1. A K-Wishart Markov Random Field
Model for Clustering of Polarimetric
SAR Imagery
Vahid Akbari ∗
Gabriele Moser ∗∗
Anthony Paul Doulgeris ∗
Stian Normann Anfinsen ∗
Torbjørn Eltoft ∗
Sebastian Serpico ∗∗
∗
DEPARTMENT OF PHYSICS AND TECHNOLOGY, UNIVERSITY OF TROMSØ,
NO-9037, TROMSØ, NORWAY
∗∗
DEPARTMENT OF BIOPHYSICAL AND ELECTRONIC ENG., UNIVERSITY OF
GENOA, IT-16145, GENOA, ITALY
IGARSS2011, 26th July 2011

2. A KW MRF model for clustering of
PolSAR Imagery
• A clustering method that combines an advanced statistical
distribution with spatial contextual information for multilook PolSAR data
• Markov Random Field (MRF) for integrating a K-Wishart distribution and
a Potts model for the spatial context
• Expectation Maximization (EM) algorithm to address parameter
estimation in the K-Wishart distribution and the spatial contextual model
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3. Presentation Outline
• Motivation
• Polarimetric SAR Imagery
• Multilook Product Model
• K-Wishart distribution
• The K-Wishart classifier
• Markovian fusion approach
• The K-Wishart MRF classifier
• Experimental results
• Conclusions
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4. Motivation
A K-Wishart Markov Random Field Model for clustering of
Polarimetric Synthetic Aperture Radar (PolSAR) Imagery
Non-Gaussian models for pixel-wise statistical analysis
non-Gaussian statistics gives better representation of the SAR images
Contextual PolSAR image clustering
Spatial contextual information improves the segmentation results
Combination of non-Gaussian statistics and contextual information
Combining non-Gaussian models for potential textural differences in the
classes and MRF for accounting the contextual information in the PolSAR
data together yields homogeneous classification results
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5. Polarimetric SAR Imagery
• Polarisation transformation equation in horizontal-vertical
(HV) polarisation basis
EH e−2πjr/λ SHH SHV EH
= (1)
EV r SV H SV V EV
r t
• Complex scattering coefficients as backscattering of a
monostatic Polarimetric SAR system (SLC data format)
 
S
√ HH
k =  2SHV 
SV V
• Multilook complex covariance (MLC) matrix data
L
1
C = k kH (2)
L
i=1
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6. Multilook Product Model
• The product model for multilook PolSAR data
C
C = T W; W ∼ Wd (L, Σ). (3)
- W represents speckle and Wd (L, Σ) denotes the scaled complex
C
Wishart distribution with L degrees of freedom and scale matrix Σ.
- T models texture and determines the non-Gaussian nature of
product model.
Figure 1: Non-Gaussianity of the product model is determined from texture
term with different shape parameter values
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7. K-Wishart distribution
Assuming the gamma distribution for the texture term of the
product model
tα−1
α
pT (t; α) = α exp (−αt) , (4)
Γ(α)
The marginal distribution of C may be obtained by
∞
C
pC (C) = pC|T (C|t)pT (t)dt; C|T ∼ Wd (L, Σ). (5)
0
The resulting distribution is the K-Wishart for the MLC data
PC(C) = KW (C; L, α, Σ) =
2|C|L−d α+Ld α−Ld
(Lα) 2 tr(Σ−1C) 2
Kα−Ld 2 Lαtr(Σ−1C) , (6)
I(L, d)Γ(α)
where the PDF is parametrized by the shape (non-Gaussianity)
parameter α, the number of looks L and the scale matrix Σ.
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8. Parameter Estimation
• Matrix log-cumulants equations to estimate the parameters
of K-Wishart model with least squares method
κ1{C} = ln |Σ| + ψd(L) + d ψ 0(α) − ln(αL)
0
ν−1
κν {C} = ψd (L) + dν ψ ν−1(α), ν>1 (7)
Matrix log-cumulants are related to the log-moments
κ1{C} = µ1
κ2{C} = µ2 − µ2
1
κ3{C} = µ3 − 3µ1µ2 + 2µ3
1
κ4{C} = µ4 − 4µ1µ3 − 3µ2 + 12µ2µ2 − 6µ4
2 1 1 (8)
where sample matrix log-moments with different orders:
n
1
µν = (log |Ci|)ν (9)
n i=1
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9. The K-Wishart Classifier
• Assume a Mixture of K-Wishart PDFs to multi-look covariance
matrix data
K
PC(C) = πk Pk (C), (10)
k=1
where K is the number of classes, πk are the class priors
• Unsupervised classification using EM-algorithm in three
steps:
E-step: Estimate class likelihoods using the K-Wishart distribution
M-step: Update all class parameters {αk , Σk } via log-cumulant method
G-step: Goodness-of-fit test to automatically determine the appropriate
number of classes by split and merge options
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10. Markovian Fusion Approach
Aims:
• Including contextual information disregarded by pixel-wise EM algorithm
• Gaining robustness against speckle
How to do?
• Integrating in Bayesian theory by formulating the maximum-a-posteriori
(MAP) decision rule as the minimization of suitable energy functions
• Modeling the prior distribution of the class labels by MRF
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11. Markov Random Field
• Let S = {si,j ; 1 ≤ i ≤ M, 1 ≤ j ≤ N } be 2-D pixel lattice and
k = {1, 2, ..., K} be the set of all possible labels in the clustering
map
• A label random field X = {Xs, Xs ∈ k, s ∈ S}, treated as an MRF
with a given neighborhood system
• Isotropic second-order neighborhood system (eight surrounding
pixels for each site)
Figure 2: Second-order neighborhood system and pairwise cliques.
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12. • According to the Hammersley-Clifford theorem, the joint probability
distribution of a Markov field is a Gibbs distribution.
PG(X) = Z −1 exp [−U (X)] = Z −1 exp[− Vc(Xc)] (11)
c
where c indicates a clique of a neighborhood system.
• With the assumption of isotropy , there is a single MRF parameter and
potential function is simplified to
−β if Xs = Xr
Vc(Xc) = Vc(Xs, Xr ) = (12)
0 otherwise
where Xr is a neighborhood of central pixel Xs in the clustering map.
• An approximation of the likelihood (11) is the pseudo-likelihood (PL)
P L(X|β) = PG(Xs|Xr ; β) (13)
s∈S
where:
exp − c s Vc (Xc )
PG(Xs|Xr ; β) = (14)
Xs ∈k exp − c s Vc (Xc )
• Simulated annealing algorithm to estimate MRF parameter β
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13. The K-Wishart MRF classifier
• Non-contextual stage: initial non-contextual clustering map
by using K-Wishart classifier. The Bayesian rule for each
class is
PX|C(X|C; θ) ∝ PC|X(C|X; θ)P (X), (15)
• MRF stage: modeling prior probabilities of each class label
by MRF in terms of the Bayesian rule in the mode-field EM
algorithm (E and M step with fixed number of classes found
from the first stage)
PX|C(X|C; θ, β) ∝ PC|X(C|X; θ)PG(X|β), (16)
the complete likelihood is given by
P (C, X|θ, β) ≈ KW(Cs|Xs; θ)PG(Xs|Xr ; β) (17)
s∈S
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14. Simulated Test Pattern Results
• A test image generated with 8-look, dual-pol K-Wishart
distributed data. The KW parameters taken from a real data.
Figure 3: Quasi-Pauli RGB image (left) and Non-contextual (center) vs.
contextual K-Wishart (right): 91% to 98% overall classification accuracies
Table 1: Classification accuracies of classes for simulated data.
Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Overall
KW 99.93% 100% 84.34% 89.78% 99.98% 76.02% 93.00% 80.71% 91%
KW MRF 100% 100% 99.71% 99.85% 100% 98.79% 100% 99.28% 98%
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15. Real Data Example: Foulum, DK
• Real PolSAR data: Farmland area near Foulum, Denmark.
Airborne, fully polarimetric, L-band EMISAR data from April 1998.
Compare K-Wishart classifier to K-Wishart MRF classifier
Figure 4: Pauli RGB image (top) and Non-contextual and contextual K-
Wishart clustering of Foulum dataset, 18 class found.
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16. Conclusions
• We presented a novel unsupervised clustering algorithm
for PolSAR imagery by combining the MRF approach to
Bayesian image classification and a finite mixture model
technique for PDF estimation.
• We showed improvement of results w.r.t. segmentation of
pixel-wise K-Wishart clustering.
- There is a visible and quantitative improvement in
terms of spatial regularity, accuracy, and smoothing of
homogeneous areas.
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