.2. Set theoretical operations . . . . . . . . . . . . . . . . . . . . . . . 158 6.3. Fuzzy dilations and erosions from duality principle . . . . . . . . . . . 160 6.3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.3.2. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.3.3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.4. Fuzzy dilations and erosions from adjunction principle . . . . . . . . . 165 6.4.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.4.2. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.5. Links between approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.5.1. Dual and adjoint operators . . . . . . . . . . . . . . . . . . . . . . 167 6.5.2. Equivalence condition between the two approaches . . . . . . . . 167 6.5.3. Illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.5.4. General form of fuzzy morphological dilations and erosions . . . 169 6.6. Application to the definition of spatial relations . . . . . . . . . . . . . . 170 6.6.1. Fuzzy topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.6.2. Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.6.3. Directional relative position between two objects . . . . . . . . . . 174 6.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 PART III. FILTERING AND CONNECTIVITY . . . . . . . . . . . . . . . . . . 177 Chapter 7. Connected Operators based on Tree Pruning Strategies . . . . 179 Philippe SALEMBIER 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.2. Connected operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.3. Tree representation and connected operator . . . . . . . . . . . . . . . . 182 7.3.1. Max-tree, min-tree and inclusion tree . . . . . . . . . . . . . . . . 182 7.3.2. Binary partition tree . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.4. Tree pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.4.1. Pruning with increasing criterion . . . . . . . . . . . . . . . . . . . 187 7.4.2. Non-increasing criterion . . . . . . . . . . . . . . . . . . . . . . . . 189 7.4.3. Pruning by global constrained optimization . . . . . . . . . . . . . 196 7.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
- 13. Contents ix Chapter 8. Levelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Jean SERRA, Corinne VACHIER, Fernand MEYER 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.2. Set-theoretical leveling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 8.2.1. Set-theoretical leveling by marker . . . . . . . . . . . . . . . . . . 201 8.2.2. Leveling as supremum of activity and as a strong filter . . . . . . . 201 8.2.3. Leveling as function of the marker . . . . . . . . . . . . . . . . . . 204 8.2.4. Multimarker leveling . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.3. Numerical levelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.3.1. Geometrical interpretation in terms of flat zones . . . . . . . . . . 211 8.3.2. The two orders for numerical activity . . . . . . . . . . . . . . . . 212 8.4. Discrete levelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.4.1. Local behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.4.2. Two leveling algorithms using geodesic iterations . . . . . . . . . 217 8.4.3. Multimarked levelings and scale-space . . . . . . . . . . . . . . . 220 8.4.4. Chaining levelings and scale-space representation of images . . . 222 8.5. Bibliographical comment . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.5.1. On grains reconstructions . . . . . . . . . . . . . . . . . . . . . . . 227 8.5.2. On extinction functions . . . . . . . . . . . . . . . . . . . . . . . . 227 8.5.3. On connected operators . . . . . . . . . . . . . . . . . . . . . . . . 228 8.5.4. On levelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Chapter 9. Segmentation, Minimum Spanning Tree and Hierarchies . . . 229 Fernand MEYER, Laurent NAJMAN 9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.2. Preamble: watersheds, floodings and plateaus . . . . . . . . . . . . . . . 230 9.2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 9.2.2. The question of contours representation . . . . . . . . . . . . . . . 231 9.2.3. Minimum spanning forests and watersheds . . . . . . . . . . . . . 232 9.2.4. Floodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 9.2.5. The question of plateaus . . . . . . . . . . . . . . . . . . . . . . . . 236 9.3. Hierarchies of segmentations . . . . . . . . . . . . . . . . . . . . . . . . 237 9.3.1. Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9.3.2. Hierarchies of watershed segmentations . . . . . . . . . . . . . . . 239 9.3.3. Contour saliency maps, subdominant ultrametric and floodings . . 240 9.3.4. Some families of floodings . . . . . . . . . . . . . . . . . . . . . . 245 9.3.5. Other hierarchical schemes: the example of scale-sets . . . . . . . 251 9.4. Computing contours saliency maps . . . . . . . . . . . . . . . . . . . . . 252 9.4.1. Minimum spanning tree . . . . . . . . . . . . . . . . . . . . . . . . 252 9.4.2. Hierarchy of markers . . . . . . . . . . . . . . . . . . . . . . . . . . 253 9.4.3. Hierarchies driven by a geometrical criterion . . . . . . . . . . . . 253 9.4.4. Cataclysmic hierarchies . . . . . . . . . . . . . . . . . . . . . . . . 254
- 14. x Mathematical Morphology 9.5. Using hierarchies for segmentation . . . . . . . . . . . . . . . . . . . . . 255 9.5.1. Local resegmentation or split-and-merge . . . . . . . . . . . . . . 255 9.5.2. Magic wand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 9.5.3. Lasso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 9.5.4. Intelligent brush . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.6. Lattice of hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 9.6.1. Infimum of two segmentations . . . . . . . . . . . . . . . . . . . . 258 9.6.2. Infimum of two hierarchies . . . . . . . . . . . . . . . . . . . . . . 259 9.6.3. Lexicographical infimum of hierarchies . . . . . . . . . . . . . . . 260 PART IV. LINKS AND EXTENSIONS . . . . . . . . . . . . . . . . . . . . . . . 263 Chapter 10. Distance, Granulometry and Skeleton . . . . . . . . . . . . . . 265 Michel COUPRIE, Hugues TALBOT 10.1. Skeletons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 10.1.1. Maximal balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 10.1.2. Firefronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 10.1.3. Properties of the skeleton in the continuum . . . . . . . . . . . . 268 10.2. Skeletons in discrete spaces . . . . . . . . . . . . . . . . . . . . . . . . 269 10.3. Granulometric families and skeletons . . . . . . . . . . . . . . . . . . . 270 10.3.1. Granulometric family . . . . . . . . . . . . . . . . . . . . . . . . . 270 10.3.2. Applications of granulometries . . . . . . . . . . . . . . . . . . . 271 10.3.3. Ultimate eroded formula . . . . . . . . . . . . . . . . . . . . . . . 272 10.3.4. Lantuéjoul formula . . . . . . . . . . . . . . . . . . . . . . . . . . 273 10.4. Discrete distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 10.5. Bisector function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 10.6. Homotopic transformations . . . . . . . . . . . . . . . . . . . . . . . . 280 10.6.1. Neighborhoods and connectedness . . . . . . . . . . . . . . . . . 283 10.6.2. Connectivity numbers and simple points . . . . . . . . . . . . . . 284 10.6.3. Homotopic thinning . . . . . . . . . . . . . . . . . . . . . . . . . 285 10.6.4. Sequential and parallel thinning algorithms . . . . . . . . . . . . 286 10.6.5. Skeleton based on the Euclidean distance . . . . . . . . . . . . . 287 10.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Chapter 11. Color and Multivariate Images . . . . . . . . . . . . . . . . . . 291 Jesus ANGULO, Jocelyn CHANUSSOT 11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 11.1.1. Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 11.1.2. Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 11.2. Basic notions and notation . . . . . . . . . . . . . . . . . . . . . . . . . 292 11.2.1. A brief reminder about color spaces . . . . . . . . . . . . . . . . 292 11.2.2. Other multivariate images . . . . . . . . . . . . . . . . . . . . . . 295 11.2.3. Color and spectral distances . . . . . . . . . . . . . . . . . . . . . 296
- 15. Contents xi 11.2.4. Taxonomy of the vector orders . . . . . . . . . . . . . . . . . . . 297 11.3. Morphological operators for color filtering . . . . . . . . . . . . . . . . 299 11.3.1. General formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 299 11.3.2. Total orders by bit interlacing . . . . . . . . . . . . . . . . . . . . 301 11.3.3. Total orders by lexicographic cascades . . . . . . . . . . . . . . . 305 11.3.4. Total orders through a distance supplemented by a reference to a lexicographic cascade . . . . . . . . . . . . . . . . . . . . . . . . 307 11.3.5. Marginal processing and combination: the case of chromat- ic/achromatic top-hats . . . . . . . . . . . . . . . . . . . . . . . . . 311 11.4. Mathematical morphology and color segmentation . . . . . . . . . . . 312 11.4.1. Marginal segmentation and combination: the case of HLS fusion controlled by the saturation . . . . . . . . . . . . . . . . . . . . . . 312 11.4.2. Color gradients and watershed applications . . . . . . . . . . . . 313 11.4.3. Using watershed based on a vector lattice . . . . . . . . . . . . . 318 11.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 Chapter 12. Algorithms for Mathematical Morphology . . . . . . . . . . . . 323 Thierry GÉRAUD, Hugues TALBOT, Marc VAN DROOGENBROECK 12.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 12.2. Translation of definitions and algorithms . . . . . . . . . . . . . . . . . 324 12.2.1. Data structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 12.2.2. Shape and size of the function domain . . . . . . . . . . . . . . . 325 12.2.3. Structure of a set of points . . . . . . . . . . . . . . . . . . . . . . 326 12.2.4. Notation abbreviations . . . . . . . . . . . . . . . . . . . . . . . . 327 12.2.5. From a definition to an implementation . . . . . . . . . . . . . . 327 12.3. Taxonomy of algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 329 12.3.1. Criteria for a taxonomy . . . . . . . . . . . . . . . . . . . . . . . . 330 12.3.2. Tradeoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 12.3.3. Classes of algorithms and canvases . . . . . . . . . . . . . . . . . 332 12.4. Geodesic reconstruction example . . . . . . . . . . . . . . . . . . . . . 334 12.4.1. The mathematical version: parallel algorithm . . . . . . . . . . . 334 12.4.2. Sequential algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 336 12.4.3. Queue-based algorithm . . . . . . . . . . . . . . . . . . . . . . . . 337 12.4.4. Hybrid algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 12.4.5. Algorithm based on union-find . . . . . . . . . . . . . . . . . . . 341 12.4.6. Algorithm comparison . . . . . . . . . . . . . . . . . . . . . . . . 343 12.5. Historical perspectives and bibliography notes . . . . . . . . . . . . . . 344 12.5.1. Before and around morphology . . . . . . . . . . . . . . . . . . . 345 12.5.2. History of mathematical morphology algorithmic developments 347 12.6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
- 16. xii Mathematical Morphology PART V. APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Chapter 13. Diatom Identification with Mathematical Morphology . . . . 357 Michael WILKINSON, Erik URBACH, Andre JALBA, Jos ROERDINK 13.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 13.2. Morphological curvature scale space . . . . . . . . . . . . . . . . . . . 358 13.3. Scale-space feature extraction . . . . . . . . . . . . . . . . . . . . . . . 359 13.4. 2D size-shape pattern spectra . . . . . . . . . . . . . . . . . . . . . . . 359 13.4.1. Shape and size pattern spectra . . . . . . . . . . . . . . . . . . . . 360 13.4.2. Attribute thinnings . . . . . . . . . . . . . . . . . . . . . . . . . . 361 13.4.3. Computing 2D shape-size pattern spectra . . . . . . . . . . . . . 362 13.5. Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 13.6. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 13.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Chapter 14. Spatio-temporal Cardiac Segmentation . . . . . . . . . . . . . 367 Jean COUSTY, Laurent NAJMAN, Michel COUPRIE 14.1. Which objects of interest? . . . . . . . . . . . . . . . . . . . . . . . . . 368 14.1.1. Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 14.1.2. Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 14.1.3. Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 14.2. How do we segment? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 14.2.1. Endocardial border . . . . . . . . . . . . . . . . . . . . . . . . . . 369 14.2.2. Epicardial border . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 14.3. Results, conclusions and perspectives . . . . . . . . . . . . . . . . . . . 372 Chapter 15. 3D Angiographic Image Segmentation . . . . . . . . . . . . . . 375 Benoît NAEGEL, Nicolas PASSAT, Christian RONSE 15.1. Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 15.2. Anatomical knowledge modeling . . . . . . . . . . . . . . . . . . . . . 376 15.3. Hit-or-miss transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 15.4. Application: two vessel segmentation examples . . . . . . . . . . . . . 378 15.4.1. Liver vascular network segmentation from X-ray CT-scan . . . . 380 15.4.2. Brain vessel segmentation from MRI data . . . . . . . . . . . . . 382 15.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Chapter 16. Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Beatriz MARCOTEGUI, Philippe SALEMBIER 16.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 16.2. Morphological multiscale decomposition . . . . . . . . . . . . . . . . . 385 16.3. Region-based decomposition . . . . . . . . . . . . . . . . . . . . . . . 389 16.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
- 17. Contents xiii Chapter 17. Satellite Imagery and Digital Elevation Models . . . . . . . . . 393 Pierre SOILLE 17.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 17.2. On the specificity of satellite images . . . . . . . . . . . . . . . . . . . 394 17.3. Mosaicing of satellite images . . . . . . . . . . . . . . . . . . . . . . . 398 17.4. Applications to digital elevation models . . . . . . . . . . . . . . . . . 400 17.5. Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . 405 Chapter 18. Document Image Applications . . . . . . . . . . . . . . . . . . . 407 Dan BLOOMBERG, Luc VINCENT 18.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 18.2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 18.2.1. Word extraction from a music score . . . . . . . . . . . . . . . . 410 18.2.2. Page segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . 410 18.2.3. Skew detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 18.2.4. Text orientation detection . . . . . . . . . . . . . . . . . . . . . . 415 18.2.5. Pattern matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 18.2.6. Background estimation for grayscale images . . . . . . . . . . . 419 Chapter 19. Analysis and Modeling of 3D Microstructures . . . . . . . . . 421 Dominique JEULIN 19.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 19.2. 3D morphological analysis . . . . . . . . . . . . . . . . . . . . . . . . . 422 19.2.1. Segmentation of 3D images . . . . . . . . . . . . . . . . . . . . . 422 19.2.2. Morphological classification of particles of complex shapes . . . 425 19.2.3. Morphological tortuosity . . . . . . . . . . . . . . . . . . . . . . . 430 19.3. Models of random multiscale structures . . . . . . . . . . . . . . . . . 431 19.3.1. Boolean models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 19.3.2. Percolation of tridimensional microstructures . . . . . . . . . . . 434 19.4. Digital materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 19.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 Chapter 20. Random Spreads and Forest Fires . . . . . . . . . . . . . . . . . 445 Jean SERRA 20.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 20.2. Random spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 20.2.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 20.2.2. Characteristic functional . . . . . . . . . . . . . . . . . . . . . . . 450 20.3. Forecast of the burnt zones . . . . . . . . . . . . . . . . . . . . . . . . . 451 20.3.1. Spontaneous extinction . . . . . . . . . . . . . . . . . . . . . . . . 451 20.3.2. An example of prediction . . . . . . . . . . . . . . . . . . . . . . 452 20.4. Discussion: estimating and choosing . . . . . . . . . . . . . . . . . . . 453
- 18. xiv Mathematical Morphology 20.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
- 19. Preface Mathematical morphology is a discipline of image analysis that was introduced in the mid-1960s by two researchers at the École des Mines in Paris: Georges Matheron [MAT 75] and Jean Serra [SER 82, SER 88c]. Historically, this was the first consistent nonlinear image analysis theory, which from the very start included not only theoretical results but also many practical aspects. Its initial objective was to facilitate studies of mineral deposits via sampling. It was implemented using dedicated image processing hardware, akin to analog computers in many ways. Mathematical morphology was endowed from the very beginning with the three pillars which ensured its success: a solid theoretical foundation, a large body of applications and an efficient implementation. Since this heroic era, many developments have been proposed. Indeed, many unforeseen applications have been developed: in materials science and in the life sciences, for example. The techniques eventually become popular internationally and improved to the level where they are now, more than 40 years after their beginning. Since 1993, a regular and well-attended series of international symposiums dedicated to the discipline have taken place and many journals have mathematical morphology tracks and special issues. Mathematical morphology is now part of the basic body of techniques taught to any student of image processing courses anywhere; most image processing software packages feature morphology toolboxes and filters, including the most popular programs such as Photoshop or Matlab. Far from being an academic pursuit, morphology is used in industry and businesses at many levels, for example: quality control in industrial production, medical imaging, document processing and much more. In spite of this popularity, researchers and practitioners in mathematical morphol- ogy often find that their operators and functions are not understood as well as they could be. For instance, many newcomers to the discipline think it only applies to binary images (images featuring only two levels: pure black and pure white). On the contrary, mathematical morphology is a complete theory capable of handling the most xv
- 20. xvi Mathematical Morphology varied image types in a way that is often subtle yet efficient. Morphology can be used to process certain types of noise in images, but can also be used more generally in filtering, segmentation, classification, analysis and coding of visual-type data. It can also be used to process general graphs, surfaces, implicit and explicit volumes, manifolds and time or spectral series in both deterministic and stochastic contexts. One of the reasons for this lack of understanding might be the relative lack of recent and comprehensive books on the topic [DOU 93, DOU 03b, HEI 94a, SOI 03a]. We were therefore very honored when Henri Maître, director of the Image and Signal collection at Hermès Publishing in France, asked us to propose, compile and edit contributions from some of the best-known researchers and practitioners in the field in order to showcase the capabilities of mathematical morphology. Thanks to ISTE and John Wiley and Sons, we are now pleased to provide this book in English. Its content has been thoroughly revised and significantly expanded from the French language version. The primary goal of this book is to expose the state of the art in mathematical morphology in a didactic fashion. However, our authors did not limit themselves to this exercise, but also developed some original and novel content. They took advantage of this opportunity to reformulate, rework and rethink the themes they work with most often, in order to make them available to a greater audience in a unique format. We are also of course very honored by the confidence afforded to us by all our numerous contributors. We take this opportunity to thank them and applaud their efforts. This book has taken a very long time to come to fruition, but our authors have been a pleasure to work with all along. We hope the end result meets their expectation. Among our authors, we particularly wish to thank Christian Ronse and Jean Serra, who have both helped us immensely to improved the general quality of the book. The 20 chapters are divided into 5 parts as follows: – The first part explains the fundamental aspects of the discipline. Starting with a general introduction, two more theoretical chapters follow. The first of these is concerned with mathematical structure, including a modernized formalism which is the result of several decades of work. – The second part extends morphology into image analysis, in particular detailing how estimations, choices and measurements can be made. This is achived through links with other disciplines such as stereology, geostatistics and fuzzy logic. – The third part concerns the theory of morphological filtering and segmentation, insisting on modern connected approaches from both the theoretical and practical aspects. – The fourth part exposes some practical aspects of mathematical morphology, in particular, how to deal with color and multivariate data. Links to discrete geometry and topology and some algorithmic aspects are included, without which applications would be impossible.
- 21. Preface xvii – Finally, the fifth part illustrates all the previous work via a sampling of interesting, representative and varied applications. In more detail, the first part introduces the theoretical foundations and general principles of mathematical morphology: – Chapter 1, written by both of us, is a didactic introduction to mathematical morphology that does not require any specific knowledge and should be accessible to any person with a general scientific background. – Chapter 2, written by Christian Ronse and Jean Serra, deals with the algebraic foundations of mathematical morphology. It introduces basic operators though the framework of complete lattice. It provides the notion of adjunction, necessary for operator composition. It illustrates the generality of the lattice framework applied to filtering and introduces the notions of segmentation by connection and by filtering of hierarchies. – Chapter 3, written by Gilles Bertrand, Michel Couprie, Jean Cousty and Laurent Najman, analyses the watershed line operator in discrete spaces. The watershed line is the premier mathematical morphology tool for segmentation. In this chapter, several definitions are proposed with varied fields of applications from a purely discrete point of view. These definitions draw from concepts originating from topology and mathematical optimization, in pixel images but also graphs and complexes. The second part deals with analysis, estimations and measurements: – Chapter 4, written by Jean Serra and ourselves, is an introduction to the theory of measurements in image analysis and mathematical morphology, with a stereological perspective. The goal of this approach is to endow mathematical morphology with the ability to extract reliable, quantitative measurements from visual information. – Chapter 5, written by Christian Lantuéjoul, describes some of the probabilistic aspects of mathematical morphology. In particular, the chapter discusses sampling, simulations and border effects. – Chapter 6, written by Isabelle Bloch, describes the state of the art in fuzzy morphology. This extension makes it possible to manage uncertainty and imprecision in a complementary matter to probabilistic approaches. The third part concerns the theory of morphological filtering and segmentation: – Chapter 7, written by Philippe Salembier, studies connected morphological filtering using the component tree. The component tree is a fundamental notion in modern morphology, allowing powerful operators to be implemented efficiently. – Chapter 8, written by Jean Serra, Corinne Vachier and Fernand Meyer, is about levelings. This class of connected operators has increasing importance in image filtering. Like all connected operators, they reduce noise while preserving contours.
- 22. xviii Mathematical Morphology – Chapter 9, written by Fernand Meyer and Laurent Najman, is about hierarchical morphological segmentation. The main tool is again the watershed line. The chapter describes this tool in a coherent manner, which makes it possible to build segmentation hierarchies. This notion is important when dealing with multiresolution issues, for parameter optimization or in order to propose fast interactive segmentations. The fourth part contains a subset of interesting topics in morphology that are applied more in nature. This includes granulometries and skeletonization, multivariate and color morphology and some algorithmic aspects of morphology: – Chapter 10, written by Michel Couprie and Hugues Talbot, discusses granulometries, distances and topological operators. Combined, these notions lead to efficient and interesting skeletonization operators. These operators reduce the amount of information needed to represent objects while conserving topological properties. – Chapter 11, written by Jesus Angulo and Jocelyn Chanussot, deals with the way multivariate and color data might be processed using mathematical morphology. As this type of data is becoming increasingly prevalent, this is of particular importance. – Chapter 12, written by Thierry Géraud, Hugues Talbot and Marc Van Droogen- broeck, deals with the implementation aspects of the discipline and with associated algorithmic matters. This aspect is of crucial importance for applications. Finally, the fifth and last part illustrates the previous chapters with detailed applications and applications fields: – Chapter 13, written by Michael Wilkinson, Erik Urbach, Andre Jalba and Jos Roerdink, concerns a methodology for the analysis of diatoms which uses morphological texture analysis very effectively. – Chapter 14, written by Jean Cousty, Laurent Najman and Michel Couprie, shows an application to the 3D+t spatio-temporal segmentation of the left ventricle of the human heart using magnetic resonance imaging (MRI). – Chapter 15, written by Benoît Naegel, Nicolas Passat and Christian Ronse, is a description of a segmentation and analysis method of the brain vascular network. – Chapter 16, written by Beatriz Marcotegui and Philippe Salembier, concerns image coding and compression using morphological segmentation. – Chapter 17, written by Pierre Soille, shows applications of mathematical morphology techniques to remote sensing. – Chapter 18, written by Dan Bloomberg and Luc Vincent, is a description of a vast array of morphological techniques applied to scanned document analysis. – Chapter 19, written by Dominique Jeulin, outlines recent progress in the analysis of materials, in particular using microtomography techniques. – Chapter 20, written by Jean Serra, combines random sets and deterministic morphological operators to analyze the spread of forest fires in Malaysia.
- 23. Preface xix A web site is dedicated to this book at the following URL: http://www. mathematicalmorphology.org/books/najman-talbot.Supplementary material is available there, including color versions of many of our illustrations. We sincerely hope that this presentation of modern mathematical morphology will allow a larger public to understand, appreciate, explore and exploit this rich and powerful discipline of image analysis. Laurent NAJMAN Hugues TALBOT June 2010
- 25. Chapter 1 Introduction to Mathematical Morphology In this chapter we endeavor to introduce in a concise way the main aspects of Mathematical Morphology, as well as what constitutes its field. This question is difficult, not so much as a technical matter but as a question of starting point. Historically, mathematical morphology began as a technique to study random sets with applications to the mining industry. It was rapidly extended to work with two- dimensional (2D) images in a deterministic framework first with binary images, then gray-level and later to color and multispectral data and in dimensions > 2. The framework of mathematical morphology encompasses many various mathematical disciplines from set theory including lattice theory, random sets, probabilities, measure theory, topology, discrete and continuous geometry, as well as algorithmic considera- tions and finally applications. The main principle of morphological analysis is to extract knowledge from the response of various transformations which are generally nonlinear. One difficulty in the way mathematical morphology has been developed and expanded [MAT 75, SER 82, SER 88c] (see also [HEI 94a, SCH 94, SOI 03a]) is that its general properties do not fall within the general topics taught at school and universities (with the exception of relatively advanced graduate-level courses). Classical mathematics define a function as an operator associating a single point in a domain with a single value. A contrario, in morphology we associate whole sets with other whole sets. The consequences of this are important. For instance, if a point generally has zero measure, this is not generally the case for sets. Consequently, while a probability of the presence of a point may be zero, this is not the case for a set. Chapter written by Laurent NAJMAN and Hugues TALBOT. 3 Mathematical Morphology: From Theory to Applications Edited by Laurent Najman and Hugues Talbot © 2010 ISTE Ltd. Published 2010 by ISTE Ltd.
- 26. 4 Mathematical Morphology In addition, we can compare morphology to other image processing disciplines. For instance, linear operator theory assumes that images are merely a multidimen- sional signal. We also assume that signals combine themselves additively. The main mathematical structure is the vector space and basic operators are those that preserve this structure and commute with basic rules (in this case, addition and multiplication by a constant). From this point deriving convolution operators is natural; hence it is also natural to study Fourier or wavelet transforms. It is also natural to study decomposition by projections on basis vectors. This way is of course extremely productive and fruitful, but it is not the complete story. Indeed, very often a 2D image is not only a signal but corresponds to a projection of a larger 3D ‘reality’ onto a sensor via an optical system of some kind. Two objects that overlap each other due to the projections do not add their feature but, on the contrary, create occlusions. The addition is not the most natural operator in this case. It makes more sense to think in terms of overlapping objects and therefore, in terms of sets, their union, intersections and so on. With morphology, we characterize what is seen via geometrical transforms, taking into account shapes, connectivity, orientation, size, etc. The mathematical structure that is most adapted to this context is not the vector space, but the generalization of set theory to complete lattices [BIR 95]. 1.1. First steps with mathematical morphology: dilations and erosions In order to be able to define mathematical morphology operators, we need to introduce the abstract notion of complete lattice. We shall then be able to ‘perform’ morphology on any instance of such a lattice. 1.1.1. The notion of complete lattice A lattice [BIR 95] (E, ≤) is a set E (the space) endowed with an ordering relationship ≤ which is reflexive (∀x ∈ E, x ≤ x), anti-symmetric (x ≤ y and y ≤ x ⇒ x = y) and transitive (x ≤ y and y ≤ z ⇒ x ≤ z). This ordering is such that for all x and y, we can define both a larger element x ∨ y and a smaller element x ∧ y. Such a lattice is said to be complete if any subset P of E has a supremum W P and an infimum V P that both belong to E. The supremum is formally the smallest of all elements of E that are greater than all the elements of P. Conversely, the infimum is the largest element of E that is smaller than all the elements of P. In a lattice, supremum and infimum play symmetric roles. In particular, if we consider the lattice P[E] constituted by the collection of all the subsets of set E, two operators ψ and ψ∗ are dual if, for all X, ψ(Xc = [ψ∗ (X)]) where Xc = E X is the complement of X in E.
- 27. Introduction to Mathematical Morphology 5 1.1.2. Examples of lattices Figure 1.1 is an example of a lattice. This instance is simple but informative, as it corresponds to the lattice of primary additive colors (red, green and blue). Each element of the lattice is a binary 3-vector, where 0 represents the absence of a primary color and 1 its presence. The color black is represented by [0, 0, 0] and white by [1, 1, 1]. Pure red is [1, 0, 0], pure green is [0, 1, 0], and so on. Magenta is represented by [1, 0, 1]. In this lattice, there does not exist a way to directly compare pure green and pure blue or magenta and yellow: the order is not total. However, white is greater (brighter) than all colors and black is smaller (darker). Whatever subset of colors is chosen, it is always possible to define a supremum by selecting the maximal individual component among the colors of the set (e.g. the supremum of [1, 0, 0] and [0, 0, 1] is [1, 0, 1]). This supremum may not be in the subset, but it belongs to the original lattice. Similarly, the infimum is defined by taking the minimal individual component. 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1 1 0 1 0 1 1 Figure 1.1. An example of a lattice: the lattice of additive primary colors Another example of a lattice is the set of real numbers R endowed with the usual order relation. This lattice is not complete since, for instance, the subset of integer numbers has +∞ as supremum but +∞ is not part of R. In contrast, R = R ∪ {−∞, +∞} is a complete lattice. Through these examples, we can see that the notion of complete lattice is not fundamentally difficult. 1.1.2.1. Lattice and order Many morphological operators preserve the ordering structure. We call such operators Φ increasing and express it by ∀x, y ∈ (E, ≤), x ≤ y ⇒ Φ(x) ≤ Φ(y). Others will transform input lattice elements into larger or smaller elements. If we have an operator Ψ which is such that ∀x ∈ (E, ≤), x ≤ Ψ(x) then the operator is called extensive: it will enlarge elements. Conversely, if Ψ(x) ≤ x, then the operator is anti- extensive: it will shrink them. The simplest operators we can introduce on a lattice are those that commute with the supremum or the infimum. Respectively, these operators are called abstract dilation and erosion. Under various conditions such operators can combine some of these properties, as we will see shortly. While these definitions are straightforward and relatively easily understood after some period of familiarization, there is a legitimate question as to why morphologists
- 28. 6 Mathematical Morphology like to propose such abstract concepts. In order to answer this question, it is useful to think one level deeper and come back to the definition of an image. Let us consider Figure 1.2a, which is a simple gray-level image. The content of this image may be technically interesting – it consists of glass fibers observed in an electron microscope – but it has no bearing here. We consider this image as a function F : E → T , where E is the set of image points and T the set of possible values of F. In this case F is perhaps a set of discrete gray levels, possibly coded over 8 bits or 256 gray levels. The space T might instead be a subset of R = R ∪ {−∞, +∞}. Conversely, the space E can be seen as continuous (for instance E = Rn ) or discrete (for instance E = Zn or a suitable subset). We will denote the set of functions from E to T by T E . (a) (b) Figure 1.2. (a) A gray-level image and (b) a binary image obtained by thresholding (a) Depending on our application, it might be useful to consider one or the other of these definitions. How can we define operators that are in some way ‘generic’ and which will work irrespective of the precise definition of E and T ? A benefit of using the lattice framework is precisely that we can define operators acting on images without specifying further the space of definition of these images. A more detailed description of lattices and algebraic morphology can be found in Chapter 2. 1.1.3. Elementary operators It is possible to define morphological operators in many different ways. It is useful to consider the very simple case of binary images i.e. image that possess only two levels: strictly black with value 0 and strictly white with value 1. This framework is not the only one over which we can express morphology, but it has several advantages: it is relatively simple and intuitive but it is also sufficiently flexible for the further generalization of most operators to more complex lattices.
- 29. Introduction to Mathematical Morphology 7 One of the simplest operators applicable to a gray-level image F is the threshold- ing. The threshold of F at level t is the set Xt(F) defined by: Xt(F) = {p ∈ E|F(p) ≤ t}. (1.1) A threshold of image of Figure 1.2a is given in Figure 1.2b. The former is called a gray-level image and the latter a binary image. We can consider a binary image either as a subset of the continuous or discrete plane or, alternatively, as a function with values in {0, 1}. Once again, if we use the lattice framework this choice has little effect. If we consider binary images as subsets of E the corresponding structure is the lattice P(E) endowed with the inclusion comparison operator, i.e. let X and Y be two subsets of E, then X ≤ Y ⇔ X ⊆ Y . The supremum of a collection of sets {A, B, . . .} is given by the union operator W {A, B, . . .} = S {A, B, . . .} and the infimum by the inclusion. This set lattice is very commonly used in practice, but it is not the only possible choice. For instance, if we seek to only work with convex sets it is much more appropriate to choose the convex set lattice with the usual inclusion operator as the infimum, but the convex hull of the union as the supremum. 1.1.3.1. Structuring elements In the day-to-day practice of morphology, we often study binary or gray-level images using families of special sets B that are known a priori and can be adapted to our needs (in terms of size, orientation, etc.). These sets B are called structuring elements. They allow us to define the operators we evoked earlier (erosions and dilations) in a practical way. For instance, let X be a binary image i.e. a subset of E. The translate of X by p ∈ E is the set Xp = {x + p|x ∈ X}. Here p defines a translation vector. The morphological dilation of X by B is given by: δB(X) = X ⊕ B = [ b∈B Xb = [ x∈X Bx = {x + b|x ∈ X, b ∈ B}. (1.2) The resulting dilation is the union of the Bp such that p belongs to X: δB(X) = S {Bp|p ∈ X}. As a consequence, the dilation of X by B ‘enlarges’ X, hence the name of the transform. In the formula, X and B play symmetric roles. Note also that when B is untranslated, (i.e. Bo), it is located somewhere relative to the origin of the
- 30. 8 Mathematical Morphology (a) (b) (c) Figure 1.3. The dilation of a cross by a triangle. The origin or the structuring element is one of the vertices of triangle B and is shown as a small black disk: (a) the original X (the light-gray cross) and B (the dark triangle); (b) the dilation taking place; and (c) the final result with the original set X overlaid coordinate system. We usually associate this point with B itself and call it the origin of the structuring element. When B is translated, so is its origin. An example of a dilation is shown in Figure 1.3. The erosion of X by B is defined: εB(X) = X ⊖ B = b∈B X−b = {p ∈ E|Bp ⊆ X}. (1.3) The erosion of X by B is the locus of the points p such that Bp is entirely included in X. An erosion ‘shrinks’ sets, hence its name. This is illustrated in Figure 1.4. (a) (b) (c) Figure 1.4. The erosion of a cross by the same triangle structuring element as in Figure 1.3: (a) the original X (the light-gray cross) and B (the dark triangle); (b) the erosion taking place; and (c) the final result, overlaid within the original set X Erosion and dilation have opposite effects on images. More formally, they are dual by complementation: the dilation of a set X by B is the erosion of its complementary
- 31. Introduction to Mathematical Morphology 9 set Xc using the symmetric structuring element of B, denoted B̌. Let p, q be two points where p ∈ Bq ⇔ q ∈ B̌p. This amounts to B̌ = {−b|b ∈ B}: (X ⊕ B)c = Xc ⊖ B̌, and (X ⊖ B)c = Xc ⊕ B̌. We illustrate this property in Figure 1.5 using the erosion as an example. Figure 1.5. The erosion of the cross of Figure 1.3, using the property that the dilation with the symmetric structuring element is the dual of this operation One way to extend the binary operators to the gray-level case is to take the hypograph SG(F) of a function F: SG(F) = {(x, t) ∈ E × T |t ≤ F(x)}. Using this approach, dilating (respectively, eroding) a gray-level image is equiva- lent to dilating (respectively, eroding) each of its thresholds. An equivalent approach consists of using the lattice of functions, using the order structure provided by the order on T . In particular, for two functions F, G ∈ T E , we obtain: F ≤ G ⇐⇒ ∀x ∈ E, F(x) ≤ G(x). In this way, equations (1.2) and (1.3) translate in the following manner: δG(F)(x) = (F ⊕ G)(x) = sup y∈E {F(y) + G(x − y)} (1.4) and εG(F)(x) = (F ⊖ G)(x) = inf y∈E {F(y) − G(y − x)}. (1.5) In these equations, function G is a structuring function. This function may be arbitrary, for instance sometimes parabolic functions are used in operations [BOO 96] such as the Euclidean distance transform [MEH 99].
- 32. 10 Mathematical Morphology 1.1.3.2. Flat structuring elements In practice, the most common structuring functions are the flat structuring elements (SEs). These are structuring functions which are identically equal to zero on a compact support K and that take the value V T elsewhere. In this case, equations (1.4) and (1.5) reduce to: εK(F)(x) = inf y∈E,y−x∈K F(y) = inf y∈Kx F(y) (1.6) δK(F)(x) = sup y∈E,x−y∈K F(y) = sup y∈(Ǩ)x F(y). (1.7) In this case, the alternative viewpoint is helpful: applying a flat morphological operator on a function F is equivalent to applying a morphological operator on all the thresholds Xt(F) of F. For instance, in the case of the dilation by a flat structuring element K, this amounts to: δK(F) = _ {t ∈ T |p ∈ δK(Xt(F))}. (1.8) Figure 1.6 depicts an example of the dilation of a 1D signal by a structuring function. Figures 1.7a and b illustrate the 2D case. Figure 1.6. Dilation of a signal (a 1D image) by a non-flat structuring element (a structuring function) and a flat structuring element. A dilation by a flat SE is the same as taking at every point the maximum of the function over the window defined by the symmetric SE We see here that morphological operators can readily be extended from the binary to the grayscale case. It is often easier to understand intuitively what an operator does in the binary case. It is also the case that, when working on gray-level images, it can be preferable to work in this mode for as long as possible and defer any thresholding. This way, the parameter of this operator can be chosen at a later stage when this decision might be easier.
- 33. Introduction to Mathematical Morphology 11 (a) (b) (c) (d) (e) (f) Figure 1.7. Gray-level dilations and erosions of the images in Figure 1.2 by a symmetric 5 × 5 square structuring element: (a) gray-level original; (b) dilation; (c) erosion; (d) binary original; (e) dilation; and (f) erosion 1.1.4. Hit-or-miss transforms The erosion and dilation operators are useful by themselves (for instance to suppress some kinds of noise) but they are even more powerful when combined. For instance, we might want to consider some transforms that take into account both points that belong to a set and those that do not belong to it. We then need two structuring elements with a common origin. The first, denoted T1, is applied to a set and the second, denoted T2, is applied to its complementary set. We write: X ⊛ T = (X ⊖ T1) ∩ (Xc ⊖ T2). (1.9) These operators are called hit-or-miss transforms or HMT. (Some authors also refer to this as the hit-and-miss transform. Both are acceptable and, as expressions, mean approximately the same thing. However, in the context of morphology, even if hit-and-miss is arguably better because we require one structuring element to fit in the foreground and the other to fit in the background, hit-or-miss is more usual.) The operators are denoted X ⊛ T , which is the locus of the points such that T1 is entirely included in set X while T2 is entirely included in the complement of X. These transforms can be used for pattern recognition, and many classical shape simplification procedures, such as skeletonization, use such techniques. Chapter 18 on document image processing describes some uses of HMTs. Chapter 15 presents an extension of HMT to grayscale images, and applies it in the context of medical image segmentation.
- 34. 12 Mathematical Morphology More generally, composing morphological operators such as dilations and erosions leads to morphological filtering. 1.2. Morphological filtering In classical signal processing, the term ‘filter’ may mean any arbitrary processing procedure. In mathematical morphology, this terminology has a more precise mean- ing: a morphological filter is an operator that is both increasing and idempotent. We encountered the former insection 1.1.2.1: it means the order is preserved. The latter term means that if we repeat the operator, the result does not change after the first time. In other words, morphological filters respect the ordering and converge in one iteration. In this context, the two most important operators are the opening and the closing. The opening is often denoted by γ and is a morphological filter (therefore increasing and idempotent) that is also anti-extensive. The closing is the complement of the opening; it is denoted most often by ϕ and is extensive. We also encountered extensivity and anti-extensivity in section 1.1.2.1. Respectively, they mean that the result is greater than the initial image, or smaller. In other words, openings make sets smaller and images darker, while closings make sets larger and images lighter. We shall now see examples of such operators. 1.2.1. Openings and closings using structuring elements It is possible, as a particular case, to define morphological filters by composing dilations and erosions using structuring elements. For instance, the opening of set X by structuring element B may be defined: γB(X) = X ◦ B = (X ⊖ B) ⊕ B = [ {Bp|p ∈ E et Bp ⊆ X} . (1.10) The closing of X by B is defined: ϕB(X) = X • B = (X ⊕ B) ⊖ B. (1.11) These formulae are similar in the gray-level case. In general terms, an opening will have a tendency to destroy the small, extruding and thin parts of objects; closing will tend to fill small holes and thin intruding parts of objects. This is illustrated in Figure 1.8.
- 35. Introduction to Mathematical Morphology 13 (a) (b) (c) (d) (e) (f) Figure 1.8. Openings and closings in the binary and gray-level cases, using the initial images from Figure 1.2 using a a 5 × 5 structuring element: (a) gray-level original; (b) opening; (c) closing; (d) binary original; (e) opening; and (f) closing These structuring element-based openings and closings are called morphological openings or closings. This is to distinguish them from the more general case of the operators that satisfy all the properties of the opening or closing, but are not necessarily the result of the composition of an erosion and a dilation. Most importantly, we generally cannot combine any arbitrary erosion on the one hand and dilation on the other and call the result an opening or a closing. The two operators that compose a morphological opening or closing are called adjunct operators, by reference to the very specific duality that links the erosion and the dilation that are effectively used. This duality is generally not the same as taking the complement set and the symmetric structuring element. Much more detail about this is given in Chapter 2. 1.2.2. Geodesy and reconstruction Let us now introduce the conditional dilation of a set X by a structuring element B, using a reference set R: δ (1) R,B(X) = (X ⊕ B) ∩ R. (1.12)
- 36. 14 Mathematical Morphology The result of this transform will always be included in the reference set R. Successive dilations are obtained by iteration of a (usually small) structuring element. Often the fundamental SE of the underlying grid is used (see section 1.2.2.2). δ (n+1) R,B (X) = (δ (n) R,B(X) ⊕ B) ∩ R. (1.13) At convergence, we have δ∞ R,B(X) = δ (n+1) R,B (X) = δ (n) R,B(X). (1.14) This type of operator is illustrated in Figure 1.9. X R δn R(X) (a) R X δn R(X) x t (b) Figure 1.9. Geodesic dilation: (a) the binary case and (b) the gray-level case with a flat structuring element 1.2.2.1. Openings and closings by reconstruction One of the first applications of geodesic dilation is the reconstruction operator. We refer to the reconstruction of X under R by B as the set δ∞ R,B(X), i.e. what we obtain by iterating the geodesic dilation operator to infinity or equivalently to idempotence. Starting from ‘markers’ that designate the parts of an image we would like to retain in some way, a geodesic reconstruction allows us to regain the original shape of those parts even although they might have been damaged in order to obtain the markers. In gray level, a reconstruction operator will reconstruct the edges of the objects of interest. We illustrate this concept in Figure 1.10. For a given fixed set of markers, a geodesic reconstruction by dilation has all the properties of an opening. As the name implies, the reconstruction operator is able to rebuild the shape of objects after they have been altered due to some other filtering operation. This
- 37. Introduction to Mathematical Morphology 15 R X (a) R X δn R(X) (b) Figure 1.10. The reconstruction operator: (a) the shape of the initial sets and (b) the 1D gray-level case operator is illustrated in Figure 1.11. The composition of an erosion followed by a reconstruction by dilations is a simple example of an algebraic opening, i.e. an opening which is not the composition of a single erosion followed by a single dilation. However, this kind of opening possesses all the properties of the opening. It is also a connected filter. Chapter 8 provides more information on this topic. (a) (b) Figure 1.11. (a) Opening by reconstruction of the map of Australia, consisting of an erosion followed by reconstruction. Note that the initial erosion deletes the island of Tasmania such that (b) the reconstruction cannot recover. However, the shape of the Australian continent is preserved By complementation, it is also possible to define in the same way a geodesic reconstruction by erosion that will result in a closing. All these operators also work on gray-level images, as illustrated in Figure 1.12.
- 38. 16 Mathematical Morphology (a) (b) (c) Figure 1.12. Gray-level closing by reconstruction: (a) the original image of particle tracks in a detection chamber; (b) the dilation by a 5 × 5 SE; and (c) a reconstruction by erosions. Most of the scintillation noise has been deleted, while retaining the general shape of the tracks 1.2.2.2. Space structure, neighborhood Until now, we have not approached the subject of the spatial structure of E. The operators we have defined previously do not really depend on it. However, the conditional dilation example illustrates the fact that specifying a structuring element for the dilation also specifies a connectivity. We shall now express this more carefully in the Zn case (but our discussion could also be carried out in a similar case in the continuous domain). Let us begin with the notion of local neighborhood Γ on space E. In the discrete case, Γ is a binary relation on E, i.e. is reflexive ((x, x) ∈ Γ) and symmetric ((x, y) ∈ Γ ↔ (y, x) ∈ Γ). We say that (E, Γ) is a (non-oriented)graph. Γ denotes the transform from E to 2E which associates x ∈ E with Γ(x) = {y ∈ E|(x, y) ∈ Γ}, i.e. the set of neighbors of x. If y ∈ Γ(x), we say that x and y are adjacent. In image processing, the more classical relations are defined on a subset of E ⊂ Z2 . For instance, in the 4-connected case, for all x = (x1, x2) ∈ E, Γ(x) = {(x1, x2), (x1 +1, x2), (x1 −1, x2), (x1, x2 + 1), (x1, x2 − 1)} ∩ E. We can define in the same way the 8- or 6- connectivity (see Figure 1.13). The transform Γ is really a dilation, and conversely, from every symmetric dilation defined on a discrete space, we can define a non-oriented graph. If a dilation is not symmetric, this is still true, but we need to involve oriented graphs. 1.2.2.3. Paths and connectivity With the square grid, which is used most often in practice in 2D, it is not possible to use a single definition of neighborhood in all cases. Indeed, we would like to retain in the discrete case the Jordan property of the Euclidean case. This states that any
- 39. Introduction to Mathematical Morphology 17 a b (a) (b) Figure 1.13. The local grid. In the square grid case, we can specify that each point is connected to its four nearest neighbors as in (a), or its 8 neighbors including the diagonal pixels as in (b). In the case of the hexagonal grid in (c), each pixel has 6 neighbors simple closed curve (a closed curve that does not self-intersect) divides the plane into two distinct regions which are connected within themselves: one is of finite extent and the other not. In the discrete case, this property is not true by default. The Jordan problem is illustrated in Figure 1.14. (a) (b) (c) Figure 1.14. The discrete Jordan property (not true in the square grid by default): (a) the non-degenerate, simple path separates the discrete plane into three connected components; (b) the path does not separate anything at all; and (c) the Jordan property is true (always the case with the hexagonal grid) If the grid in Figure 1.14a is 4-connected, the subset of the plane delimited by the path is not connected. If the grid in Figure 1.14b is 8-connected the path does not separate the inside of the curve from the outside. In contrast, with the hexagonal grid it is possible to show that these problems never occur. In order to solve this problem in a pragmatic way, image analysts often consider two kinds of connectivity [ROS 73, ROS 75] concurrently: one for the foreground ob- jects (inside the curves) and one for the background (outside). A more mathematically meaningful way of solving this problem is to consider a more complete topology for the discrete grid, e.g. following Khalimski [KHA 90].
- 40. 18 Mathematical Morphology 1.2.3. Connected filtering and levelings Combinations of openings and closings by reconstruction make it possible to define new operators which tend to extend flat zones in images. These combinations are called levelings. For more details, see Chapter 8 which is dedicated to this topic. From a more general point of view, levelings are part of a larger family of operators called connected filters. An efficient image representation for connected filtering is the component tree. This is studied in detail in Chapter 7 with applications in biology and image compression in Chapters 13 and 16, respectively. A particular case of a connected operator is the area opening, which we present in the following section. 1.2.4. Area openings and closings An opening or a closing using a particular structuring element (SE) modifies the filtered objects or image towards the shape of this SE. For instance, using a disk as an SE tends to round corners. Area openings or closings do not exhibit this drawback. Let X ⊆ E, and x0, xn ∈ X. A path from x0 to xn in X is a sequence π = (x0, x1, . . . , xn) of points of X such that xi+1 ∈ Γ(xi). In this case, n is the length of the path π. We say that X is connected if for all x and y in X there exists a path from x to y in X. We say that Y ⊆ E is a connected component of X if Y ⊆ X, Y is connected and Y is maximal for this property (i.e. Y = Z when Y ⊆ Z ⊆ X and Z is connected). In an informal fashion, an area opening will eliminate small connected components of arbitrary shape of area smaller than a given parameter λ. In a complementary manner, an area closing will fill small arbitrary holes of area smaller than λ. It is easy to verify that an area opening has the three fundamental properties of an algebraic opening: it is anti-extensive (it eliminates small connected components but leaves the others untouched); it is increasing; and it is idempotent (the small components that are eliminated at the first iteration of the opening remain eliminated, and the large components remain untouched). Area closings are of course extensive instead of being anti-extensive. An area closing is illustrated in Figure 1.15. 1.2.5. Algebraic filters Area filtering can be expressed in a different manner. Let us consider the case of the binary opening. A connected component C with area A will be preserved by any area opening of parameter λ < A. Clearly, there exists at least one morphological
- 41. Introduction to Mathematical Morphology 19 (a) (b) Figure 1.15. Area closing using a parameter of 20 square pixels. Small minima in the image were filled adaptively. Maxima in the image are unaffected (e.g. the small fiber) opening by a structuring element of area λ that preserves C, for instance the opening that uses C itself as a structuring element (or any subset of C with area λ). Knowing that we should preserve all connected components with area at least λ, we deduce that we can consider (at least conceptually) all possible openings with all connected structuring elements of area λ. It is easy to show that the supremum of these openings, i.e. the operator that at each point preserves the maximum of all these openings, is itself an opening and that it preserves all connected components with area at least λ. The supremum is therefore the area opening with parameter λ. It would be theoretically possible to implement the area opening operator by computing the result of all the possible openings using all connected structuring elements with area λ. However, this would be very inefficient as the size of the family of structuring elements increases exponentially with λ. However, the representation of an opening (or a closing) by such a morphological family is useful from the theoretical point of view. There exists a theorem by Matheron [MAT 75] that demonstrates the existence of a morphological decomposition for all openings and closings. For more details, see section 2.4.3. From the practical point of view, it is useful to remember that a combination by a supremum of openings is itself an opening. Respectively, a composition by an infimum of closings is also a closing. These filters are called algebraic openings (respectively, closings). In Chapter 12, we study how to implement some algebraic filters in practice. As an illustration, Figure 1.16 depicts an application of various algebraic filters to the denoising of thin objects. We used a closing by infimum combination of closings using various structuring elements families, either line segments or adaptive paths [HEI 05]. The objective here is to preserve the object of interest while filtering
- 42. 20 Mathematical Morphology out the background. In this particular case, the object is not sufficiently locally straight and so paths are better suited to this problem. (a) (b) (c) Figure 1.16. Algebraic closing by infimum composition: (a) an image of a strand of DNA seen in electron microscopy; (b) the infimum of closing by a sequence of segments spanning all orientations; and (c) the result of the infimum by a sequence of paths 1.2.6. Granulometric families The idea behind granulometries is inspired from sand sifting. When sifting sand through a screen (or sieve), particles that are larger than the dimension of the screen stay on top of the screen while smaller particles sift through. By using a family of screens of various sizes, we can sort the content of a sand pile by particle size. In the same manner, we can use a family of sieves that are compatible in order to obtain reproducible results. In mathematical morphology, we must use particular families of openings and closings of increasing sizes. These families are indexed by a parameter λ (often an integer) such that: λ ≥ µ ⇒ γλ ≤ γµ and ϕλ ≥ ϕµ.
- 43. Introduction to Mathematical Morphology 21 This property is called the absorption property. We often impose that γ0 = ϕ0 = Id. As an example of a granulometric family, it is possible to take a sequence of morphological openings or closings. For instance, in 8-connectivity in the square grid, we can use the family of squares Bn of size (2n+1)×(2n+1) as structuring elements. The resulting family of openings γBn or closings ϕBn , indexed by n, verifies the absorption property. We note here that B1 is the structuring element that corresponds to the basic neighborhood of a pixel. For this reason we refer to it as the unit ball of the grid. We can also use the corresponding openings or closings by reconstruction or take a family of area openings and closings, with increasing parameters. We shall use granulometric families in section 1.4.2 in this chapter; more details are also given in Chapter 10. 1.2.7. Alternating sequential filters Openings and closings are both increasing and idempotent; they only differ with respect to extensivity. This motivates us to study the class of operators that verify the former properties. We refer to these operators as morphological filters. This is both unfortunate and confusing because morphological openings and closings as well as algebraic openings and closings are morphological filters. However, this is to distinguish morphological filters from ‘plain’ filters which, in image processing, is often a generic term for an image operator. The theory of morphological filtering allows morphological operators to be efficiently composed. In particular, we can introduce alternating sequential filters (ASF) which are, as the name indicates, a composition of openings and closings which form granulometric families of increasing sizes. For instance, the white ASF, i.e. the ASF beginning with an opening, can be written: Φn(xi) = φnγnφn−1γn−1 . . . φ1γ1. (1.15) The black ASF (that begins with a closing) is defined: Ψn(xi) = γnφnγn−1φn−1 . . . γ1φ1. (1.16) The theory of morphological filtering is relatively involved and cannot be de- scribed adequately here. We simply illustrate it with some elementary applications in Figure 1.17.
- 44. 22 Mathematical Morphology (a) (b) (c) (d) Figure 1.17. Using alternating sequential filters: (a) a binary image; (b) the result of a size 2 white ASF; (c) an eye angiogram; and (d) the result of a black size 1 ASF Alternating sequential filters can be used to denoise both binary and grayscale images. The result is often easier to segment and analyze. In addition to the size parameter, the structuring element family used also has an impact and can be used to select shapes. Contrary to many filtering methods, these morphological filters allow practitioners to tune their denoising operator to the semantic content of the image and not be affected by the statistical properties of the noise. Morphological filters are therefore generally tailor-made to specific problems, depending on the content of the image under study. Morphological filtering theory is further developed in Chapters 2, 7 and 8. 1.3. Residues The operators we have seen until now are generally increasing, meaning that they preserve ordering. In contrast, the operators we present now do not.
- 45. Introduction to Mathematical Morphology 23 What makes the morphological approach different from and complementary to many other approaches is the fact that morphological operators do not seek to preserve information present in an image. Indeed, since the basic operators of morphology are not invertible, we expect a reduction in information content after each operator application. The key to success with morphology is to realize this, and to use this defining characteristic to our advantage. We can achieve this by selectively destroying the undesirable content of the image: noise, background irregularities, etc. while preserving the desired content for as long as possible. Figures 1.15 and 1.16 are direct illustrations of this philosophy. It is sometimes necessary to destroy undesirable content in an image, but not practical to do so. A complementary tactic is to effectively erase the desirable portion of an image, but to restore it through a difference with the original image. This gives rise to the idea of residues. Simply put, residues are transforms that involve combinations of morphological operators with the differences (or subtractions). Top-hat transforms, morphological gradients [RIV 93] and other similar transforms that we present in the next section are all examples of residues. Residues are generally well behaved in morphology, precisely because the basic properties of morphological operators are in our favour. For instance, because openings are anti-extensive, the difference between the original image and any opening derived from that image will always be positive. 1.3.1. Gradients The gradient of an image is basically its first derivative. For a 2D or generally nD image (n > 1), the gradient operator produces an n-vector at each point, where each component corresponds to the slope along the n principal directions of the grid in the discrete space. With morphology, we most often only consider the magnitude of the gradient at each point, which is a scalar irrespective of the dimension of the image. Gradients show the amount of local variation in the image. Zones of high gradient typically correspond to object contours or texture. Using erosions and dilations, we can define the morphological gradient as follows: Grad(F) = δB(F) − εB(F). (1.17) There are two other definitions: Grad+ (F) = δB(F) − F Grad− (F) = F − εB(F). (1.18)
- 46. 24 Mathematical Morphology These are the external and internal gradients, respectively. In general, B is taken to be the unit ball of the grid. We shall see an example of the use of the morphological gradient operator when we study the watershed line in section 1.5. 1.3.2. Top-hat transforms So-called top-hat transforms are the pixel-wise difference between an original image and an opening of this image (for white top-hats) or between the closing of an image and its original (for black top-hats). Since top-hats essentially show what the opening or closing has deleted from the original image, the former makes it possible to detect peaks and bright small areas in the original image; the latter finds valleys and small troughs in the image. The black top-hat is the white top-hat of the complementary image. Figure 1.18 is an illustration of the principles of white top- hats, while Figure 1.19 depicts an application to cell fluorescence microscopy. There are as many top-hats as there are different openings and closings. t x Structuring element Original function Opening White top-hat Figure 1.18. White top-hat of a 1D signal 1.4. Distance transform, skeletons and granulometric curves Let X ⊆ E. The distance from x to y in X is either the length of the smallest path from x to y within X, or +∞ if there does not exist a path from x to y that stays within X. The concept of distance makes it possible to introduce the related idea of distance transform. This associates each point x from a set X with the distance from this point to the nearest point in the complementary set of X (see Figure 1.20).
- 47. Introduction to Mathematical Morphology 25 (a) (b) Figure 1.19. Top-hat on an image of cells: (a) original image (small bright spots are vesicles in the cells, made fluorescent through the use of a bio-molecular marker); (b) the result of a thresholding of the white top-hat overlaid over the original image (a) (b) (c) Figure 1.20. The distance transform of a set X is the application that from each point of X associates its distance to the complementary set. In this illustration we consider the 8-distance but this is by no means a rule: (a) set; (b) distance transform; (c) level sets 1.4.1. Maximal balls and skeletons A maximal ball with radius R is the set of points located at a distance less than or equal to R from a central point p. It is obvious this definition depends on the distance used. For instance, using the 8-distance in 2D, the ball of radius 3 is a 7 × 7 square. When the Euclidean distance is used, the ball is a discrete disk. A maximal ball B relative to a set E is a ball such that there does not exist a ball B′ such that B ⊂ B′ ⊂ E (see Figure 1.21). This apparently simple notion is useful to define some interesting residues, in particular the skeleton. This notion has been known since the 1930s [BOU 32, DUR 30, DUR 31], but was popularized in image processing contexts in the 1960s [BLU 61, CAL 68] under the name of medial axis of E. The medial axis is defined as the collection of all centers of maximal balls of E. In Euclidean space, the medial axis is called the skeleton. The skeleton of a set O that is connected, open, non-empty and bounded has many interesting properties from
- 48. 26 Mathematical Morphology Figure 1.21. Two balls included in a binary set. The ball centered at point M is maximal because no other ball from the same family (here a family of disks) can contain it and simultaneously be included in the binary set. The ball centered in N is not maximal the geometrical and topological point of view: it is connected, homotopic to O and negligible from the point of view of the Lebesgue measure (although it may be dense everywhere in O) [RIV 87]. In the discrete case, the centers of maximal balls in O are well defined but are not necessarily located on the grid; the medial axis may therefore not be connected. We then define informally the discrete skeleton S(O) as a connected set, included in O, homotopic to O and as thin as possible. This notion is disjoint from the notion of medial axis, but it is possible to constrain the discrete skeleton to contain the medial axis. In practice, both discrete skeletons and medial axes are often noisy. To be able to use them in practice, we must be able to filter them. In Chapter 10, these notions are more precisely defined and algorithms are given to compute them. An example of filtered medial axis and filtered skeleton are given in Figure 1.22. (a) (b) (c) Figure 1.22. An example of a skeleton: (a) a binary image; (b) its filtered medial axis; and (c) a filtered skeleton of (a) that contains the medial axis
- 49. Introduction to Mathematical Morphology 27 1.4.2. Granulometric curves Mathematical morphology, even within itself, is capable of providing information on the size of objects in images in several different ways. One of these methods relies on the notion of granulometries, which is directly derived from the notion of morphological filter (see section 1.2.6 and Chapter 10). The granulometric curve of an image is a representation of the distribution of sizes in an image. This is based on the observation that intermediate residues of a granulometric family γn, indexed by n, are characteristic of the size of objects in images. More formally, the granulometric curve is the function Gf (λ) defined on the interval of λ, such that: GI(λ) = X I − X γλ(I) λ ∈ [0, 1, 2, . . ., R] (1.19) where I is the input image, P I is the sum (or integral) of all the pixels in the image and R is the value of λ for which no further change occurs due to γR (since the image has become constant). For finite discrete images, the value R always exists. If we use closings instead of openings, the operands on either side of the subtraction sign in equation (1.19) are exchanged (since all closings are extensive, and so ϕ(I) ≥ I). Granulometric curves by openings make it possible to estimate the size of peaks in images, while granulometries by closings measure the size of troughs. It is possible to build a granulometric curve using both openings and closings. In order to obtain a single continuous curve, convention states that the parameter of the closings is given in abscissa from the origin towards the negative value and the parameter of the openings is given from the origin towards the positive values. The same sign convention is used for both openings and closings; such a curve is depicted in Figure 1.23. In this example, we illustrate the fact that the granulometric curve records the volume of image (i.e. the area of features times their gray level) that is erased beyond a certain size, both for openings and closings. 1.4.2.1. Applications The granulometric curve summarizes the distribution of size of objects in an image, without necessitating a segmentation step. Consequently, the notion of object is not well defined in this context. We can only talk about volumes of gray levels. In addition, since the granulometric curve is one-dimensional, the information content is necessarily reduced from that of the whole image. The interpretation of this content is not always easy. It can sometimes be interpreted in terms of texture energy, as in the example given in Figure 1.24
- 50. 28 Mathematical Morphology (a) -10 -5 0 5 10 15 20 25 30 35 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Arbitrary area units Closing size Opening size Granulometry curve (b) Figure 1.23. A granulometric curve by openings and closings of an artificial image, using a family of Euclidean disks as structuring elements (indexed by their radii). The increasing sizes of openings go from the origin towards +∞ and the increasing size of closings go from the origin to −∞: (a) image and (b) granulometric curve in the context of a study involving the aging of steam pipes used in electricity production. Many applications use granulometries to estimate size-related parameters for subsequent procedures; see [COM 07]. 1.4.2.2. Granulometries by erosions and dilations It is also possible to produce granulometries by using only erosion or dilations. There is a strong link between these and skeletons [MAT 92]. 1.4.3. Median set and morphological interpolation Another application of distances worth mentioning, also related to skeletons, is their capacity for computing a median set used as an interpolation algorithm. In the literature, median sets appeared in the work of Casas [CAS 96] and Meyer [MEY 96]. The equation of the underlying operation and its basic properties were given by Serra in [SER 98b]. Iwanowski has successfully developed it for various morphings on still images and video sequences, in black and white and in color [IWA 00a]. More recently, Vidal et al. used a recursive technique for improving the interpolations [VID 07]. Recall that the Hausdorff distance is the maximum distance of a set from the nearest point in the other set. It measures how far two subsets of a metric space are from each other. Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set.
- 51. Introduction to Mathematical Morphology 29 (a) (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 rate of loss radius (c) 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 rate of loss radius (d) Figure 1.24. Application of granulometries: granulometric curves of surface microscopy images of pig iron steam pipes, in the case of a young pipe and an old pipe. Age deteriorates grain boundaries, which results in a larger number of small grains compared with younger samples: (a) young sample; (b) old sample; (c) young sample curve; (d) old sample curve Consider an ordered pair of closed sets {X, Y } with X ⊆ Y and such that their Hausdorff distance is finite. Their median element is the closed set M(X, Y ), composed of X and Y and whose boundary points are equidistant from X and the complement Y c to Y . In other words, the boundary of M is nothing but the skeleton by zone of influence (also known as the generalized Voronoï diagram) between X and Y c . The set M can depend on a parameter α which weights the relative importances of X and Y in the interpolation. The analytic expression of the weighted median set Mα(X, Y ) is obtained from its two primitives X, Y by taking the union [SER 98b]: Mα(X, Y ) = ∪λ {(X ⊕ αλB) ∩ (Y ⊖ (1 − α)λB)} . (1.20) In the general case, for two sets A and B with non-empty intersection (i.e. A ∩ B 6= ∅), we set X = A ∩ B and Y = A ∪ B so that X ⊂ Y and apply equation (1.20). Figure 1.25, whose steps are described in more detail in [IWA 00b], depicts the progressive passage from a bird to a plane as α varies from 0 to 1. As the map defined by equation (1.20) is increasing for both X and Y , it extends directly to digital numerical functions by simply replacing union and intersection by supremum and infimum, respectively.
- 52. 30 Mathematical Morphology Figure 1.25. Series of morphological interpolations from a bird to a plane, by means of equation (1.20) 1.5. Hierarchies and the watershed transform If we consider anew the analogy between grayscale images and a terrain topogra- phy, we can define an interesting transform called the watershed line. By analogy with hydrology, imagine a drop of water falling on the terrain represented by the image. Assuming sufficient regularity of the image, this drop will fall towards a local minimum in the image. With each local minimum M, we can refer to the set of points p such that a drop of water falling on p ends up in M. This set is call a catchment basin. The points located at the border of at least two such basins constitute a set of closed contours called the watershed line; see Figure 1.26 for an illustration of this. Catchment basins Watershed line Minima Figure 1.26. The watershed line A different view of the watershed line consists of not considering the points p, but starting from the minima M. We imagine that the image is inundated starting from the bottom (as if every minimum in the image is hollow and the whole image was dipped in water from the bottom). In this case every local minimum gradually fills with water, and the watershed line is the locus of the points where at least two water
- 53. Introduction to Mathematical Morphology 31 bodies meet. Although both visions are equivalent in the continuous domain under sufficient regularity assumptions [NAJ 94b], they are not compatible in all discrete frameworks and, notably, not compatible in the pixel framework [COU 07c, NAJ 05]. Although the previous explanation may not appear to be very formal, the literature on the topic of watershed properties and algorithms is abundant; see [BEU 79b, COU 05, MEY 94b, ROE 01, VIN 91c]. The formalization of the various concepts derived from the watershed in the discrete case, as well as the mathematical properties of the objects so obtained, are presented in Chapter 3. The watershed line transform forms the basis of a powerful and flexible segmen- tation methodology introduced at the Centre de Morphologie Mathematique in the 1970s [BEU 79b] and further developed in the 1990s [MEY 90b, VIN 91c]. This methodology was later unified using hierarchical approaches [BEU 94, NAJ 96]. The general idea is that we first need to produce internal markers of the objects under study. These are binary sets which can be labeled (i.e. given a distinct gray level for each connected component), that are included in the objects sought. The shape of these markers is unimportant; only their position and their extent matter. In a similar manner, we seek markers that are external to the objects, i.e. totally included in the background. A function which exhibits high values near contours and low values in a near-constant area of the image is used. Usually some regularized version of the gradient operator can be employed. This function is then reconstructed using the geodesic reconstruction operator of section 1.2.2 by imposing all markers (both internal and external) as minima in this function, and by eliminating the original minima present in the function. A single watershed line is then present. This separates internal and external markers, and tends to place itself on the contour of objects to be segmented. Many chapters of the Applications part of this book (Part V, notably Chapters 14 and 17) use one of the many variations of the watershed. It is therefore useful to illustrate the above procedure on a simple example due to Gratin [GRA 93]. Here we seek to segment a 2D magnetic resonance image (MRI) of an egg. On this image (Figure 1.27), markers for the exterior of the egg, the white and the yolk are set manually, but it is of course possible to obtain these through an automated procedure. Contrary to expectations and despite the simple nature of the problem, a simple thresholding does not yield good results due to the high level of noise. In contrast, the watershed segmentation procedure result is almost perfect. The general methodology for morphological segmentation is developed further in Chapter 9. It relies on defining some criterion that induces a hierarchy of segmenta- tions, i.e. a nested sequence of connected partitions. Any hierarchy of segmentations is equivalent to a specific watershed referred to as a saliency map [NAJ 96] or
- 54. 32 Mathematical Morphology (a) (b) (c) (d) Figure 1.27. MRI of an egg: (a) original image (courtesy of N. Roberts, University of Liverpool); (b) the gradient of this image (seen as a 3D terrain); (c) manually set markers; and (d) result of the segmentation ultrametric watershed [NAJ 09a, NAJ 09b]. Filtering such a watershed amounts to transforming the hierarchy into another watershed. Figure 1.28 illustrates the principle on an image of uranium oxide. We want to extract the cells but, unfortunately, a brute-force watershed application gives an oversegmented image. Instead of trying to find some markers, we can filter the image to remove the background noise. Here the chosen filter depends on a depth criterion (see Chapters 7 and 9). Rather than setting a fixed level of noise reduction for the filtering, it is better to compute the whole hierarchy of segmentations that can be obtained by varying the parameter. The resulting hierarchy is represented as a saliency map in Figure 1.28b. Any threshold of Figure 1.28b gives a segmentation. The more a contour is present in the hierarchy of segmentations, the more visible it is. It can be seen that there is a large difference between the noise contours and the ‘true’ contours; choosing the correct level of thresholding is therefore easy (Figure 1.28c). It is even possible to
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- 56. On no other planet in the Galaxy could any human experience such triumph as Burl felt now because never before had human beings been so completely subjugated by their environment. On no other planet had such an environment existed, with humans flung so helplessly upon its mercy. Burl had been normal among his fellows when he was as frightened and furtive as they. Now he had been given shock treatment by fate. He was very close to normal for a human being newly come to the forgotten planet, save that he had the detailed information which would enable a normal man to cope with the nightmare environment. What he lacked now was the habit. But it would be intolerable for him to return to his former state of mind. He walked almost thoughtfully after his fled followers. And he was still a savage in that he was remarkably matter-of-fact. He paused to break off a huge piece of the edible golden mushrooms his fellow- men had noticed on the way up. Lugging it easily, he went back down over the ground that had looked so astonishingly free of inimical life—which it was because of the spider that had used it as a hunting-preserve. Burl began to see that it was not satisfactory to be one of a tribe of men who ran away all the time. If one man with a spear or stone could kill spiders, it was ridiculous for half a dozen men to run away and leave that one man the job alone. It made the job harder. It occurred to Burl that he had killed ants without thinking too much about it, but nobody else had. Individual ants could be killed. If he got his followers to kill foot-long ants, they might in time battle the smaller, two-foot beetles. If they came to dare so much, they might attack greater creatures and ultimately attempt to resist the real predators. Not clearly but very dimly, the Burl who had been shocked back to the viewpoint which was normal to the race of men saw that human
- 57. beings could be more than the fugitive vermin on which other creatures preyed. It was not easy to envision, but he found it impossible to imagine sinking back to his former state. As a practical matter, if he was to remain as leader his tribesmen would have to change. It was a long time before he reached the neighborhood of the hiding-place of which he had not been told the night before. He sniffed and listened. Presently he heard faint, murmurous noises. He traced them, hearing clearly the sound of hushed weeping and excited, timid chattering. He heard old Tama shrilly bewailing fate and the stupidity of Burl in getting himself killed. He pushed boldly through the toadstool-growth and found his tribe all gathered together and trembling. They were shaken. They chattered together—not discussing or planning, but nervously recalling the terrifying experience they had gone through. Burl stepped through the screen of fungi and men gaped at him. Then they leaped up to flee, thinking he might be pursued. Tet and Dik babbled shrilly. Burl cuffed them. It was an excellent thing for him to do. No man had struck another man in Burl's memory. Cuffings were reserved for children. But Burl cuffed the men who had fled from the cliff-edge. And because they had not been through Burl's experiences, they took the cuffings like children. He took Jon and Jak by the ear and heaved them out of the hiding- place. He followed them, and then drove them to where they could see the base of the cliff from whose top they had tumbled stones— and then run away. He showed them the carcass of the spider, now being carted away piecemeal by ants. He told them angrily how it had been killed. They looked at him fearfully. He was exasperated. He scowled at them. And then he saw them shifting uneasily. There were clickings. A single, foraging black ant— rather large, quite sixteen inches long—moved into view. It seemed
- 58. to be wandering purposelessly, but was actually seeking carrion to take back to its fellows. It moved toward the men. They were alive, therefore, it did not think of them as food—though it could regard them as enemies. Burl moved forward and struck with his club. It was butchery. It was unprecedented. When the creature lay still he commanded one of his typo for followers to take it up. Inside its armored legs there would be meat. He mentioned the fact, pungently. Their faces expressed amazed wonderment. There was another clicking. Another solitary ant. Burl handed his club to Dor, pushing him forward. Dor hesitated. Though he was not afraid of one wandering ant, he held back uneasily. Burl barked at him. Dor struck clumsily and botched the job. Burl had to use his spear to finish it. But a second bit of prey lay before the men. Then, quite suddenly, this completely unprecedented form of foraging became understandable to Burl's followers. Jak giggled nervously. An hour later Burl led them back to the tribe's hiding-place. The others had been terror-stricken, not knowing where the men had gone. But their terror changed to mute amazement when the men carried huge quantities of meat and edible mushroom into the hiding-place. The tribe held what amounted to a banquet. Dik and Tet swaggered under a burden of ant-carcass. This was not, of course, in any way revolting. Back on Earth, even thousands of years before, Arabs had eaten locusts cooked in butter and salted. All men had eaten crabs and other crustaceans, whose feeding habits were similar to those of ants. If Burl and his tribesmen had thought to be fastidious, ants on the forgotten planet would still have been considered edible, since they had not lost the habits of extreme cleanliness which made them notable on Earth.
- 59. This feast of all the tribe, in which men had brought back not only mushroom to be eaten, but actual prey—small prey—of their hunting, was very probably the first such occasion in at least thirty generations of the forty-odd since the planet's unintended colonization. Like the other events, which began with Burl trying to spear a fish with a rhinoceros-beetle's horn, it was not only novel, on that world, but would in time have almost incredibly far-reaching consequences. Perhaps the most significant thing about it was its timing. It came at very nearly the latest instant at which it could have done any good. There was a reason which nobody in the tribe would ever remember to associate with the significance of this banquet. A long time before —months in terms of Earth-time—there had been a strong breeze that blew for three days and nights. It was an extremely unusual windstorm. It had seemed the stranger, then, because during all its duration everyone in the tribe had been sick, suffering continuously. When the windstorm had ended, the suffering ceased. A long time passed and nobody remembered it any longer. There was no reason why they should. Yet, since that time there had been a new kind of thing growing among the innumerable moulds and rusts and toadstools of the lowlands. Burl had seen them on his travels, and the expeditionary force against the clotho spider had seen them on the journey up to the cliff-edge. Red puffballs, developing first underground, were now pushing the soil aside to expose taut, crimson parchment spheres to the open air. The tribesmen left them alone because they were strange; and strange things were always dangerous. Puffballs they were familiar with— big, misshapen things which shot at a touch a powder into the air. The particles of powder were spores—the seed from which they grew. Spores had remained infinitely small even on the forgotten planet where fungi grew huge. Only their capacity for growth had increased. The red growths were puffballs, but of a new and different kind.
- 60. As the tribe ate and admired, the hunters boasting of their courage, one of the new red mushrooms reached maturity. This particular growing thing was perhaps two feet across, its main part spherical. Almost eighteen inches of the thing rose above- ground. A tawny and menacing red, the sphere was contained in a parchment-like skin that was pulled taut. There was internal tension. But the skin was tough and would not yield, yet the inexorable pressure of life within demanded that it stretch. It was growing within, but the skin without had ceased to grow. This one happened to be on a low hillside a good half-mile from the place where Burl and his fellows banqueted. Its tough, red parchment skin was tensed unendurably. Suddenly it ripped apart with an explosive tearing noise. The dry spores within billowed out and up like the smoke of a shell-explosion, spurting skyward for twenty feet and more. At the top of their ascent they spread out and eddied like a cloud of reddish smoke. They hung in the air. They drifted in the sluggish breeze. They spread as they floated, forming a gradually extending, descending dust-cloud in the humid air. A bee, flying back toward its hive, droned into the thin mass of dust. It was preoccupied. The dust-cloud was not opaque, but only a thick haze. The bee flew into it. For half a dozen wing-beats nothing happened. Then the bee veered sharply. Its deep-toned humming rose in pitch. It made convulsive movements in mid-air. It lost balance and crashed heavily to the ground. There its legs kicked and heaved violently but without purpose. The wings beat furiously but without rhythm or effect. Its body bent in paroxysmic flexings. It stung blindly at nothing. After a little while the bee died. Like all insects, bees breathe through spiracles—breathing-holes in their abdomens. This bee had flown into the cloud of red dust which was the spore-cloud of the new mushrooms.
- 61. The cloud drifted slowly along over the surface of yeasts and moulds, over toadstools and variegated fungus monstrosities. It moved steadily over a group of ants at work upon some bit of edible stuff. They were seized with an affliction like that of the bee. They writhed, moved convulsively. Their legs thrashed about. They died. The cloud of red dust settled as it moved. By the time it had travelled a quarter-mile, it had almost all settled to the ground. But a half-mile away there was another skyward-spurting uprush of red dust which spread slowly with the breeze. A quarter-mile away another plumed into the air. Farther on, two of them spouted their spores toward the clouds almost together. Living things that breathed the red dust writhed and died. And the red-dust puffballs were scattered everywhere. Burl and his tribesmen feasted, chattering in hushed tones of the remarkable fact that men ate meat of their own killing.
- 62. 6. RED DUST It was very fortunate indeed that the feast took place when it did. Two days later it would probably have been impossible, and three days later it would have been too late to do any good. But coming when it did, it made the difference which was all the difference in the world. Only thirty hours after the feasting which followed the death of the clotho spider, Burl's fellows—from Jon to Dor to Tet and Dik and Saya—had come to know a numb despair which the other creatures of his world were simply a bit too stupid to achieve. It was night. There was darkness over all the lowlands, and over all the area of perhaps a hundred square miles which the humans of Burl's acquaintance really knew. He, alone of his tribe, had been as much as forty miles from the foraging-ground over which they wandered. At any given time the tribe clung together for comfort, venturing only as far as was necessary to find food. Although the planet possessed continents, they knew less than a good-sized county of it. The planet owned oceans, and they knew only small brooks and one river which, where they knew it, was assuredly less than two hundred yards across. And they faced stark disaster that was not strictly a local one, but beyond their experience and hopelessly beyond their ability to face. They were superior to the insects about them only in the fact they realized what was threatening them. The disaster was the red puffballs. But it was night. The soft, blanketing darkness of a cloud-wrapped world lay all about. Burl sat awake, wrapped in his magnificent velvet cloak, his spear beside him and the yard-long golden plumes
- 63. of a moth's antennae bound to his forehead for a headdress. About him and his tribesmen were the swollen shapes of fungi, hiding the few things that could be seen in darkness. From the low-hanging clouds the nightly rain dripped down. Now a drop and then another drop; slowly, deliberately, persistently moisture fell from the skies. There was other sounds. Things flew through the blackness overhead—moths with mighty wing-beats that sometimes sent rhythmic wind-stirrings down to the tribe in its hiding-place. There were the deep pulsations of sound made by night-beetles aloft. There were the harsh noises of grasshoppers—they were rare— senselessly advertising their existence to nearby predators. Not too far from where Burl brooded came bright chirrupings where relatively small beetles roamed among the mushroom-forests, singing cheerfully in deep bass voices. They were searching for the underground tidbits which took the place of truffles their ancestors had lived on back on Earth. All seemed to be as it had been since the first humans were cast away upon this planet. And at night, indeed, the new danger subsided. The red puffballs did not burst after sunset. Burl sat awake, brooding in a new sort of frustration. He and all his tribe were plainly doomed—yet Burl had experienced too many satisfying sensations lately to be willing to accept the fact. The new red growths were everywhere. Months ago a storm-wind blew while somewhere, not too far distant, other red puffballs were bursting and sending their spores into the air. Since it was only a windstorm, there was no rain to wash the air clean of the lethal dust. The new kind of puffball—but perhaps it was not new: it could have thriven for thousands of years where it was first thrown as a sport from a genetically unstable parent—the new kind of puffball would not normally be spread in this fashion. By chance it had. There were dozens of the things within a quarter-mile, hundreds within a mile, and thousands upon thousands within the area the tribe normally foraged in. Burl had seen them even forty miles away,
- 64. as yet immature. They would be deadly at one period alone—the time of their bursting. But there were limitations even to the deadliness of the red puffballs, though Burl had not yet discovered the fact. But as of now, they doomed the tribe. One woman panted and moaned in her exhausted sleep, a little way from where Burl tried to solve the problem presented by the tribe. Nobody else attempted to think it out. The others accepted doom with fatalistic hopelessness. Burl's leadership might mean extra food, but nothing could counter the doom awaiting them—so their thoughts seemed to run. But Burl doggedly reviewed the facts in the darkness, while the humans about him slept the sleep of those without hope and even without rebellion. There had been many burstings of the crimson puffballs. As many as four and five of the deadly dust-clouds had been seen spouting into the air at the same time. A small boy of the tribe had breathlessly told of seeing a hunting-spider killed by the red dust. Lana, the half-grown girl, had come upon one of the gigantic rhinoceros-beetles belly-up on the ground, already the prey of ants. She had snatched a huge, meat-filled joint and run away, faster than the ants could follow. A far-ranging man had seen a butterfly, with wings ten yards across, die in a dust-cloud. Another woman—Cori—had been nearby when a red cloud settled slowly over long, solid lines of black worker-ants bound on some unknown mission. Later she saw other workers carrying the dead bodies back to the ant-city to be used for food. Burl still sat wakeful and frustrated and enraged as the slow rain fell upon the toadstools that formed the tribe's lurking-place. He doggedly went over and over the problem. There were innumerable red puffballs. Some had burst. The others undoubtedly would burst. Anything that breathed the red dust died. With thousands of the puffballs around them it was unthinkable that any human in this place could escape breathing the red dust and dying. But it had not always been so. There had been a time when there were no red puffballs here.
- 65. Burl's eyes moved restlessly over the sleeping forms limned by a patch of fox-fire. The feathery plumes rising from his head were outlined softly by the phosphorescence. His face was lined with a frown as he tried to think his own and his fellows' way out of the predicament. Without realizing it, Burl had taken it upon himself to think for his tribe. He had no reason to. It was simply a natural thing for him to do so, now that he had learned to think—even though his efforts were crude and painful as yet. Saya woke with a start and stared about. There had been no alarm, —merely the usual noises of distant murders and the songs of singers in the night. Burl moved restlessly. Saya stood up quietly, her long hair flowing about her. Sleepy-eyed, she moved to be near Burl. She sank to the ground beside him, sitting up—because the hiding- place was crowded and small—and dozed fitfully. Presently her head drooped to one side. It rested against his shoulder. She slept again. This simple act may have been the catalyst which gave Burl the solution to the problem. Some few days before, Burl had been in a far-away place where there was much food. At the time he'd thought vaguely of finding Saya and bringing her to that place. He remembered now that the red puffballs flourished there as well as here—but there had been other dangers in between, so the only half-formed purpose had been abandoned. Now, though, with Saya's head resting against his shoulder, he remembered the plan. And then the stroke of genius took place. He formed the idea of a journey which was not a going-after-food. This present dwelling-place of the tribe had been free of red puffballs until only recently. There must be other places where there were no red puffballs. He would take Saya and his tribesmen to such a place. It was really genius. The people of Burl's tribe had no purposes, only needs—for food and the like. Burl had achieved abstract thought— which previously had not been useful on the forgotten planet and, therefore, not practised. But it was time for humankind to take a
- 66. more fitting place in the unbalanced ecological system of this nightmare world, time to change that unbalance in favor of humans. When dawn came, Burl had not slept at all. He was all authority and decision. He had made plans. He spoke sternly, loudly—which frightened people conditioned to be furtive—holding up his spear as he issued commands. His timid tribesfolk obeyed him meekly. They felt no loyalty to him or confidence in his decisions yet, but they were beginning to associate obedience to him with good things. Food, for one. Before the day fully came, they made loads of the remaining edible mushroom and uneaten meat. It was remarkable for humans to leave their hiding-place while they still had food to eat, but Burl was implacable and scowling. Three men bore spears at Burl's urging. He brandished his long shaft confidently as he persuaded the other three to carry clubs. They did so reluctantly, even though previously they had killed ants with clubs. Spears, they felt, would have been better. They wouldn't be so close to the prey then. The sky became gray over all its expanse. The indefinite bright area which marked the position of the sun became established. It was part-way toward the center of the sky when the journey began. Burl had, of course, no determined course, only a destination—safety. He had been carried south, in his misadventure on the river. There were red puffballs to southward, therefore he ruled out that direction. He could have chosen the east and come upon an ocean, but no safety from the red spore-dust. Or he could have chosen the north. It was pure chance that he headed west. He walked confidently through the gruesome world of the lowlands, holding his spear in a semblance of readiness. Clad as he was, he made a figure at once valiant and rather pathetic. It was not too sensible for one young man—even one who had killed two spiders— to essay leading a tiny tribe of fearful folk across a land of monstrous ferocity and incredible malignance, armed only with a
- 67. spear from a dead insect's armor. It was absurd to dress up for the enterprise in a velvety cloak made of a moth's wing, blue moth-fur for a loin-cloth, and merely beautiful golden plumes bobbing above his forehead. Probably, though, that gorgeousness had a good effect upon his followers. They surely could not reassure each other by their numbers! There was a woman with a baby in her arms—Cori. Three children of nine or ten, unable to resist the instinct to play even on so perilous a journey, ate almost constantly of the lumps of foodstuff they had been ordered to carry. After them came Dik, a long-legged adolescent boy with eyes that roved anxiously about. Behind him were two men. Dor with a short spear and Jak hefting a club, both of them badly frightened at the idea of fleeing from dangers they knew and were terrified by, to other dangers unknown and, consequently, more to be feared. The others trailed after them. Tet was rear-guard. Burl had separated the pair of boys to make them useful. Together they were worthless. It was a pathetic caravan, in a way. In all the rest of the Galaxy, man was the dominant creature. There was no other planet from one rim to the other where men did not build their cities or settlements with unconscious arrogance—completely disregarding the wishes of lesser things. Only on this planet did men hide from danger rather than destroy it. Only here could men be driven from their place by lower life-forms. And only here would a migration be made on foot, with men's eyes fearful, their bodies poised to flee at sight of something stronger and more deadly than themselves. They marched, straggling a little, with many waverings aside from a fixed line. Once Dik saw the trap-door of a trapdoor-spider's lair. They halted, trembling, and went a long way out of their intended path to avoid it. Once they saw a great praying-mantis a good half- mile off, and again they deviated from their proper route. Near midday their way was blocked. As they moved onward, a great, high-pitched sound could be heard ahead of them. Burl stopped; his
- 68. face grew pinched. But it was only a stridulation, not the cries of creatures being devoured. It was a horde of ants by the thousands and hundreds of thousands, and nothing else. Burl went ahead to scout. And he did it because he did not trust anybody else to have the courage or intelligence to return with a report, instead of simply running away if the news were bad. But it happened to be a sort of action which would help to establish his position as leader of his tribe. Burl moved forward cautiously and presently came to an elevation from which he could see the cause of the tremendous waves of sound that spread out in all directions from the level plain before him. He waved to his followers to join him, and stood looking down at the extraordinary sight. When they reached his side—and Saya was first—the spectacle had not diminished. For quite half a mile in either direction the earth was black with ants. It was a battle of opposing armies from rival ant- cities. They snapped and bit at each other. Locked in vise-like embraces, they rolled over and over upon the ground, trampled underfoot by hordes of their fellows who surged over them to engage in equally suicidal combat. There was, of course, no thought of surrender or of quarter. They fought by thousands of pairs, their jaws seeking to crush each other's armor, snapping at each other's antennae, biting at each other's eyes.... The noise was not like that of army-ants. This was the agonizing sound of ants being dismembered while still alive. Some of the creatures had only one or two or three legs left, yet struggled fiercely to entangle another enemy before they died. There were mad cripples, fighting insanely with head and thorax only, their abdomens sheared away. The whining battle-cry of the multitude made a deafening uproar. From either side of the battleground a wide path led back toward separate ant-cities which were invisible from Burl's position. These
- 69. highways were marked by hurrying groups of ants—reinforcements rushing to the fight. Compared to the other creatures of this world the ants were small, but no lumbering beetle dared to march insolently in their way, nor did any carnivores try to prey upon them. They were dangerous. Burl and his tribesfolk were the only living things remaining near the battle-field—with one single exception. That exception was itself a tribe of ants, vastly less in number than the fighting creatures, and greatly smaller in size as well. Where the combatants were from a foot to fourteen inches long, these guerilla- ants were no more than the third of a foot in length. They hovered industriously at the edge of the fighting, not as allies to either nation, but strictly on their own account. Scurrying among the larger, fighting ants with marvelous agility, they carried off piecemeal the bodies of the dead and valiantly slew the more gravely wounded for the same purpose. They swarmed over the fighting-ground whenever the tide of battle receded. Caring nothing for the origin of the quarrel and espousing neither side, these opportunists busily salvaged the dead and still- living debris of the battle for their own purposes. Burl and his followers were forced to make a two-mile detour to avoid the battle. The passage between bodies of scurrying reinforcements was a matter of some difficulty. Burl hurried the others past a route to the front, reeking of formic acid, over which endless regiments and companies of ants moved frantically to join in the fight. They were intensely excited. Antennae waving wildly, they rushed to the front and instantly flung themselves into the fray, becoming lost and indistinguishable in the black mass of fighting creatures. The humans passed precariously between two hurrying battalions— Dik and Tet pausing briefly to burden themselves with prey—and hurried on to leave as many miles as possible behind them before nightfall. They never knew any more about the battle. It could have started over anything at all—two ants from the different cities may
- 70. have disputed some tiny bit of carrion and soon been reinforced by companions until the military might of both cities was engaged. Once it had started, of course, the fighters knew whom to fight if not why they did so. The inhabitants of the two cities had different smells, which served them as uniforms. But the outcome of the war would hardly matter. Not to the fighters, certainly. There were many red mushrooms in this area. If either of the cities survived at all, it would be because its nursery-workers lived upon stored food as they tended the grubs until the time of the spouting red dust had ended. Burl's folk saw many of the red puffballs burst during the day. More than once they came upon empty, flaccid parchment sacs. More often still they came upon red puffballs not yet quite ready to emit their murderous seed. That first night the tribe hid among the bases of giant puffballs of a more familiar sort. When touched they would shoot out a puff of white powder resembling smoke. The powder was harmless fortunately and the tribe knew that fact. Although not toxic, the white powder was identical in every other way to the terrible red dust from which the tribe fled. That night Burl slept soundly. He had been without rest for two days and a night. And he was experienced in journeying to remote places. He knew that they were no more dangerous than familiar ones. But the rest of the tribe, and even Saya, were fearful and terrified. They waited timorously all through the dark hours for menacing sounds to crash suddenly through the steady dripping of the nightly rain around them. The second day's journey was not unlike the first. The following day, they came upon a full ten-acre patch of giant cabbages bigger than a family dwelling. Something in the soil, perhaps, favored vegetation over fungi. The dozens of monstrous vegetables were the setting for riotous life: great slugs ate endlessly of the huge green leaves—and
- 71. things preyed on them; bees came droning to gather the pollen of the flowers. And other things came to prey on the predators in their turn. There was one great cabbage somewhat separate from the rest. After a long examination of the scene, Burl daringly led quaking Jon and Jak to the attack. Dor splendidly attacked elsewhere, alone. When the tribe moved on, there was much meat, and everyone— even the children—wore loin-cloths of incredibly luxurious fur. There were perils, too. On the fifth day of the tribe's journey Burl suddenly froze into stillness. One of the hairy tarantulas which lived in burrows with a concealed trap-door at ground-level, had fallen upon a scarabeus beetle and was devouring it only a hundred yards ahead. The tribesfolk trembled as Burl led them silently back and around by a safe detour. But all these experiences were beginning to have an effect. It was becoming a matter of course that Burl should give orders which others should obey. It was even becoming matter-of-fact that the possession of food was not a beautiful excuse to hide from all danger, eating and dozing until all the food was gone. Very gradually the tribe was developing the notion that the purpose of existence was not solely to escape awareness of peril, but to foresee and avoid it. They had no clear-cut notion of purpose as yet. They were simply outgrowing purposelessness. After a time they even looked about them with, dim stirrings of an attitude other than a desperate alertness for danger. Humans from any other planet, surely, would have been astounded at the vistas of golden mushrooms stretching out in forests on either hand and the plains with flaking surfaces given every imaginable color by the moulds and rusts and tiny flowering yeasts growing upon them. They would have been amazed by the turgid pools the journeying tribe came upon, where the water was concealed by a thick layer of slime through which enormous bubbles of foul-smelling gas rose to enlarge to preposterous size before bursting abruptly.
- 72. Had they been as ill-armed as Burl's folk, though, visitors from other planets would have been at least as timorous. Lacking highly specialized knowledge of the ways of insects on this world even well- armed visitors would have been in greater danger. But the tribe went on without a single casualty. They had fleeting glimpses of the white spokes of symmetrical spider-webs whose least thread no member of the tribe could break. Their immunity from disaster—though in the midst of danger—gave them a certain all-too-human concentration upon discomfort. Lacking calamities, they noticed their discomforts and grew weary of continual traveling. A few of the men complained to Burl. For answer, he pointed back along the way they had come. To the right a reddish dust-cloud was just settling, and to the rear rose another as they looked. And on this day a thing happened which at once gave the complainers the rest they asked for, and proved the fatality of remaining where they were. A child ran aside from the path its elders were following. The ground here had taken on a brownish hue. As the child stirred up the surface mould with his feet, dust that had settled was raised up again. It was far too thin to have any visible color. But the child suddenly screamed, strangling. The mother ran frantically to snatch him up. The red dust was no less deadly merely because it had settled to the ground. If a storm-wind came now—but they were infrequent under the forgotten planet's heavy bank of clouds—the fallen red dust could be raised up again and scattered about until there would be no living thing anywhere which would not gasp and writhe—and die. But the child would not die. He would suffer terribly and be weak for days. In the morning he could be carried. When night began to darken the sky, the tribe searched for a hiding- place. They came upon a shelf-like cliff, perhaps twenty or thirty feet
- 73. high, slanting toward the line of the tribesmen's travel. Burl saw black spots in it—openings. Burrows. He watched them as the tribe drew near. No bees or wasps went in or out. He watched long enough to be sure. When they were close, he was certain. Ordering the others to wait, he went forward to make doubly sure. The appearance of the holes reassured him. Dug months before by mining-bees, gone or dead now, the entrances to the burrows were weathered and bedraggled. Burl explored, first sniffing carefully at each opening. They were empty. This would be shelter for the night. He called his followers, and they crawled into the three-foot tunnels to hide. Burl stationed himself near the outer edge of one of them to watch for signs of danger. Night had not quite fallen. Jon and Dor, hungry, went off to forage a little way beyond the cliff. They would be cautious and timid, taking no risks whatever. Burl waited for the return of his explorers. Meanwhile he fretted over the meaning of the stricken child. Stirred-up red dust was dangerous. The only time when there would be no peril from it would be at night, when the dripping rainfall of the dark hours turned the surface of this world into thin shine. It occurred to Burl that it would be safe to travel at night, so far as the red dust was concerned. He rejected the idea instantly. It was unthinkable to travel at night for innumerable other reasons. Frowning, he poked his spear idly at a tumbled mass of tiny parchment cup-like things near the entrance of a cave. And instantly movement became visible. Fifty, sixty, a hundred infinitesimal creatures, no more than half an inch in length, made haste to hide themselves among the thimble-sized paperlike cups. They moved with extraordinary clumsiness and immense effort, seemingly only by contortions of their greenish-black bodies. Burl had never seen any creature progress in such a slow and ineffective fashion. He drew one of the small creatures back with the point of his spear and examined it from a safe distance.
- 74. He picked it up on his spear and brought it close to his eyes. The thing redoubled its frenzied movements. It slipped off the spear and plopped upon the soft moth-fur he wore about his middle. Instantly, as if it were a conjuring-trick, the insect vanished. Burl searched for minutes before he found it hidden deep in the long, soft hairs of his garment, resting motionless and seemingly at ease. It was the larval form of a beetle, fragments of whose armor could be seen near the base of the clayey cliffside. Hidden in the remnants of its egg-casings, the brood of minute things had waited near the opening of the mining-bee tunnel. It was their gamble with destiny when mining-bee grubs had slept through metamorphosis and come uncertainly out of the tunnel for the first time, that some or many of the larvae might snatch the instant's chance to fasten to the bees' legs and writhe upward to an anchorage in their fur. It happened that this particular batch of eggs had been laid after the emergence of the grubs. They had no possible chance of fulfilling their intended role as parasites on insects of the order hymenoptera. They were simply and matter-of-factly doomed by the blindness of instinct, which had caused them to be placed where they could not possibly survive. On the other hand, if one or many of them had found a lurking- place, the offspring of their host would have been doomed. The place filled by oil-beetle larvae in the scheme of things is the place— or one of the places—reserved for creatures that limit the number of mining-bees. When a bee-louse-infested mining-bee has made a new tunnel, stocked it with honey for its young, and then laid one egg to float on that pool of nourishment and hatch and feed and ultimately grow to be another mining-bee—at that moment of egg- laying, one small bee-louse detaches itself. It remains zestfully in the provisioned cell to devour the egg for which the provisions were accumulated. It happily consumes those provisions and, in time, an oil-beetle crawls out of the tunnel a mining-bee so laboriously prepared.
- 75. Burl had no difficulty in detaching the small insect and casting it away, but in doing so he discovered that others had hidden themselves in his fur without his knowledge. He plucked them away and found more. While savages can be highly tolerant of vermin too small to be seen, they feel a peculiar revolt against serving as host to creatures of sensible size. Burl reacted violently—as once he had reacted to the discovery of a leech clinging to his heel. He jerked off his loin-cloth and beat it savagely with his spear. When it was clean, he still felt a wholly unreasonable sense of humiliation. It was not clearly thought out, of course. Burl feared huge insects too much to hate them. But that small creatures should fasten upon him produced a completely irrational feeling of outrage. For the first time in very many years or centuries a human being upon the forgotten planet felt that he had been insulted. His dignity had been assailed. Burl raged. But as he raged, a triumphant shout came from nearby. Jon and Dor were returning from their foraging, loaded down with edible mushroom. They, also, had taken a step upward toward the natural dignity of men. They had so far forgotten their terror as to shout in exultation at their find of food. Up to now, Burl had been the only man daring to shout. Now there were two others. In his overwrought state this was also enraging. The result of hurt vanity on two counts was jealousy, and the result of jealousy was a crazy foolhardiness. Burl ground his teeth and insanely resolved to do something so magnificent, so tremendous, so utterly breathtaking that there could be no possible imitation by anybody else. His thinking was not especially clear. Part of his motivation had been provided by the oil-beetle larvae. He glared about him at the deepening dusk, seeking some exploit, some glamorous feat, to perform immediately, even in the night. He found one.
- 76. 7. JOURNEY THROUGH DEATH It was late dusk and the reddened clouds overhead were deepening steadily toward black. Dark shadows hung everywhere. The clay cliff cut off all vision to one side, but elsewhere Burl could see outward until the graying haze blotted out the horizon. Here and there, bees droned homeward to hive or burrow. Sometimes a slender, graceful wasp passed overhead, its wings invisible by the swiftness of their vibration. A few butterflies lingered hungrily in the distance, seeking the few things they could still feast upon. No moth had wakened yet to the night. The cloud-bank grew more sombre. The haze seemed to close in and shrink the world that Burl could see. He watched, raging, for the sight that would provide him with the triumph to end all triumphs among his followers. The soft, down- reaching fingers of the night touched here and there and the day ended at those spots. Then, from the heart of the deep redness to the west a flying creature came. It was a beautiful thing—a yellow emperor butterfly—flapping eastward with great sail-like velvet wings that seemed black against the sunset. Burl saw it sweep across the incredible sky, alight delicately, and disappear behind a mass of toadstools clustered so thickly they seemed nearly a hillock and not a mass of growing things. Then darkness closed in completely, but Burl still stared where the yellow emperor had landed. There was that temporary, utter quiet when day-things were hidden and night-things had not yet ventured out. Fox-fire glowed. Patches of pale phosphorescence—luminous mushrooms—shone faintly in the dark. Presently Burl moved through the night. He could imagine the yellow emperor in its hiding-place, delicately preening slender limbs before
- 77. it settled down to rest until the new day dawned. He had noted landmarks, to guide himself. A week earlier and his blood would have run cold at the bare thought of doing what he did now. In mere cool-headed detachment he would have known that what he did was close to madness. But he was neither cool-headed nor detached. He crossed the clear ground before the low cliff. But for the fox-fire beacons he would have been lost instantly. The slow drippings of rain began. The sky was dead black. Now was the time for night- things to fly, and male tarantulas to go seeking mates and prey. It was definitely no time for adventuring. Burl moved on. He found the close-packed toadstools by the process of running into them in the total obscurity. He fumbled, trying to force his way between them. It could not be done; they grew too close and too low. He raged at this impediment. He climbed. This was insanity. Burl stood on spongy mushroom-stuff that quivered and yielded under his weight. Somewhere something boomed upward, rising on fast-beating wings into blackness. He heard the pulsing drone of four-inch mosquitos close by. He moved forward, the fungus support swaying, so that he did not so much walk as stagger over the close-packed mushroom heads. He groped before him with spear and panted a little. There was a part of him which was bitterly afraid, but he raged the more furiously because if once he gave way even to caution, it would turn to panic. Burl would have made a strange spectacle in daylight gaudily clothed as he was in soft blue fur and velvet cloak, staggering over swaying insecurity, coddling ferocity in himself against the threat of fear. Then his spear told him there was emptiness ahead. Something moved, below. He heard and felt it stirring the toadstool-stalks on which he stood. Burl raised his spear, grasping it in both hands. He plunged down with it, stabbing fiercely.
- 78. The spear struck something vastly more resistant than any mushroom could be. It penetrated. Then the stabbed thing moved as Burl landed upon it, flinging him off his feet, but he clung to the firmly imbedded weapon. And if his mouth had opened for a yell of victory as he plunged down, the nature of the surface on which he found himself, and the kind of movement he felt, turned that yell into a gasp of horror. It wasn't the furry body of a butterfly he had landed on; his spear hadn't pierced such a creature's soft flesh. He had leaped upon the broad, hard back of a huge, meat-eating, nocturnal beetle. His spear had pierced not the armor, but the leathery joint-tissue between head and thorax. The giant creature rocketed upward with Burl clinging to his spear. He held fast with an agonized strength. His mount rose from the blackness of the ground into the many times more terrifying blackness of the air. It rose up and up. If Burl could have screamed, he would have done so, but he could not cry out. He could only hold fast, glassy-eyed. Then he dropped. Wind roared past him. The great insect was clumsy at flying. All beetles are. Burl's weight and the pain it felt made its flying clumsier still. There was a semi-liquid crashing and an impact. Burl was torn loose and hurled away. He crashed into the spongy top of a mushroom and came to rest with his naked shoulder hanging halfway over some invisible drop. He struggled. He heard the whining drone of his attempted prey. It rocketed aloft again. But there was something wrong with it. With his weight applied to the spear as he was torn free, Burl had twisted the weapon in the wound. It had driven deeper, multiplying the damage of the first stab. The beetle crashed to earth again, nearby. As Burl struggled again, the mushroom-stalk split and let him gently to the ground.
- 79. He heard the flounderings of the great beetle in the darkness. It mounted skyward once more, its wing-beats no longer making a sustained note. It thrashed the air irregularly and wildly. Then it crashed again. There was seeming silence, save for the steady drip-drip of the rain. And Burl came out of his half-mad fear: he suddenly realized that he had slain a victim even more magnificent than a spider, because this creature was meat. He found himself astonishedly running toward the spot where the beetle had last fallen. But he heard it struggle aloft once more. It was wounded to death. Burl felt certain of it this time. It floundered in mid-air and crashed again. He was within yards of it before he checked himself. Now he was weaponless, and the gigantic insect flung itself about madly on the ground, striking out with colossal wings and limbs, fighting it knew not what. It struggled to fly, crashed, and fought its way off the ground—ever more weakly—then smashed again into mushrooms. There it floundered horribly in the darkness. Burl drew near and waited. It was still, but pain again drove it to a senseless spasm of activity. Then it struck against something. There was a ripping noise and instantly the close, peppery, burning smell of the red dust was in the air. The beetle had floundered into one of the close-packed red puffballs, tightly filled with the deadly red spores. The red dust would not normally have been released at night. With the nightly rain, it would not travel so far or spread so widely. Burl fled, panting. Behind him he heard his victim rise one last time, spurred to impossible, final struggle by the anguish caused by the breathed-in
- 80. red dust. It rose clumsily into the darkness in its death-throes and crashed to the ground again for the last time. In time to come, Burl and his followers might learn to use the red- dust puffballs as weapons—but not how to spread them beyond their normal range. But now, Burl was frightened. He moved hastily sidewise. The dust would travel down-wind. He got out of its possible path. There could be no exultation where the red dust was. Burl suddenly realized what had happened to him. He had been carried aloft an unknown though not-great distance, in an unknown direction. He was separated from his tribe, with no faintest idea how to find them in the darkness. And it was night. He crouched under the nearest huge toadstool and waited for the dawn, listening dry-throated for the sound of death coming toward him through the night. But only the wind-beats of night-fliers came to his ears, and the discordant notes of gray-bellied truffle-beetles as they roamed the mushroom thickets, seeking the places beneath which—so their adapted instincts told them—fungoid dainties, not too much unlike the truffles of Earth, awaited the industrious miner. And, of course, there was that eternal, monotonous dripping of the raindrops, falling at irregular intervals from the sky. Red puffballs did not burst at night. They would not burst anyhow, except at one certain season of their growth. But Burl and his folk had so far encountered the over-hasty ones, bursting earlier than most. The time of ripeness was very nearly here, though. When day came again, and the chill dampness of the night was succeeded by the warmth of the morning, almost the first thing Burl saw in the gray light was a tall spouting of brownish-red stuff leaping abruptly into the air from a burst red parchment-like sphere. He stood up and looked anxiously all around. Here and there, all over the landscape, slowly and at intervals, the plumes of fatal red
- 81. sprang into the air. There was nothing quite like it anywhere else. An ancient man, inhabiting Earth, might have likened the appearance to that of a scattered and leisurely bombardment. But Burl had no analogy for them. He saw something hardly a hundred yards from where he had hidden during the night. The dead beetle lay there, crumpled and limp. Burl eyed it speculatively. Then he saw something that filled him with elation. The last crash of the beetle to the ground had driven his spear deeply between the joints of the corselet and neck. Even if the red dust had not finished the creature, the spear-point would have ended its life. He was thrilled once more by his superlative greatness. He made due note that he was a mighty slayer. He took the antennae as proof of his valor and hacked off a great barb-edged leg for meat. And then he remembered that he did not know how to find his fellow- tribesmen. He had no idea which way to go. Even a civilized man would have been at a loss, though he would have hunted for an elevation from which to look for the cliff hiding- place of the tribe. But Burl had not yet progressed so far. His wild ride of the night before had been at random, and the chase after the wounded beetle no less dictated by chance. There was no answer. He set off anxiously, searching everywhere. But he had to be alert for all the dangers of an inimical world while keeping, at the same time, an extremely sharp eye out for bursting red puffballs. At the end of an hour he thought he saw familiar things. Then he recognized the spot. He had come back to the dead beetle. It was already the center of a mass of small black bodies which pulled and hacked at the tough armor, gnawing out great lumps of flesh to be carried to the nearest ant-city. Burl set off again, very carefully avoiding any place that he recognized as having been seen that morning. Sometimes he walked through mushroom-thickets—dangerous places to be in—and
- 82. sometimes over relatively clear ground colored exotically with varicolored fungi. More than once he saw the clouds of red stuff spurting in the distance. Deep anxiety filled him. He had no idea that there were such things as points of the compass. He knew only that he needed desperately to find his tribesfolk again. They, of course, had given him up for dead. He had vanished in the night. Old Tama complained of him shrilly. The night, to them, meant death. Jon quaked watchfully all through it. When Burl did not come to the feast of mushroom that Jon and Dor had brought back, they sought him. They even called timidly into the darkness. They heard the throbbing of huge wings as a great creature rose desperately into the sky, but they did not associate that sound with Burl. If they had, they would have been instantly certain of his fate. As it was, the tribe's uneasiness grew into terror which rapidly turned to despair. They began to tremble, wondering what they would do with no bold chieftain to guide them. He was the first man to command allegiance from others in much too long a period, on the forgotten planet, but the submission of his followers had been the more complete for its novelty. His loss was the more appalling. Burl had mistaken the triumphant shout of the foragers. He'd thought it independence of him—rivalry. Actually, the men dared to shout only because they felt secure under his leadership. When they accepted the fact that he had vanished—and to disappear in the night had always meant death—their old fears and timidity returned. To them it was added despair. They huddled together and whispered to one another of their fright. They waited in trembling silence through all the long night. Had a hunting-spider appeared, they would have fled in as many directions as there were people, and undoubtedly all would have perished. But day came again, and they looked into each other's eyes and saw the self-same fear. Saya was probably the most pitiful of the group. Her face was white and drawn beyond that of any one else.
- 83. They did not move when day brightened. They remained about the bee-tunnels, speaking in hushed tones, huddled together, searching all the horizon for enemies. Saya would not eat, but sat still, staring before her in numbed grief. Burl was dead. Atop the low cliff a red puffball glistened in the morning light. Its tough skin was taut and bulging, resisting the pressure of the spores within. Slowly, as the morning wore on, some of the moisture that kept the skin stretchable dried. The parchment-like stuff contracted. The tautness of the spore-packed envelope grew greater. It became insupportable. With a ripping sound, the tough skin split across and a rush of the compressed spores shot skyward. The tribesmen saw and cried out and fled. The red stuff drifted down past the cliff-edge. It drifted toward the humans. They ran from it. Jon and Tama ran fastest. Jak and Cori and the other were not far behind. Saya trailed, in her despair. Had Burl been there, matters would have been different. He had already such an ascendancy over the minds of the others that even in panic they would have looked to see what he did. And he would have dodged the slowly drifting death-cloud by day, as he had during the night. But his followers ran blindly. As Saya fled after the others she heard shrieks of fright to the left and ran faster. She passed by a thick mass of distorted fungi in which there was a sudden stirring and panic lent wings to her feet. She fled blindly, panting. Ahead was a great mass of stuff—red puffballs—showing here and there among great fanlike growths, some twelve feet high, that looked like sponges. She fled past them and swerved to hide herself from anything that might be pursuing by sight. Her foot slipped on the slimy body of a shell-less snail and she fell heavily, her head striking a stone. She lay still.
- 84. Almost as if at a signal a red puffball burst among the fanlike growths. A thick, dirty-red cloud of dust shot upward, spread and billowed and began to settle slowly toward the ground again. It moved as it settled flowing over the inequalities of die ground as a monstrous snail or leach might have done, sucking from all breathing creatures the life they had within them. It was a hundred yards away, then fifty, then thirty.... Had any member of the tribe watched it, the red dust might have seemed malevolently intelligent. But when the edges of the dust- cloud were no more than twenty yards from Saya's limp body, an opposing breeze sprang up. It was a vagrant, fitful little breeze that halted the red cloud and threw it into some confusion, sending it in a new direction. It passed Saya without hurting her, though one of its misty tendrils reached out as if to snatch at her in slow-motion. But it passed her by. Saya lay motionless on the ground. Only her breast rose and fell shallowly. A tiny pool of red gathered near her head. Some thirty feet from where she lay, there were three miniature toadstools in a clump, bases so close together that they seemed but one. From between two of them, however, two tufts of reddish thread appeared. They twinkled back and forth and in and out. As if reassured, two slender antennae followed, then bulging eyes and a small, black body with bright-red scalloped markings upon it. It was a tiny beetle no more than eight inches long—a sexton or burying-beetle. Drawing near Saya's body it scurried onto her flesh. It went from end to end of her figure in a sort of feverish haste. Then it dived into the ground beneath her shoulder, casting back a little shower of hastily-dug dirt as it disappeared. Ten minutes later, another small creature appeared, precisely like the first. Upon the heels of the second came a third. Each made the same hasty examination and dived under her unmoving form.
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