Professor William Hoff
Dept of Electrical Engineering &Computer Science http://inside.mines.edu/~whoff/

Colorado School of Mines Department of Electrical Engineering and Computer Science
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• “Morphological” = shape, form, structure
• Notes
– Useful for extracting and describing image component regions
– Usually applied to binary (black & white, or 0 & 1) images
– Based on set theory
• Key topics:
– Set theory
– Binary operations: dilation, erosion, opening, closing
– Connected components
– Morphological algorithms
– Gray scale morphology
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• Obtained from
– Thresholding gray level images (e.g., Matlab’s “ im2bw ”)
– Or, as a resulting product of feature detectors
• Often want to count or measure shape of 2D binary image regions
• Typical applications
– Objects on a conveyor belt
– Characters and text, maps
– Chromosomes
– Fingerprints
– Circuit boards
– Overhead aerial images
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• For 2D binary images, A is the (unordered) set of pairs ( x,y ) such that the image value at ( x,y ) is equal to 1
A

  
| I
A x y

The vertical bar means “such that”

• “Union” of two sets A and B
– The set of elements belonging to A, B , or both
A

B
•
“Intersection” of two sets A and B
– The set of elements belonging to both A and B
A

B
• w “is an element of” set A w

A
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• “Complement”
– The set of elements that are not in A
A c
  w | w

A

• “Difference” of two sets A and B
– The set of elements belonging to
A , but not to B
  
|

,

B

Comma means “and”
• “Subset”
– A is a subset of B if every element of A is also in B
A

B
• The empty set { }

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A

  
| I
A x y


B

  
| I
B x y


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ˆ 
 w w
  b , for b

B
 z

 c c
  z , for b

B

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• Morphological operations are defined based on “structuring elements”
• A structuring element is a small set or subimage, used to probe for structure
• Examples
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• The erosion of set A by set (i.e., structuring element) B is
AӨB =
 z |
  z

A

• Interpretation: shift B by z , if it is completely inside A , output a 1
• Example
Origin of B
1 1 1
1
1
1
1 1 1 1 1
B
A
A ӨB
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w w w/2
A ӨB
Radius r r
A ӨB w w w
A ӨB
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• Erosion enlarges holes, breaks thin parts, shrinks object
• Is not commutative
AӨB ≠ BӨA
A
B
A ӨB
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• The dilation of set A by set (structuring element) B is
A
 
 z |
  z
A


• Interpretation: reflect B , shift by z , if it overlaps with A , output a 1 at the center of B
• Example
Origin of B
1 1
1 1
B
1
1 1
1 1
1
B
ˆ
1
A
A

B
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• Example
Origin of B
1
1
1
1 1
A
1
1
1
B
A

B
Colorado School of Mines Department of Electrical Engineering and Computer Science
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w w w/2
A

B
Radius r w w r
A

B w
A

B
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A
B A

B
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• Dilation fills in holes, thickens thin parts, grows object
A
B
A

B
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• Is commutative
A
  
• and associative
B A
A
 
B

C
 
A

B
 
C
• Proof of commutative property
A
 
 z |
  z
A


– If we translate both sets, the intersection area is the same


|
ˆ  
 z


– If we reflect both sets, the intersection area is the same


|
  z

B A
B
A
B
ˆ z
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( A
Ө
B) c = A c 
• Proof
AӨB =
 z |
  z

A


 z |
  z
A c  

(AӨB) c =


 z |
  z
A c  
 c z |
  z
A c  


A c  ˆ
Recall that A c means the complement of set A
If set ( B ) z
is contained in A , then the intersection of ( B ) z
with the complement of A is empty
Complement both sides
By the definition of dilation
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Origin of A
1
1
1
1
A
Origin of B
1 1
1
B
AӨB
(AӨB) c
A c
A c 
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• Make a structuring element
– SE = strel('square', W) or
– SE = strel('disk', R )
• Dilation and erosion
– imdilate(IM, SE)
– imerode(IM, SE)
• Try image ‘blobs.png’
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• Opening – an erosion followed by a dilation
– Use the same structuring element B for both
A  B


A

B


B
• Is the union of all translates of B, that fit into A
A B

     z z

A

The opening of the dark-blue square by a disk, resulting in the light-blue square with round corners.
Colorado School of Mines http://en.wikipedia.
org/wiki/Mathemat ical_morphology
Department of Electrical Engineering and Computer Science
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• Intuitive description
– Let B be a disk
– The boundary of the opening is the points in B that reach the farthest into A as B is rolled around inside of A
Colorado School of Mines Department of Electrical Engineering and Computer Science
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• Closing – a dilation followed by an erosion
– Use the same structuring element B for both
A

B


A

B


B
• Take the union of all the translates of B that do not intersect A; the closing is the complement of that
The closing of the dark-blue shape (union of two squares) by a disk, resulting in the union of the dark-blue shape and the light-blue areas. http://en.wikipedia.
org/wiki/Mathemat ical_morphology
Colorado School of Mines Department of Electrical Engineering and Computer Science
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• Intuitive description
– Let B be a disk
– We roll B around the outside of A
– The boundary of the closing is the points of B that just touch A
Colorado School of Mines Department of Electrical Engineering and Computer Science
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B
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B
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• Are duals of each other

A

B
 c 
• Properties
A c 
A B A
A is a subset of
• Once an image has been opened (or closed), the re-application of the same operation has no effect


A
A


B
B




B
B


A

A  B
B
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Colorado School of Mines Department of Electrical Engineering and Computer Science

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• Make a structuring element
– SE = strel('square', W) or
– SE = strel('disk', R, N)
• Dilation and erosion
– imopen(IM, SE)
– imclose(IM, SE)
• Try image ‘blobs.png’
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4-connected
• Define adjacency
– 4-adjacent
– 8-adjacent
• Two pixels are connected in S if there is a path between them consisting entirely of pixels in S
• S is a (4- or 8-) connected component (“blob”) if there exists a path between every pair of pixels
• “Labeling” is the process of assigning the same label number to every pixel in a connected component
8-connected
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• Hand label simple binary image
1
1
1 1
1 1
1 1
1 1
1 1
Binary image Labeled image
(4-connected)
Labeled image
(8-connected)
Colorado School of Mines Department of Electrical Engineering and Computer Science
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• One pass through image to assign temporary labels and record equivalences
• Second pass to replace temporary labels with equivalence labels
• Let
– B(r,c) is the input binary image
– L(r,c) is the output image of labels
• Side note – faster labeling algorithms do exist. They use 2x2 blocks to search for connected components and use the fact that all the pixels within the block are 8-connected.
• C. Grana, D. Borghesani, and R. Cucchiara. “Fast block based connected components labeling.”
Proc of ICIP, pages 4061-4064, 2009.
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for c=1 to MAXCOL for r=1 to MAXROW
{ if B(r,c) == 0 then else
L(r,c) = 0;
{
{
% if pixel not white, assign no label
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1 1 1
1 1 1
1 1 1
1 1 1
Binary image Temporary labels after 1 st pass
Final (equivalence) labels after 2 nd pass
Colorado School of Mines Department of Electrical Engineering and Computer Science
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• Labeling connected components
(white blobs)
– im2bw
• threshold to convert to binary image
– bwlabel
• do connected component labeling
• generate an image of labels
– label2rgb
• for visualization
• converts each label to a random color
• If we want to find black blobs
– imcomplement
• Flip black and white regions
– then repeat steps
Image Fig9-16(a).jpg
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>> I = imread('Fig9.16(a).jpg');
>> imshow(I,[])
>> whos
Name Size Bytes Class Attributes
I 512x512 262144 uint8
>> BW = im2bw(I, graythresh(I));
>> figure, imshow(BW)
>> [L,num] = bwlabel(BW);
>> figure, imshow(L,[])
>> num num =
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>> RGB = label2rgb(L);
>> figure, imshow(RGB)
>> BW = imcomplement(BW);
>> [L,num] = bwlabel(BW);
>> RGB = label2rgb(L);
>> figure, imshow(RGB)
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• Morphological operations on binary images involve a “structuring element”.
– A structuring element is a small subimage, used to probe for structure.
• Morphological operations are useful for:
– Eliminating small regions or filling in “holes”.
– Finding “connected components”.
• What is a “connected component”?
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