Image Segmentation with Fundamentals, Point, Line, and Edge Detection, Thresholding, Segmentation

1
Image Segmentation
Image segmentation is the operation of partitioning an
image into a collection of connected sets of pixels.
1. into regions, which usually cover the image
2. into linear structures, such as
- line segments
- curve segments
3. into 2D shapes, such as
- circles
- ellipses
- ribbons (long, symmetric regions)

4
Example 3:
Lines and Circular Arcs

5
Region Segmentation:
Segmentation Criteria
From Pavlidis
A segmentation is a partition of an image I into
a set of regions S satisfying:
1.  Si = S Partition covers the whole image.
2. Si  Sj = , i  j No regions intersect.
3.  Si, P(Si) = true Homogeneity predicate is
satisfied by each region.
4. P(Si  Sj) = false, Union of adjacent regions
i  j, Si adjacent Sj does not satisfy it.

6
So
So all we have to do is define and implement the
similarity predicate.
But, what do we want to be similar in each region?
Is there any property that will cause the regions to
be meaningful objects?

7
Main Methods of Region
Segmentation
1. Region Growing
2. Clustering
3. Split and Merge

8
Region Growing
Region growing techniques start with one pixel of a
potential region and try to grow it by adding adjacent
pixels till the pixels being compared are too disimilar.
• The first pixel selected can be just the first unlabeled
pixel in the image or a set of seed pixels can be chosen
from the image.
• Usually a statistical test is used to decide which pixels
can be added to a region.

9
The RGGROW Algorithm
• Let R be the N pixel region so far and P be a neighboring
pixel with gray tone y.
• Define the mean X and scatter S (sample variance) by
X = 1/N  I(r,c)
S = 1/N  (I(r,c) - X)
2
2
(r,c)  R
(r,c)  R
2

10
The RGGROW Statistical Test
The T statistic is defined by
(N-1) * N
T = -------------- (y - X) / S
(N+1)
2
2
1/2
It has a T distribution if all the pixels in R and the
test pixel y are independent and identically distributed
normals (IID assumption) .
N-1

11
Decision and Update
• For the T distribution, statistical tables give us the
probability Pr(T  t) for a given degrees of freedom
and a confidence level. From this, pick suitable
threshold t.
• If the computed T  t for desired confidence level,
add y to region R and update X and S .
• If T is too high, the value y is not likely to have arisen
from the population of pixels in R. Start a new region.
2

12
RGGROW Example
image
segmentation
Not great!
What do you think this would
do on wallpaper texture?

13
Clustering
• There are K clusters C1,…, CK with means m1,…, mK.
• The least-squares error is defined as
• Out of all possible partitions into K clusters,
choose the one that minimizes D.
Why don’t we just do this?
If we could, would we get meaningful objects?
D =   || xi - mk || .
k=1 xi  Ck
K
2

14
Some Clustering Methods
• K-means Clustering and Variants
• Isodata Clustering
• Histogram-Based Clustering and Recursive Variant
• Graph-Theoretic Clustering

17
Meng-Hee Heng’s K-means Variant
1. Pick 2 points Y and Z that are furthest apart in the
measurement space and make them initial cluster means.
2. Assign all points to the cluster whose mean they are
closest to and recompute means.
3. Let d be the max distance from each point to its cluster mean
and let X be the point with this distance.
4. Let q be the average distance between each pair of means.
5. If d > q / 2, make X a new cluster mean.
6. If a new cluster was formed, repeat from step 2.

18
Illustration of Heng Clustering
Y
D>q/2
q
X
1 2 3
We used this for segmentation of textured scenes.
Z

19
Heng Clustering with Texture
Feature

20
Isodata Clustering
1. Select several cluster means and form clusters.
2. Split any cluster whose variance is too large.
3. Group together clusters that are too small.
4. Recompute means.
5. Repeat till 2 and 3 cannot be applied.
We used this to cluster normal vectors in 3D data.

21
Comparison
Original
K-means, K=6
Isodata, K became 5

22
Ohlander’s Recursive Histogram-
Based Clustering
• color images of real indoor and outdoor scenes
• starts with the whole image
• selects the R, G, or B histogram with largest peak
and finds clusters from that histogram
• converts to regions on the image and creates masks for each
• pushes each mask onto a stack for further clustering

23
Ohlander’s Method
sky
tree2
tree1
Ohta suggested using
(R+G+B)/3, (R-B)/2
and (2G-R-B)/4
instead
of (R, G, B).
separate
R, G, B

24
Jianbo Shi’s Graph-Partitioning
• An image is represented by a graph whose nodes
are pixels or small groups of pixels.
• The goal is to partition the vertices into disjoint sets so
that the similarity within each set is high and
across different sets is low.

25
Minimal Cuts
• Let G = (V,E) be a graph. Each edge (u,v) has a weight w(u,v)
that represents the similarity between u and v.
• Graph G can be broken into 2 disjoint graphs with node sets
A and B by removing edges that connect these sets.
• Let cut(A,B) =  w(u,v).
• One way to segment G is to find the minimal cut.
uA, vB

26
Cut(A,B)
w1
w2
A
B
cut(A,B) =  w(u,v).
uA, vB

27
Normalized Cut
Minimal cut favors cutting off small node groups,
so Shi proposed the normalized cut.
cut(A, B) cut(A,B)
Ncut(A,B) = ------------- + -------------
asso(A,V) asso(B,V)
asso(A,V) =  w(u,t)
uA, tV
How much is A connected
to the graph as a whole.
normalized
cut

28
Example Normalized Cut
2
2 2
2 2
4
1 3
2
2 2
3
2
2
2
1
3 3
Ncut(A,B) = ------- + ------
21 16
A B

30
How Shi used the procedure
Shi defined the edge weights w(i,j) by
w(i,j) = e *
e if ||X(i)-X(j)||2 < r
0 otherwise
||F(i)-F(j)||2 / I
||X(i)-X(j)||2 / X
where X(i) is the spatial location of node i
F(i) is the feature vector for node I
which can be intensity, color, texture, motion…
The formula is set up so that w(i,j) is 0 for nodes that
are too far apart.

31
Examples of
Shi Clustering
See Shi’s Web Page
http://www-2.cs.cmu.edu/~jshi

32
Lines and Arcs
Segmentation
In some image sets, lines, curves, and circular arcs
are more useful than regions or helpful in addition
to regions.
Lines and arcs are often used in
• object recognition
• stereo matching
• document analysis

33
Edge Detection
Basic idea: look for a neighborhood with strong signs
of change.
81 82 26 24
82 33 25 25
81 82 26 24
Problems:
• neighborhood size
• how to detect change

34
Differential Operators
Differential operators
• attempt to approximate the gradient at a pixel via masks
• threshold the gradient to select the edge pixels

35
Example: Sobel Operator
-1 0 1 1 2 1
Sx = -2 0 2 Sy = 0 0 0
-1 0 1 -1 -2 -1
On a pixel of the image
• let gx be the response to Sx
• let gy be the response to Sy
Then g = (gx + gy ) is the gradient magnitude.
 = atan2(gy,gx) is the gradient direction.
2 2 1/2

36
Java Toolkit’s Sobel Operator
original image gradient thresholded
magnitude gradient
magnitude

37
Zero Crossing Operators
Motivation: The zero crossings of the second derivative
of the image function are more precise than
the peaks of the first derivative.
step edge
smoothed
1st derivative
2nd derivative
zero crossing

38
Marr/Hildreth Operator
• First smooth the image via a Gaussian convolution
• Apply a Laplacian filter (estimate 2nd derivative)
• Find zero crossings of the Laplacian of the Gaussian
This can be done at multiple resolutions.

39
Haralick Operator
• Fit the gray-tone intensity surface to a piecewise
cubic polynomail approximation.
• Use the approximation to find zero crossings of the
second directional derivative in the direction that
maximizes the first directional derivative.
The derivatives here are calculated from direct
mathematical expressions wrt the cubic polynomial.

40
Canny Edge Detector
• Smooth the image with a Gaussian filter.
• Compute gradient magnitude and direction at each pixel of
the smoothed image.
• Zero out any pixel response  the two neighboring pixels
on either side of it, along the direction of the gradient.
• Track high-magnitude contours.
• Keep only pixels along these contours, so weak little
segments go away.

42
Best Canny on Kidney from Hw1

43
Best Canny on Blocks from Hw1

44
Finding Line and Curve Segments
from Edge Images
Given an edge image, how do we find line and arc segments?
Method 1: Tracking
Use masks to identify the following events:
1. start of a new segment
2. interior point continuing a segment
3. end of a segment
4. junction between multiple segments
5. corner that breaks a segment into two
junction
corner

45
Edge Tracking Procedure
for each edge pixel P {
classify its pixel type using masks
case
1. isolated point : ignore it
2. start point : make a new segment
3. interior point : add to current segment
4. end point : add to current segment and finish it
5. junction or corner : add to incoming segment
finish incoming segment
make new outgoing segment(s)
The ORT package uses a fancier corner finding approach.

46
Hough Transform
• The Hough transform is a method for detecting
lines or curves specified by a parametric function.
• If the parameters are p1, p2, … pn, then the Hough
procedure uses an n-dimensional accumulator array
in which it accumulates votes for the correct parameters
of the lines or curves found on the image.
y = mx + b
image
m
b
accumulator

47
Finding Straight Line Segments
• y = mx + b is not suitable (why?)
• The equation generally used is: d = r sin  + c cos 
d

r
c
d is the distance from the line to origin
 is the angle the perpendicular makes
with the column axis

48
Procedure to Accumulate Lines
• Set accumulator array A to all zero.
Set point list array PTLIST to all NIL.
• For each pixel (R,C) in the image {
• compute gradient magnitude GMAG
• if GMAG > gradient_threshold {
• compute quantized tangent angle THETAQ
• compute quantized distance to origin DQ
• increment A(DQ,THETAQ)
• update PTLIST(DQ,THETAQ) } }

49
Example
0 0 0 100 100
0 0 0 100 100
0 0 0 100 100
100 100 100 100 100
100 100 100 100 100
- - 0 0 -
- - 0 0 -
90 90 40 20 -
90 90 90 40 -
- - - - -
- - 3 3 -
- - 3 3 -
3 3 3 3 -
3 3 3 3 -
- - - - -
360
.
6
3
0
- - - - - - -
- - - - - - -
- - - - - - -
4 - 1 - 2 - 5
- - - - - - -
0 10 20 30 40 …90
360
.
6
3
0
- - - - - - -
- - - - - - -
- - - - - - -
* - * - * - *
- - - - - - -
(1,3)(1,4)(2,3)(2,4)
(3,1)
(3,2)
(4,1)
(4,2)
(4,3)
gray-tone image DQ THETAQ
Accumulator A PTLIST
distance
angle

50
How do you extract the line
segments from the accumulators?
pick the bin of A with highest value V
while V > value_threshold {
order the corresponding pointlist from PTLIST
merge in high gradient neighbors within 10 degrees
create line segment from final point list
zero out that bin of A
pick the bin of A with highest value V }

51
Finding Circles
Equations:
r = r0 + d sin 
c = c0 + d cos 
r, c, d are parameters
Main idea: The gradient vector at an edge pixel points
to the center of the circle.
*(r,c)
d

52
Why it works
Filled Circle:
Outer points of circle have gradient
direction pointing to center.
Circular Ring:
Outer points gradient towards center.
Inner points gradient away from center.
The points in the away direction don’t
accumulate in one bin!

53
Procedure to Accumulate Circles
• Set accumulator array A to all zero.
Set point list array PTLIST to all NIL.
• For each pixel (R,C) in the image {
For each possible value of D {
- compute gradient magnitude GMAG
- if GMAG > gradient_threshold {
. Compute THETA(R,C,D)
. R0 := R - D*cos(THETA)
. C0 := C - D* sin(THETA)
. increment A(R0,C0,D)
. update PTLIST(R0,C0,D) }}

54
The Burns Line Finder
1. Compute gradient magnitude and direction at each pixel.
2. For high gradient magnitude points, assign direction labels
to two symbolic images for two different quantizations.
3. Find connected components of each symbolic image.
1
2
3
4
5
6 7
8
1
2
3
4
5
6 7 8
• Each pixel belongs to 2 components, one for each symbolic image.
• Each pixel votes for its longer component.
• Each component receives a count of pixels who voted for it.
• The components that receive majority support are selected.
-22.5
+22.5
0
45

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