we4.l09 - mean-shift and hierarchical clustering for textured polarimetric sar image...
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MEAN-SHIFT AND HIERARCHICAL CLUSTERING FOR TEXTURED POLARIMETRIC SAR
IMAGE SEGMENTATION/CLASSIFICATION
Jean-Marie BeaulieuComputer Science Department
Laval University
Ridha TouziCanada Centre for Remote Sensing
Natural Resources Canada
•Clustering - attributes - segmentation
•The segment clustering approach
•Mean-shift clustering
•Distance measures for PolSAR images
•Results with the K distribution
Exploration in Segmentation - Clustering
Utilization of texture information
• Clustering is the partition of data points into groups or clusters (unsupervised classification)
• Iterative and hierarchical techniques
• Iterative clustering
• Move group centers (K-means algorithm)
• The number of groups is fix
• Hierarhical clustering • Sequential merging of clusters
• Merge the best pair
• Represented by a tree
• Attributes or feature space (many dimensions)• Radiometric information (or color/spectral)
|hh|
|vv|
|hv|
|hv|
hhx hv
vv
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
Radar 1-look
3 2D plotshh-vvhv-vvhh-hv
Radar multi-look* * *
* * *
* * *
hh hh hh hv hh vv
Z hv hh hv hv hv vv
vv hh vv hv vv vv
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
• Spatial information - position in the image
• Clustering -- distance between points D(Gi,Gj)
• Segmentation -- only adjacent regions
Si
Sj
• Exploring the space between clustering --- and --- segmentation
spatial information
Subpart of image Whole image
• Exploring the space between clustering --- and --- segmentation
spatial information
• Hierarchical segmentation of the image• Clustering of regions-segments
region groups or aggregates
• Use only large regions-segments
• Mean-shift clustering (iterative)
• Followed by hierarchical clustering
• Assign a small segment to the most similar group
• Combining hierarchical / iterative segmentation / clustering
• Different ways to explore the partition space
• Hierarchical segmentation - spatial information
• Iterative Mean-Shift clustering - spatial information
• Hierarchical clustering
• Mean-Shift clustering move every data points toward higher probability density zones (modes)
• Density point count over a window (histogram)• Direction toward higher density
position of weighted mean (window)
Dspectral = D(Gi,Gj) / Fspectral
Dspatial = Distance between centers / Fspatial
Weight = EXP [ - (Dspectral2 +Dspatial2) ]
Mean = weighted point mean
Fhift = α value + (1-α) Mean
MEAN-SHIFT
• Distance measure D(Gi,Gj) for PolSar images
• Maximum Log Likelihood criterion (MLL)
{ } ( )
( )
, ( | )
( ) ln ( | ) ( )
( , ) ( ) ( ) ( )k i
i i i i k G k
k G k iZ I G P
i j i j i j
P G p Z
MLL P p Z MLL G
D G G MLL G MLL G MLL G G∈ ∈
= → θ = Σ α → θ
= θ =
= + − ∪
∑ ∑
• Non textured PolSAR image• Zk follows a complex Wishart distribution
( ){ }33 1
3
exp( | )
( ) ( 1) ( 2)
LLk k
k L
L Z L tr Zp Z
L L L
− −− ΣΣ =
π Γ Γ − Γ − Σ
$ $ $( , ) ( ) ln ln lnGi Gj Gi Gji j i j i jD G G n n n n∪= + Σ − Σ − Σ
• Textured PolSAR image (Zk = μk Zk-homogeneous)• Zk follows a complex K distribution
( )( )
( ){ }
( 3 ) / 23(3 ) / 2 1
3
13
( ) 2( | , )
( ) ( 1) ( 2) ( )
2
LLLk k
k L
L k
L Z tr Zp Z
L L L
K L tr Z
α−−+α −
−−α
α Σα Σ =
π Γ Γ − Γ − Γ α Σ
α Σ
$
$( )$( )
32
132
13
( ) ln( ) ln( ( )) ln( )
ln
2
L
Lk
k G
L kk G
MLL G n L n nL
tr Z
K L tr Z
+α
−α−
∈
−−α
∈
α − Γ α − Σ
⎛ ⎞+ Σ⎜ ⎟⎝ ⎠
⎧ ⎫+ α Σ⎨ ⎬
⎩ ⎭
∑
∑
10k segments
10k segments
200 groups
50 groups
20 groups
• Group center positions
Initial 14804 large regions
20 groups200 groups
5000 groups
200 groups
2 rounds, 200 groups
original 2 rounds, 200 groups
original 2 rounds, 200 groups
2 rounds, 200 groups, class # 13
2 rounds, 200 groups, class # 12
2 rounds, 200 groups, class # 174
original 200 groups
50 groups2 rounds, 200 groups
original 200 groups
50 groups2 rounds, 200 groups
original 200 groups
50 groups2 rounds, 200 groups
original 200 groups
50 groups2 rounds, 200 groups
Wishart, 200 groups
K dist., 200 groups
CONCLUSION
• Combination of segmentation and clustering
• Combination of iterative (Mean-Shift) and hierarchical techniques
• K distribution for segmentation and clustering
For L-look image, a pixel k should be represented by its L-look covariance matrix, Zk
Zk follows a complex Wishart distribution
MULTILOOK IMAGE
( ){ }33 1
3
exp( | )
( ) ( 1) ( 2)
LLk k
k L
L Z L tr Zp Z
L L L
− −− ΣΣ =
π Γ Γ − Γ − Σ
SEGMENTATION BY HYPOTHESIS TESTING
Test the similarity of segment covariances Ci = Cj = C- merge segment with same covariance
Use the difference of determinant logarithms as a test statistic
{ }, ( ) ln ln lni j si sj si sj si si sj sjC K n n C n C n C∪= + − −
With the scaling factor K, the statistic is approximately distributed as a chi-squared variable as nsi and nsj become large.
False Alarm Rate (FAR) thresholding
Segmentation compare two segments
Classification compare one pixel with one class
Local decision Global segmentation result
Sequence of tests
Distribution of Ci,j FAR threshold
Design decision processes with constant FAR
S1
S2
S6 S5
S4
S31) need a partition of the image
{ } { },k kP s s i I= = ⊂
2) need statistical parameters
{ },s s P= θ ∈θ
3) need an image probability model
( | )i sp x θxi are conditionally independent
SEGMENTATION AS MAXIMUM LIKELIHOOD APPROXIMATION
S1
S2
S6 S5
S4
S3
Given an image
The segmentation problem is to find the partition that maximizes the likelihood.
Global search – too many possible partitions.
is derived from statistics calculated over a segment s.
the likelihood of
{ },ix i IX = ∈
{ },s P= θθ
is ( , | ) ( | , )L P p PX X=θ θ
( )( , | ) ( | )i s ii I P
L P p xX∈
= θθ ∏
sθ
The maximum likelihood increases with the number of segments
k number of segments
( | , )p PX θ
Can't find the optimum partition with k segments, PkToo many, except for P1 and Pnxn.
Hierarchical segmentation get Pk from Pk+1 by merging 2 segments.
HIERARCHICAL SEGMENTATION
A hierarchical segmentation begins with an initial partition P0 (with N segments) and then sequentially merges these segments.
Segment tree
level n+1
level n
level n-1
Merging criterion: merge the 2 segments producing the smallest decrease of the maximum likelihood(stepwise optimization)
number of segments
( | , )p PX θ
Sub-optimum within hierarchical merging framework.
k
Criterion cost of merging 2 segments
Log likelihood form
( ) ( )( ) ( )ln ( , | ) ln ( | ) ln ( | )i s i i s ii I i I
L P p x p xX∈ ∈
⎛ ⎞= θ = θ⎜ ⎟
⎝ ⎠θ ∏ ∑
Summation inside region
minimize Δ
( ) ( ) ( )( ) ( ) ( )
ln ( | ) ln ( | ) ln ( | )i j i j
i j i j
i j i j
S S S Sx S x S x S S
MLL S MLL S MLL S S
p x p x p x ∪∈ ∈ ∈ ∪
Δ = + − ∪
Δ = θ + θ − θ∑ ∑ ∑
( )( ) ln ( | ) ( )S P
Si SS P
ip xLLF P MLL S∈ ∈ ∈
θ= =∑∑ ∑
This is equivalent to the hypothesis testing criterion.
Hierarchical segmentation by stepwise optimisation.
, ( ) ln ln lni j si sj si sj si si sj sjC n n C n C n C∪= + − −
HOMOGENEOUS IMAGEThe stepwise criterion is
Assume that a texture value μ modifies the covariance matrix Zk = μk Zk-homogeneous
Zk follows a K distribution
TEXTURED IMAGE
( )( )
( ){ }
( 3 ) / 23(3 ) / 2 1
3
13
( ) 2( | , )
( ) ( 1) ( 2) ( )
2
LLLk k
k L
L k
L Z tr Zp Z
L L L
K L tr Z
α−−+α −
−−α
α Σα Σ =
π Γ Γ − Γ − Γ α Σ
α Σ
The maximum log likelihood for one segment is
Best α and Σ Iteration (gradient descent)
Approximation Σ = segment covariance matrix α = 1/(CVR)2 Method of Moments
( )( )( ){ }
32
132
13
( ) ln( ) ln( ( )) ln( )
ln
2
L
Lk
k S
L kk S
MLL S n L n nL
tr Z
K L tr Z
+α
−α−
∈
−−α
∈
α − Γ α − Σ
+ Σ
+ α Σ
∑
∑
, ( ) ( ) ( )i j i j i jC MLL S MLL S MLL S S= + − ∪