Digital Image Processing
Lecture 12: Image Topology
April 11, 2005
Prof. Charlene Tsai
Before Lecture …
 Mid-term exam
 Date: 4/18
 Time: 10:15~11:30
 Venue: ??? (will announce online)
 Materials: mostly from the in-class quizzes
and homeworks.
 All in-class quizzes (up to lecture 11) are
available online.
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Introduction
 Definition: The study of properties of a figure
that are unaffected by any deformation, as
long as there is no tearing or joining of the
figure.
A region with 3
connected
components
(objects)
A region with
2 holes
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Another example
 How many bacteria here?
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Neighbors and Adjacency
 We often have interests in classifying pixels into
different categories
 Neighbourhoods:
4 4-nghbr
8 8-nghbr
 Two pixels P and Q are 4-adjacent if they are 4neighbours of one another, and 8-adjacent if they are
8-neighbours of one another.
•
4-adjacent
•
P
•
•
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•
•
•
P
•
•
•
•
8-adjacent
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Paths
 Suppose that P and Q are any two pixels (not
necessarily adjacent), and suppose P and Q can be
joined by a sequence of pixels as shown:
4-adjacent
8-adjacent
4-connected
8-connected
 If the path contains only 4-adjacent pixels, then P, Q
are 4-connected.
 If the path contains 8-adjacent pixels, then P and Q
are 8-connected
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Components
 A set of pixels all of which are 4-connected to
each other is called a 4-component; if all the
pixels are 8-connected the set is an 8component.
One 8-component
4-component
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4-component
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Formal Definition of a Path
 A 4-path from P to Q is a sequence of pixels
P = p0, p1, p2, …, pn = Q
such that for each i=0,1,…, n-1, pixel pi is 4adjacent to pixel pi+1.
 An 8-path is where the pixels in the sequence
connecting P and Q are 8-adjacent
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Properties of Connectivity –
Equivalence Relations
 A relation x~y between two objects x and y is
an equivalence relation if the relation is
 Reflexive:
 Symmetric:
 Transitive:
 Examples for equivalence relation?
 Examples that are not equivalence relation?
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Equivalence Relations
 Some examples:
— Numeric equality
— Set cardinality
— Connectedness
 Examples of some relations
which are not equivalence:
— Personal relations
— Subset relation
— Pixel adjacency
 The importance of the equivalence relation concept is
that it allows us a very neat way of dealing with
issues of connectedness.
 Equivalence class: the set of all objects equivalent
to each other.
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Connected Component Labelling
 Define the components of a binary image as
being the equivalence classes of the
connectedness equivalence relation.
 Finding all equivalence classes of connected
pixels in a binary image is called connected
component labelling.
 The result of connected component labeling
is another image in which everything in one
connected region is labelled "1" (for
example), everything in another connected
region is labelled "2", etc.
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Connected Component Labelling (con’d)
 Labelling all the 4-components of a binary
image, starting a the top left and working
across and down.
 “Scan” the image row by row moving across
from left to right.
 Assign labels to pixels in the image; these
labels will be the numbers of the components
of the image.
 A pixel in the image will be called a foreground
pixel (fp); a pixel not in the image will be called
a background pixel (bp).
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Connected Component Labelling:
Algorithm
1. Check the state of p. If it is a bp, move on to the next
scanning position. If it is fp, check the state of u and
l. If they are both bp, assign p a new label.
 If only one of u and l are fp, assign the label to p
 If both u and l are fp, but different labels, assign
either label to p, and make the 2 labels
equivalent.
2. At the end of the scan, all foreground pixels have l
been labelled, but some labels may be equivalent.
Then sort the labels into equivalence classes, and
assign a different label to each class.
3. Do a second pass through the image, replacing the
label on each foreground pixel with the label
assigned to its equivalence class in the previous
step
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p
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Example 1
The first foreground pixel
Upper neighbor : non-existent
Left neighbor : background
First foreground pixel in 2nd row
Upper neighbor : background
Left neighbor : non-existent
Step 1.
2nd foreground pixel in 2nd row
Upper neighbor : foreground
Left
neighbor : foreground
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3rd foreground pixel in 2nd row
Upper neighbor : background
LeftLecture
neighbor
: background
1
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1st foreground pixel in 3rd row:
Upper neighbor :background
Left neighbor : background
1st foreground pixel in 3rd row:
Upper neighbor :foreground ( label 3)
Left neighbor : foreground (label 4)
Step 2. We have the following equivalent classes of labels
{1,2} and {3, 4,5}
Assign label 1 to the first class, and
label 2 to the second class
Step 3. Each pixel with labels 1 or 2 from the first step will be
assigned label 1, and each pixel with labels 3, 4, or 5
from the first step will be assigned label 2
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Example 2
Original Binary image
Pass 1:
Pass 2:
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Example 3
Pass 1:
Original Binary image
Pass 2:
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Matlab bwlabel Function
 Syntax:
L = bwlabel(BW,n)
[L,num] = bwlabel(BW,n)
 Description:
 L = bwlabel(BW,n) returns a matrix L, of the same size as
BW, containing labels for the connected objects in BW. n
can have a value of either 4 or 8, where 4 specifies 4connected objects and 8 specifies 8-connected objects
(defaults to 8).
 The elements of L are integer values greater than or equal to
0. The pixels labeled 0 are the background. The pixels
labeled 1 make up one object, the pixels labeled 2 make up
a second object, and so on.
 [L,num] = bwlabel(BW,n) returns in num the number of
connected objects found in BW.
 See the supplemental notes on using lookup table.
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Distances and Metrics
 A distance function d(x,y) is called a metric if
it satisfies the following:
 1. d x, y   d  y, x  (symmetry)
 2. d x, y   0 and d x, y   0 iff x  y (positivit y)
 3. d x, y   d  y, z   d x, z  (the triangle inquality)
 A standard distance metric is Euclidean
distance (a straight line between 2 points)
d x, y  
where
x1  y1 2  x2  y2 2
x  x1 , x2  and y   y1 , y2 
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Distance Metrics for Grid
d8 x, y 
d 4 x, y 
 The Euclidean distance is
52  22  5.39
 For 4-path and 8-path?
 d 4 x, y   x1  y1  x2  y2 (taxicab metric)
 d8 x, y   max x1  y1 , x2  y2 
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The Distance Transform
 How to find the distance of every pixel from a region R?
 A well-known method is Chamfering
 Algorithm:
 Given: subset S of image
 Initialization: elements of S contain 0, and  for others.
 Step1: Row by row, from top to bottom and left to right:

 
F ( p)  min F ( p), M f ( p, q)  F (q)
qN p

 Step2: Row by row, from bottom to top and right to left:
F ( p)  min F ( p), M b ( p, q)  F (q)
qN  p 
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Masks for Chamfering
 Mask1:
1
1
0
0
1
Forward mask, Mf
1
Backward mask, Mb
 Mask2: 3-4 chamfer (most popular)
4
3
3
0
4
4
Forward mask, Mf
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0
3
3
4
Backward mask, Mb
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Masks for Chamfering (con’d)
 Mask 3:
11
11
11
7
5
5
0
7
11
11
7
11
Forward mask, Mf
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0
5
5
7
11
11
Backward mask, Mb
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Example
 Initialization:
(using 3-4 chamfer)
fw
4
3
4
3
0
3
4
3 bw
4
 Step1: ???







 






  
0 0 
 0 
 0 
 0 
  






F ( p)  min F ( p), M f ( p, q)  F (q)
qN p


F ( p), M b ( p, q)  F (q)
 Step2: ??? F ( p)  qmin
N  p 
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Summary
 We discussed
 Neighbors and Adjacency
 Definition of path
 Equivalence relation
 Component labeling
 Distance metrics
 Euclidean distance
 Metrics for 4- and 8-path
 Chamfering
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