Higher dimensional image analysis using brunn minkowski theorem, convexity and mathematical morphology

Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5782 (print) ISSN 2225-0506 (online)
Vol 2, No.4, 2012

Higher Dimensional Image Analysis using Brunn-Minkowski
Theorem, Convexity and Mathematical Morphology
Ramkumar P.B (Corresponding author)
Rajagiri School of Engineering & Technology, Rajagiri Valley PO , Kerala, India ,PIN - 682039
Tel: 0484 2427835
* E-mail of the corresponding author: rkpbmaths@yahoo.co.in
Abstract
The theory of deterministic morphological operators is quite rich and has been used on set and lattice theory.
Mathematical Morphology can benefit from the already developed theory in convex analysis. Mathematical
Morphology introduced by Serra is a very important tool in image processing and Pattern recognition. The
framework of Mathematical Morphology consists in Erosions and Dilations. Fractals are mathematical sets with a
high degree of geometrical complexity that can model many natural phenomena. Examples include physical objects
such as clouds, mountains, trees and coastlines as well as image intensity signals that emanate from certain type of
fractal surfaces. So this article tries to link the relation between combinatorial convexity and Mathematical
Morphology.
Keywords: Convex bodies, convex polyhedra, homothetics, morphological cover, fractal, dilation, erosion.
1. Introduction
1.1 Types of Images
An image is a mapping denoted as I, from a set, NP, of pixel coordinates to a set, M, of values such that for every

coordinate vector, p = in NP, there is a value I(p) drawn from M. NP is also called the image plane.[1]

Under the above defined mapping a real image maps an n-dimensional Euclidean vector space into the real numbers.
Pixel coordinates and pixel values are real.
A discrete image maps an n-dimensional grid of points into the set of real numbers. Coordinates are n-tuples of
integers, pixel values are real.
A digital image maps an n-dimensional grid into a finite set of integers. Pixel coordinates and pixel values are integers.
A binary image has only 2 values. That is, M= {mfg , mbg}, where mfg, is called the foreground value and mbg is called
the background value.
The foreground value is mfg = 0, and the background is mbg = –∞. Other possibilities are {mfg, mbg} = {0,∞}, {0,1},
{1,0}, {0,255}, and {255,0}.
1.2 Definition

The foreground of binary image I is .

The background is the complement of the foreground and vice-versa
1.3 Dilation and Erosion
Morphology uses ‘Set Theory’ as the foundation for many functions [1]. The simplest functions to implement are
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‘Dilation’ and ‘Erosion’
1.3.1 Definition: Dilation of the object A by the structuring element B is given by

Usually A will be the signal or image being operated on A and B will be the Structuring Element’

1.3.2 Definition: Erosion
The opposite of dilation is known as erosion. Erosion of the object A by a structuring element B is given by

Erosion of A by B is the set of points x such that B translated by x is contained in A.

1.4 Opening and Closing

Two very important transformations are opening and closing. Dilation expands an image object and erosion shrinks
it. Opening, generally smoothes a contour in an image, breaking narrow isthmuses and eliminating thin protrusions.
Closing tends to narrow smooth sections of contours, fusing narrow breaks and long thin gulfs, eliminating small
holes, and filling gaps in contours.
1.4.1 Definition Opening
The opening of A by B, denoted by , is given by the erosion by B, followed by the dilation by B, that is

1.4.2 Closing

The opposite of opening is ‘Closing’ defined by
Closing is the dual operation of opening and is denoted by . It is produced by the dilation of A by B, followed by
the erosion by B:
2. Morphological Operators defined on a Lattice
2.1 Definition: Dilation

Let be a complete lattice, with infimum and minimum symbolized by and , respectively.[1],[2].[10]

A dilation is any operator that distributes over the supremum and preserves the least element.

, .
2.2 Definition : Erosion

An erosion is any operator that distributes over the infimum , .
2.3 Galois connections
Dilations and erosions form Galois connections. That is, for all dilation δ there is one and only one erosion that satisfies

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Vol 2, No.4, 2012

for all .

Similarly, for all erosion there is one and only one dilation satisfying the above connection.
Furthermore, if two operators satisfy the connection, then δ must be a dilation , and an erosion.
2.4 Definition :Adjunctions :
Pairs of erosions and dilations satisfying the above connection are called "adjunctions", and the erosion is said to be the
adjoint erosion of the dilation, and vice-versa.
2.5 Opening and Closing

For all adjunction , the morphological opening and morphological closing are

defined as follows:[2]

, and .

The morphological opening and closing are particular cases of algebraic opening (or simply opening) and algebraic
closing(or simply closing). Algebraic openings are operators in L that are idempotent, increasing, and anti-extensive.
Algebraic closings are operators in L that are idempotent, increasing, and extensive.
2.6 Particular cases
Binary morphology is a particular case of lattice morphology, where L is the power set of E (Euclidean space or grid),
that is, L is the set of all subsets of E, and is the set inclusion. In this case, the infimum is set intersection, and the
supremum is set union.
Similarly, grayscale morphology is another particular case, [2] where L is the set of functions mapping E into

, and , , and , are the point-wise order, supremum, and infimum, respectively. That is, is f

and g are functions in L, then if and only if ; the infimum is given by

; and the supremum is given by .[1]

g (m)
Let f ( n) be the signal and be the structuring element over a domain then the dilation and erosion are defined as
g (n )

( f ⊕ g )(n) = max m∈D [ f (n − m) + g (m)], D is any domain and

(f g )(n) = min m∈D [ f (n + m) − g (m)], D is any domain. ⊕ and are dilation and erosion operators.[2]

2.7 Morphogenetic field

Let X ≠ ϕ and W ⊆ P ( X ) such that i) φ , X ∈ W , ii) If B ∈ W then its complement B ∈ W iii) If

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Vol 2, No.4, 2012

∞
Bi ∈ W is a sequence of signals defined in X, then U Bi ∈
n =1
W.

Let A= {φ : W → U / φ (∪ Ai ) = ∨φ ( Ai ) & φ (∧ Ai ) = ∧φ ( Ai )} . Then WU is called Morphogenetic field

[16]where the family Wu is the set of all image signals defined on the continuous or discrete image Plane X and
taking values in a set U .The pair ( Wu, A ) is called an operator space where A is the collection of operators defined on
X.
2.8 Morphological space
The triplet (X, Wu, A ) consisting of a set X, a morphogenetic field Wu and an operator A(or collection of operators)
defined on X is called a Morphological space [8].
Note: If X = Z2 then it is called Discrete Morphological space
3 . Morphological Cover

Let ( X ,Wu , A) be a morphological space. Let ε be the size of the structuring element B, then

N
CB (ε ) = ∑ ( f ⊕ Bε − f Bε )(n) is called the morphological cover of f.[13]
n =1

4. Brunn-Minkowski Theorem

Let ( X ,Wu , A) be a morphological space .Let 0<t<1 and let Ho and H1 be any two compact convex

structuring bodies in .Consider the compact convex body Ht = (1-t)H0+tH1.Then the following inequality
holds true:

(v( H t ))1/ n ≥ (1 − t )(v( H 0 ))1/ n + t (v( H1 ))1/ n where v denotes any volume element in .

This inequality is reduced to the equality iff H0 and H1 are homothetic.[14]

4.1 Lemma:

Let 0<t<1 and let and be two compact convex bodies in such that Then,

for the compact convex body the inequality holds true. Furthermore,

the inequality is fulfilled iff is a translate of .(ie. for some translation h of

Rn.[14]

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4.2 Result: Let ( X ,Wu , A) be a morphological space .Let U= E be a vector space and let dim(E) =n. Let P be an

n-dimensional convex body structuring element in X taking values in E. Then a finite family of structuring
elements of n dimensional simplices such that

1)

2) The interior of these structuring elements (simplices) are pairwise disjoint.

3) For each ,the set of vertices of is contained in the set of vertices of P.

Remark: In view of the above result, we can state the following result.

4.3 Result: Let ( X ,Wu , A) be a morphological space. All convex structuring elements defined on a morphological
space can be decomposed.

4.4 Proposition: If ∪ Bi denotes the decomposition of a convex structuring element B then B = ∪ Bi .
i i

4.5 Proposition: A morphological space ( X ,Wu , A) is compact iff for every family of closed convex sets defined
by v has a non empty intersection.

4.6 Proposition: Let ( X ,Wu , A) be a morphological space. If are decomposition of a convex

structuring element B ,then f ⊕ B = f ⊕ ∪ Bi
i

4.7 Lemma ::Let ( X ,Wu , A) be a morphological space .Let A and B be any two compact convex structuring elements in

the space X = Rn and let the relation S B ( A) = n(v( A))( n−1) / n .(v( B ))1/ n be satisfied. Then A and B are homothetic.
In particular, if B is a ball then A is a ball too, where S denotes an (n-1) dimensional volume in Rn .

4.8 Definition : The cross section of f at level t is the set defined by

X t ( f ) = { x / f ( x) ≥ t} where

4.9 Proposition:[In view of the above lemma]

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Vol 2, No.4, 2012

Let ( X ,Wu , A) be a morphological space .Let and be two cross sections of a

compact convex bodies in Rn(or image in Rn) and let the relation
S X t ( X t1 ( f )) = n(v( X t1 ( f )))( n−1) / n .(v( X t2 ( f )))1/ n be satisfied. Then are
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homothetic.

5 Morphological Equivalence

If two objects are homothetic then they are morphologically equivalent.

In particular, are Morphologically equivalent.

5.1 Proposition:

Let ( X ,Wu , A) be a morphological space .Let be the given compact planar structuring set in .

Let be positive homothetic. i.e. H 0 = ε B0 = {ε b / b ∈ B0 } & H1 = ε B1 = {ε b / b ∈ B1} and

also for a compact convex body. Then the following equality holds:

(v( H t ))1/ n = (1 − t )(v( H 0 ))1/ n + t (v( H1 ))1/ n .[14]

5.2 Proposition:

Let ( X ,Wu , A) be a morphological space .Let where ,ε B = {ε b / b ∈ B} for any

Basic structuring body B, then v(Ct ) = 1 where be a compact convex body. Also

v(C0 ) = 1 and v(C1 ) = 1 .[14]

5.3 Lemma

Let ( X ,Wu , A) be a morphological space .Let M be a smooth surface in supplied with the induced metric.

That is lengths are measured along the surface: For any let be the disk of radius r centred at x.

Then (v( H t ))1/ n = (1 − t )(v( H 0 ))1/ n + t (v( H1 ))1/ n where and homothetic and

are defined as H 0 = U Dr1 ( x ), ∀x and H1 = U Dr2 ( x ), ∀x .[14]

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5.4 Result: Morphological cover of any smooth surface M=f is compact.

5.5 Morphological floating space: Let ( X ,Wu , A) be a morphological space. . Let ε be the size of any structuring

N
element B and the morphological cover of f be CB (ε ) = ∑ ( f ⊕ Bε − f Bε )(n) .Then the infima or
n =1

suprema of all morphological covers for every ε defines a Morphological floating space. So, there exist two
floating spaces which are known as upper and lower floating spaces.

5.6 Complete Morphological space

A Morphological space is said to be complete if there exist upper and lower floating spaces.

5.7 Morphological Projection

If a Morphological space ( X ,Wu , A) is complete then there exist a projection of f into the floating space.

6 Conclusion

Convex sets and convex structures play an important role in higher dimensional analysis. Convex sets can be defined in
a vector space or more generally in a Morphological space. An attempt towards the extension of work done in lower
dimension is explained in this paper. Image processing in higher dimension is more complicated. We hope that this
work will help to reduce the difficulties at least in a smaller scale.

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Vol 2, No.4, 2012

References
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Frank Y .Shih, Image Processing and Mathematical Morphology, CRC Press, 2009.

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First A. Author (Ramkumar P.B) is working as an assistant professor at Rajagiri School of Engineering &
Technology Cochin . He received the post graduate degree in mathematics from Cochin university of Science &
Technology , Cochin , India in 1999. His interests are in Mathematical Morphology, Discrete Mathematics and
Optimization Techniques.

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