Image segmentation 3 morphology

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Mathematical Morphology

 Mathematical morphology is a powerful
methodology which was initiated in the late
1960s by G.Matheron and J.Serra at the
Fontainebleau School of Mines in France.
 nowadays it offers many theoretic and
algorithmic tools inspiring the development of
research in the fields of signal
processing, image processing, machine
vision, and pattern recognition.

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Morphological Operations -1

The four most basic operations in mathematical
morphology are dilation, erosion, opening and Closing:

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Morphological Image Processing

 Boolean algebra
 Dilation and erosion
 Opening and closing
 Hit-or-miss
 Basic algorithms
 Extension to gray-scale

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Examples of Boolean Algebra
 Switching algebra
 S = {0, 1}
 Finite Boolean algebras
 Example: S = {(0, 0), (0, 1), (1, 0), (1, 1)}
 (a1, a2)’ = (a’1, a’2)
 (0, 1) (1, 0) = (0, 0)
 Set unions/intersections
 Union is like
 Intersection is like
 Empty set is like 0
 There is no 1 (universal set)

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Review: Boolean Algebra
 A Boolean algebra is a set with at least two
elements, three operations (and , or , not ‘) and two
special elements (0, 1) that have the following
properties.
 A B is an element of the set. This function is defined for all
elements A and B in the set. It is symmetric (A B = B A)
 A B has the same properties
 A’ is defined for all elements in the set.
 A A’=0, A A’=1
 The operations and + are distributive.
 A (B C)=(A B) (A C)
 A (B C)=(A B) (A C)
 0 and 1 are identities, in the following sense
 0 A=A
 1 A=A

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Some Basic Definitions
 Let A and B be sets with components a=(a1,a2) and
b=(b1,b2), respectively.
 The translation of A by x=(x1,x2) is
A + x = {c | c = a + x, for a A}
 The reflection of A is
Ar = {x | x = -a for a A}
 The complement of A is
Ac = {x | x A}
 The union of A and B is
A B = {x | x A or x B }
 The intersection of A and B is
A B = {x | x A and x B }
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Some Basic Definitions
 The difference of A and B is.
A – B = A Bc = {x | x A and x B}
 A and B are said to be disjoint or mutually exclusive if they
have no common elements.
 If every element of a set A is also an element of another set
B, then A is said to be a subset of B.

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Additional Operations
 Elements of set: points
 Points are integers (1-D discrete space)
 Points are 2-D vectors with integer components (2-D
discrete space)
 Operations
 Addition (vector addition)
 Reflection (multiply by -1)
 Integer multiplication
 A set of points can be translated or reflected
 S+x = x+S (new set consists of all points of
S, translated by x)
 S^ is the set reflected through the origin

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Continuous and Discrete Morphology

 There are morphology theories of continuous
and discrete spaces
 Example of continuous space
 Real line
 Example of discrete space
 Integers
 We will talk about the morphology of discrete
spaces

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Morphology

A binary image containing two object sets A and B

 B = {(0,0), (0,1), (1,0)}
 A = {(5,0), (3,1), (4,1), (5,1), (3,2), (4,2), (5,2)}

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Basic Morphological Operations

 Dilation
 A+B = {x| x = y+z, y in A, z in B}
 Equivalent definition
 {x, (x+B^) A is not empty}
 Erosion
 A-B = {x| x+B is a subset of A}

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Some Basic Definitions (Dilation)
 Dilation
A B = {x | (B + x) A }
 Dilation expands a region.

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More Examples (2)

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Some Basic Definitions (Erosion)
 Erosion
A B = {x | (B + x) A}
 Erosion shrinks a region.

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More Examples (3)

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Some Basic Definitions (opening)
 Opening is erosion followed by dilation:
A B = (A B) B
 Opening smoothes regions, removes spurs, breaks narrow
lines.

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Morphological Opening
A oB @(A B) B

A opened by B

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Some Basic Definitions (closing)
 Closing is dilation followed by erosion:
A B = (A B) B
 Closing fills narrow gaps and holes in a region.

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Morphological Closing

A • B @(A B) B
A closed by B

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Combination of Opening and Closing

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Hit-or-Miss

 Given: points on plane
 Template: Set of one points (foreground) and set
of zero points (background)
 Example foreground: B1=D, B2=D
 Find: Points x for which B1+x are 1, B2+x are 0
 Solution:

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Boundary Extraction

(A) A (A B)'

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Region Filling

Xk (Xk 1 B) A'
•Start with point in region A.
Keep expanding by
dilation, using points in region A
only.

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Extraction of Connected Component

Xk (Xk 1 B) A

 Start with point
on object. Keep
adding points

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Skeleton

 Morphological skeleton
Start with structuring element, B
Generate a sequence of elements Bk=kB,
B0=0 n
A k 0 Sk Bk
Construction

Sk (A Bk ) ((A Bk ) B)'

Connected skeleton

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Morphological Skeleton

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Connected Skeleton

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Some Morphological Algorithms

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 Boundary of a set, A, can be found by
A - (A B)

B

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 A region can be filled iteratively by
Xk+1 = (Xk B) Ac ,
where k = 0,1,…
and X0 is a point inside the region.

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Morphological Operations

BWMORPH Perform morphological operations on binary image.
BW2 = BWMORPH(BW1,OPERATION) applies a specific
morphological operation to the binary image BW1.

BW2 = BWMORPH(BW1,OPERATION,N) applies the operation N
times. N can be Inf, in which case the operation is repeated
until the image no longer changes.

OPERATION is a string that can have one of these values:
'skel' With N = Inf, remove pixels on the boundaries
of objects without allowing objects to break
apart
'spur' Remove end points of lines without removing
small objects completely.
'fill' Fill isolated interior pixels (0's surrounded by
1's)
...

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BW1 = imread('circbw.tif');
BW2 = bwmorph(BW1,'skel',Inf);
imshow(BW1);
figure, imshow(BW2);

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BW1 = imread('circbw.tif');
BW2 = bwperim(BW1);
imshow(BW1);
figure, imshow(BW2)

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Pixel Connectivity
Connectivity defines which pixels are connected to other pixels. A set of
pixels in a binary image that form a connected group is called an object
or a connected component.

4-connected 8-connected

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BW = [0 0 0 0 0 0 0 0;
0 1 1 0 0 1 1 1;
0 1 1 0 0 0 1 1; X = bwlabel(BW,4);
0 1 1 0 0 0 0 0; RGB = label2rgb(X, @jet, 'k');
0 0 0 1 1 0 0 0; imshow(RGB,'notruesize')
0 0 0 1 1 0 0 0;
0 0 0 1 1 0 0 0;
0 0 0 0 0 0 0 0];
X = bwlabel(BW,4)
X = 0 0 0 0 0 0 0 0
0 1 1 0 0 3 3 3
0 1 1 0 0 0 3 3
0 1 1 0 0 0 0 0
0 0 0 2 2 0 0 0
0 0 0 2 2 0 0 0
0 0 0 2 2 0 0 0
0 0 0 0 0 0 0 0

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Application example: Using
connected components to detect
foreign objects in packaged food.
There are four objects with
significant size!

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 Thinning: Thin regions iteratively; retain connections and
endpoints.
 Skeletons: Reduces regions to lines of one pixel thick;
preserves shape.
 Convex hull: Follows outline of a region except for
concavities.

 Pruning: Removes small branches.

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Extensions to Gray-Scale Images
Dilation

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Dilation

Take the maximum within the window.

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Erosion

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 Dilation:
 Makes image brighter

 Reduces or eliminates dark details

 Erosion:
 Makes image darker

 Reduces or eliminates bright details

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Opening: Narrow bright areas are reduced.
Closing: Narrow dark areas are reduced.

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Application Example

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Application Example-Segmentation

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Application Example-Granulometry

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Morphological Reconstruction (exp.algo)

 Algorithm for binary reconstruction:
1. M = V o K , where K is any SE.
2. T = M,
3. M= M Ki , where i=4 or i=8,

4. M = M∩ V, [Take only those pixels from M that are also in V .]
5. if M T then go to 2,

6. else stop;

Original (V) Opened (M) Reconstructed (T)
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Application in 2D Image Processing
Character Extraction-1

Character Extraction From Cover Image (Source)
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Character Extraction From Cover Image (Results)
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Morning Noon

Afternoon Evening
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Morning Noon

Afternoon Evening
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Organs Extraction-1

slice20 slice25 slice30

slice25 slice30 65
slice20

Organs Extraction-2

Top View Back View

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Organs Extraction-3

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Organs Extraction-4

Segmented heart beating cycle

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Organs Extraction-5

Kidney with Bones Kidney with Vessels

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Brain View snapshot-3
--Registration tool kit

Features:
1. Cut plane in 3D
2. Work in 2 data sets
3. 2D and 3D view
4. Registration methods:
• LandMark
• ThinPlateSpline
• GridTransform
• MutualInformation

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Brain View snapshot-4
--Segmenation tool kit

Features:
1. Cut plane in 3D
2. Work in 2 data sets
3. 2D and 3D view
4. Segmenation
methods:
• Morphology
• Snake
• Level Set
• Watershed

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Research Plan-1
 Medical Image Analysis
--- Segmentation and Registration
 More efforts address on Ultrasound Image (2D, 3D)

Segmented baby face from US

2D Segmentation using GDM 72
Real time US, MR integration for IGS

Image Comparison Techniques

• Image Subtraction
the simplest and most direct approach to PCB inspection
problem. The PCB is compared to an image of an ideal part. The
subtraction can be done by logical XOR operation between the
two images .
• Feature Matching
an improved form of image subtraction, in which the extracted
features from the object and those defined by the model are
compared.
• Phase Only Method

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Example PCB and Mask

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Result

short
spur
dirt
open
mousebite
pinhole

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Image segmentation 3 morphology

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Image segmentation 3 morphology