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= + − ∪