主講者:陳建齊
Outline & Content
1. Introduction
2. Thresholding
3. Edge-based segmentation
4. Region-based segmentation
5. conclusion
2
1. Introduction
 What is segmentation?
 Three major ways to do.
 Thresholding
 Edge-based segmentation
 Region-based segmentation
3
Thresholding
 Finding histogram of gray level intensity.
 Basic Global Thresholding
 Otsu’s Method
 Multiple Threshold
 Variable Thresholding
4
Edge-based segmentation
 Using mask to detect edge in image by convolution.
 Basic Edge Detection
 The Marr-Hildreth edge detector(LoG)
 Short response Hilbert transform(SRHLT)
 Watersheds
5
Region-based segmentation
 Finding region, but not finding edge.
 Region Growing
 Data Clustering (Hierarchical clustering)
 Partitional clustering
 Cheng-Jin Kuo`s method
6
2. Thresholding
Basic Global Thresholding
1)
2)
Select an initial To
Segment image use:
g ( x, y )   01
if f(x,y) T
if f(x,y)  T
3)
Compute the average intensity m 1
and m 2 for the pixels in and .
4)
Compute a new threshold: T  1 (m 1  m 2 )
5)
Until the difference between values of T is smaller than a
predefined parameter.
2
7
Otsu’s Method
 {0,1,2,…,L-1} , L means gray level intensity
 MN  n 0 n 1 n 2 ...  n L1
M*N is the total number of pixel.
n i denote the number of pixels with intensity i
 we select a threshold T (k )  k , 0  k  L  1 , and use it to
classify C : intensity in the range [0, k ] and C : [k  1, L  1]
 P (k )   p , P (k )   p  1  P (k )
 P 1 m 1  P 2 m 2  m G , P 1 + P 2 =1
2
 G2 =  i  m G  p i

, it is global variance.
i 0
1
k
1
i 0
i
2
L 1
2
i  k 1
i
1
L1
8
  (k )  P 1 P 2 (m1  m 2 ) 
2
B
2
(m G P 1 (k )  m(k )) 2
P 1 (k )(1  P 1 (k ))
it is 2between-class variance
 B (k )

=

, 0   (k *)  1
 G2
it is a measure of separability between class.
2

2

(
k
)

max

 B
B (k )
ok  L 1
 g ( x, y )  10
if f(x,y) k*
if f(x,y) k*
For x = 0,1,2,…,M-1 and y = 0,1,2…,N-1.
 Using image Smoothing/Edge to improve Global
Threshold
Smoothing
Edge
detection
What situation is Large object we Small object we
more suitable for the are interested.
are interested
method
9
Multiple Threshold
 As Otsu’s method, it takes more area and k*
  B2  P 1 (m 1 m G )2  P 2 (m 2 m G )2  P 3 (m 3 m G )2
 P 1 m1  P 2 m 2  P 3 m 3  m G
 P 1 + P 2 +P 3 =1
  B2 (k1* , k2* )  max  B2 (k 1 , k 2 )
o  k 1  k 2  L 1
2
*
*

(
k
,
k
*
*
B 1
2)

(
k
,
k
)

 1 2
 G2
 Disadvantage: it becomes too complicate when
number of area more than three.
10
Variable Thresholding
1) Image partitioning
.
 It is work when the objects of interest and the
background occupy regions of reasonably comparable
size. If not , it will fail.
11
2) Variable thresholding based on local image

properties
Let  xy and m xy denote the standard deviation and
mean value of the set of pixels contained in a
neighborhood, S xy .

Q(local parameters) is true
g ( x, y )  10 ifif Q(local
parameters) is true

Q(
xy , m xy ) 

true if f ( x , y )  a
false otherwise
xy
ANDf ( x , y ) bm xy
12
3) Using moving average


It discussed is based on computing a moving average
along scan lines of an image.
1 k 1
1
m(k  1) 
z

m
(
k
)

( z k 1  z k  n )

i
n i  k  2n
n
z k 1 denote the intensity of the point at step k+1.
n denote the number of point used in the average.
 m(1)  z 1 / n is the initial value.
 T xy  bm xy ,where b is constant and is the moving
average at point (x,y)

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3. Edge-based segmentation
Basic Edge Detection
 Why we can find edge by difference?

image
intensity
first-order deviation
second-order deviation
First-order
deviation
produce
thicker edges
Second-order
deviation
stronger
response to fine
detail
double-edge
response
Determine edge
is from light to
dark or dark to
light
14
1) Gradient
 The image gradient is to find edge strength and direction at
location (x,y) of image.

 f 
 g x   x 
f  grad( f )      
 g y   f 
 y 
 The magnitude (length) of vector , denoted as M(x,y):
mag(f )  g x  g y
 The direction of the gradient vector is given by the angle:
 ( x, y)  tan
1
gy
 
 g x 
15
 Roberts cross-gradient operators:
 Prewitt operator:
 Sobel operator:
G( x, y) 

1
0
if R ( x , y ) T
otherwise
16
The Marr-Hildreth edge
detector(LoG)
 This is second-order deviation, we call Laplacian.
2
 g ( x, y)  [ G( x, y)]  f ( x, y)
g ( x, y)  2 [G( x, y)  f ( x, y)]
 Filter the input image with an n*n Gaussian lowpass
filter. 99.7% of the volume under a 2-D Gaussian
surface lies between about the mean. So n  6 .
17
Short response Hilbert
Transform(SRHLT)
1) Hilbert transform
1
g
(

)

h
(
x
)
*
g
(
x
),
where
h
(
x
)


H
x
 GH ( f )  H ( f )G( f ) , H ( f )   j sgn( f )

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2) SRHLT
 gH    hb  x   g  x  , where hb  x   b csch( bx)

GH ( f )  H b ( f )G ( f ) where GH ( f )  FT [ g H ( )],
G ( f )  FT [ g ( )],
H b ( f )   j tanh( f / b).
HLT
b0

Choose a suitable
value
differentiation
b
Lower b
Higher b
Impulse response
Longer
Shorter
Noise robustness
Good
Bad
Type of edge
Ramp
Step
Output
Thick
Sharp
19
 Simulation result
20
Watersheds
 Algorithm:
 T [n]  {( s, t ) | g ( s, t )  n} , g(s,t) is intensity.
n= min+1 to n = max +1. And let T[n]=0, others 1.
R
 C[n]  C n ( M i ) ,C n (M i ) is minimum point beneath n.
i 1
21
Markers
 External markers:
 Points along the watershed
line along highest points.
 Internal markers:
(1) That is surrounded higher points .
(2) Points in region form a connected component
(3) All points in connected component have the same
intensity.
22
4. Region-based segmentation
Region Growing
 Algorithm:
a. Choose a random pixels
b. Use 8-connected and threshold to determine
c. Repeat a and b until almost points are classified.
23
 Simulation of region growing (90% pixels )
Threshold/second: 20/4.7 seconds.
24
Data Clustering

Using centroid to represent the huge numbers of
clusters

Hierarchical clustering, we can change the number
of cluster anytime during process if we want.

Partitional clustering , we have to decide the number
of clustering we need first before we begin the
process.
25
Hierarchical clustering
1) Algorithm of hierarchical agglomeration(built)：
A. See every single data as a cluster Ci .
B. Find out Ci ,
Cj
for the distance is the shortest.
C. Repeat the steps until satisfies our demand.
•
d (a, b) as the distance between data a and b
26
2) Algorithm of hierarchical division (break up )：

Diameter of cluster
D(Ci )  max(d (a,b)) , for a  Ci , b  Ci
27
1) See the whole database as one cluster.
2) Find out the cluster having the biggest diameter
3) max (d ( x, c)) , for x  C .
4) Split out x as a new cluster C1 , and see the rest data
points as Ci .
5) If d ( y, Ci ) > d ( y, C1 ) , for y  Ci then split y out of Ci and
classify it to C1
6) Back to step2 and continue the algorithm until C1
and Ci is not change anymore.
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Partitional clustering
 Decide the numbers of the cluster(k-means)
 Problem:
 Initial problem
29
 Number of regions are more than clusters you set.
 Determine the number of clusters.

Vadility=
intra
inter
1 k
intra-cluster distance=   x  zi
N i 1 xCi
inter-cluster distance  min( zi  z j
2
2
)
, i  1, 2,3,......, K  1
; j  i  1,......K
30
 Simulation of k-means
Clustering/time: 9 clustering/ 0.1
31
 Advantage and disadvantage of data clustering
Hierarchical
algorithm
Partitional
algorithms
advantage
1.Concept is simple 1.Computing speed
is fast.
2. Result is reliable. 2.Numbers of
cluster is fixed, so
the concept is also
simple.
disadvantage
It is consuming, so
is not suitable for
a large database.
1. Determine the
number of
clusters.
2. Initial problem
3. ….
32
Cheng-Jin Kuo`s method
 Algorithm
33
1) Adaptive threshold decision with local variance

Variance of Lena: 1943
716
899
447
1293 2470
1579 1960 2238
1497 1822 1974 1273
2314 1129 1545 1646

Small variance cause small threshold.
34
2) Adaptive threshold decision with local variance
and frequency

Variance of baboon: 1503
716
899
447
1293 2470
1579 1960 2238
1497 1822 1974 1273
2314 1129 1545 1646
6.9451
8.8807
7.2965 7.0914
8.0413 10.0076 8.4951 7.6421
9.6709 10.3219 7.9815 6.1310
9.0464 10.4821 7.1513 6.4118
35
 High frequency, high variance. Set highest threshold.
(4,1)
 High frequency, low variance. Set second high
threshold.
(4,2)
 Low frequency, high variance. Set third high
threshold.
(1,4)
 Low frequency, low variance. Set the lowest threshold.

(1,1)
36
Comparison of all algorithm by
data compression
Region
growing
Speed
bad
K-means
Watershed Cheng-Jin
Kuo`s
method
Good(worse
bad
than C.J.K’s
method)
fragmentary oversegmen
tation
intact
Shape
connectivi
ty
Good(better Good(equal
Shape
than C.J.K’s C.J.K’s
match
method)
method)
bad
Good
Intact
Good
37
Conclusion
 Speed
 Connectivity
 System reliability
38
Reference
R. C. Gonzalez, R. E. Woods, Digital Image
Processing third edition, Prentice Hall, 2010.
2. C. J. Kuo, J. J. Ding, Anew Compression-Oriented
Fast image Segmentation Technique, NTU,2009.
1.
39
Q&A
40