Colorado School of Mines
Image and Multidimensional
Signal Processing
Professor William Hoff
Dept of Electrical Engineering &Computer Science
Colorado School of Mines
Department of Electrical Engineering and Computer Science
http://inside.mines.edu/~whoff/
Morphological Image Algorithms
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Outline
• Morphological algorithms
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–
–
–
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Boundary Extraction
Hit or Miss Transform
Region Filling
Skeletonization
Function “bwmorph” in Matlab
• Grayscale morphology
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Boundary Extraction
• To find the boundary of a set A, erode
it by a small structuring element B
• Then take the set difference between A
and its erosion
A
A B
A   A B 
 ( A)  A   A B 
• Example
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Department of Electrical Engineering and Computer Science
Hit or Miss Transform
• A method to find the location of a shape B1 in an image A
– Erosion of A Ө B1 gives all places where B1 fits in A
• But B1 fits in any sufficiently large shape
• So also require the boundary around the shape, B2 to be
empty
– Erosion of Ac Ө B2 gives all places where B2 fits in empty places
of A
• Then take intersection
A  B   A B1   Ac  B2 
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Question:
• Template matching using
cross correlation also finds
shapes.
• How is the hit-or-miss
transform different?
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Department of Electrical Engineering and Computer Science
Region Filling
•
•
•
•
Let A be the set of 8connected boundary points of
a region
Start with a point inside the
region
Repeatedly dilate
At each step, set to zero the
points corresponding to the
region boundary
X k   X k 1  B   Ac , k  1,2,3,
•
Stop when no more changes
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Skeletonization
• Is the set of all points that are equally distant from
two closest points of the object boundary
• Equivalently, the union of all maximal disk centers
that are contained in the object
• A concise representation of a shape
• Analogy
– Start a fire at the boundary, let it burn inward
– Points where fire is quenched are the skeleton
Object can be
reconstructed as a
union of disks
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Matlab Example
• Use function bwmorph(BW, 'skel', Inf);
But results are
sensitive to small
perturbations in
the boundary
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Grayscale Morphology
•
•
Instead of binary images, assume we
have gray values f(x,y), where x,y are
integer pixels
Let b(x,y) be the gray level structuring
element
250
b(x,y)
200
150
100
50
•
We will only focus on flat (constant
value) structuring elements
0
0
50
100
150
200
250
Distance along profile
300
350
400
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Grayscale Morphology
• Definition of dilation with flat structuring element
 f  b( x, y )  max

f ( x  s, y  t )
( s ,t )b
• Reflect b(s,t), slide it past f(x,y); at each position, take the maximum
of f(x,y) within the window of b(s,t)
b(s,t)
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Grayscale Morphology
• Definition of dilation with flat structuring element
 f  b( x, y )  max

f ( x  s, y  t )
( s ,t )b
• Reflect b(s,t), slide it past f(x,y); at each position, take the maximum
of f(x,y) within the window of b(s,t)
b(s,t)
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Gray Scale Erosion
• Definition of erosion (with flat SE)
 f  b( x, y)  (min

f ( x  s, y  t )
s ,t )b
• Slide b(s,t) past f(x,y); at each position, take the minimum of f(x,y)
within that window
b(s,t)
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Gray Scale Erosion
• Definition of erosion (with flat SE)
 f  b( x, y)  (min

f ( x  s, y  t )
s ,t )b
• Slide b(s,t) past f(x,y); at each position, take the minimum of f(x,y)
within that window
b(s,t)
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example
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Opening and Closing
•
Similar to binary case
– Opening is erosion followed
by dilation
– Closing is dilation followed by
erosion
•
f b   f θ b  b
f  b   f  b θ b
Geometric interpretation of
opening
– Push the SE up from below
against the underside of f
– Take the highest values
achieved at every point
Opening removes small,
bright details
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Department of Electrical Engineering and Computer Science
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Opening and Closing
•
Geometric interpretation of closing
– Push the SE down from above against the topside of f
– Take the lowest values achieved at every point
Closing removes
small, dark details
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Top Hat Transform
• Open the image with a structuring element,
subtract it from the original
• Leaves only details smaller than the structuring
element
h  f  ( f  b)
• Matlab example
I = imread('rice.png');
% Pick structuring element larger than a rice grain
background = imopen(I,strel('disk',15));
subplot(1,2,1), imshow(I);
subplot(1,2,2), imshow(background);
I2 = I-background;
figure, imshow(I2, []);
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Colorado School of Mines
Department of Electrical Engineering and Computer Science
Another example
• Segment the coins from the image
I = imread('pennies.tif');
I = rgb2gray(I);
imshow(I,[]);
% Pick a disk that will just fit inside the pennies
SE = strel('disk', 6, 0);
I2 = imopen(I,SE);
figure, imshow(I2,[]);
% Threshold to get the pennies
B = Iopen > 150;
figure, imshow(B);
L = bwlabel(B);
props = regionprops(L);
figure(1);
for i=1:length(props)
rectangle('Position', props(i).BoundingBox, 'EdgeColor', 'r');
end
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Department of Electrical Engineering and Computer Science
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Summary / Questions
• Morphological algorithms are a useful set of tools for
extracting features of interest in a binary image,
including
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–
–
–
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Finding connected regions
Filtering out very large or very small regions
Finding the boundary of regions
Filling in the interior of regions
Finding the “skeleton” of a region
• Morphological algorithms can be extended to grayscale
images.
– What does “dilation” and “erosion” mean for grayscale?
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Colorado School of Mines
Department of Electrical Engineering and Computer Science