WE4.L09 - POLARIMETRIC SAR ESTIMATION BASED ON NON-LOCAL MEANS

2010 IEEE International Geoscience and Remote Sensing Symposium
Honolulu, Hawa¨ USA, July 2010
ı,

Polarimetric SAR estimation based on non-local means

Charles Deledalle1 , Florence Tupin1 , Lo¨ Denis2
ıc

1 Institut Telecom, Telecom ParisTech, CNRS LTCI, Paris, France
2
Observatoire de Lyon, CNRS CRAL, UCBL, ENS de Lyon, Universit´ de Lyon, Lyon, France
e

July 28, 2010

C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 1 / 18

Motivation

Why non-local methods ?

(a) Noisy image (b) Non-local means

Noise reduction + resolution preservation


Motivation

Why non-local methods ?

(a) Noisy image (b) Non-local means

Noise reduction + resolution preservation

Goal: to adapt non-local methods to PolSAR data,
Method: take into account the statistics of PolSAR data,


Table of contents

1 Non-local means (NL means)

2 Non-local estimation for PolSAR data
Weighted maximum likelihood
Setting of the weights

3 Results of NL-PolSAR
Results on simulations
Results on real data


Table of contents





Non-local means (NL means) (1/2)

Background
Local filters (e.g. boxcar, Gaussian filter...) ⇒ resolution loss,
To combine similar pixels instead of neighboring pixels.
2
1 X − |s−t| 1 X − sim(s,t)
us =
ˆ e ρ2 vt us =
ˆ e h2 vt
Z t Z t
Gaussian filter [Yaroslavsky, 1985]



Background
2
1 X − |s−t| 1 X − sim(s,t)
us =
ˆ e ρ2 vt us =
ˆ e h2 vt
Z t Z t

Non-local means [Buades et al., 2005]
Similarity evaluated using square patches ∆s and ∆t centered on s and t,
Consider the redundant structure of images.



Background
2
1 X − |s−t| 1 X − sim(s,t)
us =
ˆ e ρ2 vt us =
ˆ e h2 vt
Z t Z t

Non-local means [Buades et al., 2005]
Similarity evaluated using square patches ∆s and ∆t centered on s and t,
Consider the redundant structure of images.

Hyp. 1: similar neighborhood ⇒ same central pixel,
Hyp. 2: each patch is redundant (can be found many times).



Background
2
1 X − |s−t| 1 X − sim(s,t)
us =
ˆ e ρ2 vt us =
ˆ e h2 vt
Z t Z t

Algorithm of NL means



Similarity criterion [Buades et al., 2005]
Euclidean distance: X
sim(s, t) = |vs,k − vt,k |2
k

with k the k-th respective pixel of ∆s and ∆t .

Euclidean distance: comparison pixel by pixel



Similarity criterion [Buades et al., 2005]
Euclidean distance: X
sim(s, t) = |vs,k − vt,k |2
k

with k the k-th respective pixel of ∆s and ∆t .

Euclidean distance: comparison pixel by pixel

Hyp. 1: similar neighborhood ⇒ same central pixel,
Hyp. 2: each patch is redundant (can be found many times),
Hyp. 3: the noise is additive, white and Gaussian.

Table of contents





Weighted maximum likelihood for PolSAR data

Context and notations
Searched data: Ts an N × N complex matrix,
Noise model: p(.|Ts ) a circular complex Gaussian,
Noisy observations: ks a N dimensional vector such that ks ∼ p(.|Ts ),
To denoise: ˆ
to search an estimate Ts of Ts .



To denoise: ˆ

From weighted average to weighted maximum likelihood
NL means perform a weighted average of noisy pixel values,
For PolSAR data we suggest to use a weighted maximum likelihood:
P
ˆ
X w (s, t)kt k†
t
Ts = arg max w (s, t) log p(kt |T) = P
T
t
w (s, t)

where w (s, t) is a weight approaching the indicator function of the set of redundant
pixels (i.e with i.i.d values): {s, t|Ts = Tt }.
Choice of w (s, t): oriented, adaptive or non-local neighborhoods.



To denoise: ˆ

Example (Reﬁned Lee - oriented neighborhood [Lee, 1981])
Redundant pixels are located in one of these eight neighborhoods:

image extracted from [Lee, 1981]



To denoise: ˆ

Example (IDAN - adaptive neighborhood [Vasile et al., 2006])
Redundant pixels are located in an adaptive neighborhood (obtained by region
growing algorithm):

image extracted from [Vasile et al., 2006]



To denoise: ˆ

Example (NL means - non-local neighborhood)
Redundant pixels are located anywhere in the image:

image extracted from [Buades et al., 2005]


Setting of the weights for PolSAR data (1/3)

We search weights w (s, t) such that:
w (s, t) is high if Ts = Tt ,
w (s, t) is low if Ts = Tt ,



We search weights w (s, t) such that:
w (s, t) is high if Ts = Tt ,
w (s, t) is low if Ts = Tt ,

Statistical similarity between noisy patches [Deledalle et al., 2009]

Using the hypothesis of NL means:
Patches ∆s and ∆t similar ⇒ central values s and t close.
Weights are deﬁned as follows:

w (s, t) = p(T∆s,k = T∆t,k |ks,k , kt,k )1/h
Y
= p(Ts,k = Tt,k |ks,k , kt,k )1/h
k

with p(Ts,k = Tt,k |ks,k , kt,k ) statistical similarity
h regularization parameter.



Bayesian decomposition
p(T1 = T2 |k1 , k2 ) ∝ p(k1 , k2 |T1 = T2 ) × p(T1 = T2 )
| {z } | {z }
similarity likelihood a priori similarity



p(T1 = T2 |k1 , k2 ) ∝ p(k1 , k2 |T1 = T2 ) × p(T1 = T2 )
| {z } | {z }

Similarity likelihood
R
p(k1 |T1 = t)p(k2 |T2 = t)p(T1 = t)p(T2 = t) dt
p(k1 , k2 |T1 = T2 ) = R
p(T1 = t)p(T2 = t) dt
Since p(T1 = t) is unknown, we deﬁne:

p(k1 , k2 |T1 = T2 ) ˆ ˆ
p(k1 |T1 = TMV )p(k2 |T2 = TMV )



p(T1 = T2 |k1 , k2 ) ∝ p(k1 , k2 |T1 = T2 ) × p(T1 = T2 )
| {z } | {z }

R
p(k1 , k2 |T1 = T2 ) = R
p(T1 = t)p(T2 = t) dt

p(k1 , k2 |T1 = T2 ) ˆ ˆ
p(k1 |T1 = TMV )p(k2 |T2 = TMV )
ˆ
Since only two observations are available to estimate TMV , the matrix is singular.
ˆ
By enforcing TMV to be diagonal, the following similarity criterion is obtained:
„ « „ « „ «
|k1,1 | |k2,1 | |k1,2 | |k2,2 | |k1,3 | |k2,3 |
log + + log + + log + .
|k2,1 | |k1,1 | |k2,2 | |k1,2 | |k2,3 | |k1,3 |



p(T1 = T2 |k1 , k2 ) ∝ p(k1 , k2 |T1 = T2 ) × p(T1 = T2 )
| {z } | {z }

R
p(k1 , k2 |T1 = T2 ) = R
p(T1 = t)p(T2 = t) dt

p(k1 , k2 |T1 = T2 ) ˆ ˆ
p(k1 |T1 = TMV )p(k2 |T2 = TMV )
ˆ
Since only two observations are available to estimate TMV , the matrix is singular.
ˆ
By enforcing TMV to be diagonal, the following similarity criterion is obtained:
„ « „ « „ «
|k1,1 | |k2,1 | |k1,2 | |k2,2 | |k1,3 | |k2,3 |
log + + log + + log + .
|k2,1 | |k1,1 | |k2,2 | |k1,2 | |k2,3 | |k1,3 |
NB: the diagonal assumption is only used to derive a similarity criterion between two
noisy pixels. Full covariance matrices are then estimated via WMLE.



p(T1 = T2 |k1 , k2 ) ∝ p(k1 , k2 |T1 = T2 ) × p(T1 = T2 )
| {z } | {z }

Refined similarity
Refining the weights is necessary when the signal to noise ratio is low.
ˆ
Using a pre-estimate T at pixel 1 and 2 provides two estimations of the noise models:
ˆ
p(.|T1 ) and ˆ
p(.|T2 ).
ˆ ˆ
Assuming p(T1 = T2 ) depends on the proximity of p(.|T1 ) to p(.|T2 ), then we define
ˆ ˆ
SDKL(T1 ||T2 )
p(T1 = T2 ) exp(− )
“ T
” “ ”
ˆ ˆ ˆ −1 ˆ ˆ2 ˆ
with SDKL (T1 , T2 ) ∝ tr T1 T2 + tr T−1 T1 − 6.

The Kullback-Leibler divergence provides a statistical test of the hypothesis T1 = T2
[Polzehl and Spokoiny, 2006]



Iterative scheme
The refined similarity involves an iterative scheme in two steps:
ˆ
1 Estimate the weights from k and Ti −1 :
Y
w (s, t) ← ˆ
p(k1 , k2 |T1 = T2 )1/h p(T1 = T2 |Ti −1 )1/T ,

2 Maximize the weighted likelihood:
P
î w (s, t)kt k†
t
Ts ← P .
w (s, t)
The procedure converges in about ten iterations.

Scheme of the iterative filtering process


Table of contents





Results of NL-PolSAR (Simulations)

(a) Noise-free (b) Noisy (c) Boxcar 7 × 7

(d) Lee (e) IDAN (f) NL-PolSAR

|HH + VV |, |HH − VV |, |HV |
| {z }
RVB

Results of NL-PolSAR (Dresden, -Band E-SAR data c DLR)

(a) Noisy HH (b) NL-PolSAR

|HH + VV |, |HH − VV |, |HV |
| {z }
RVB



(a) Noisy HH (b) Boxcar 7 × 7 (c) Lee

(e) IDAN (f) NL-PolSAR

|HH + VV |, |HH − VV |, |HV |
| {z }
RVB




Entropy H


Conclusion

We proposed an eﬃcient estimator of local covariance matrices for PolSAR data
based on non-local approaches,
The idea is to search iteratively the most suitable pixels to combine,
Similarity criteria:
joint similarity between the noisy observations (HH, HV, VV) of surrounding patches,
and
joint similarity between the pre-estimated covariance matrices of surrounding patches.


Conclusion

and

Results on simulated and L-Band E-SAR data:
Good noise reduction without signiﬁcant loss of resolution,
Preservation of the inter-channel information,
Outperforms the state-of-the-art estimators,
Appealing for classiﬁcation purposes.


Conclusion

and

Results on simulated and L-Band E-SAR data:
Good noise reduction without signiﬁcant loss of resolution,
Preservation of the inter-channel information,
Outperforms the state-of-the-art estimators,
Appealing for classiﬁcation purposes.

More details in:
[Deledalle et al., 2009] Deledalle, C., Denis, L., and Tupin, F. (2009).
Iterative Weighted Maximum Likelihood Denoising with Probabilistic Patch-Based
Weights.
IEEE Transactions on Image Processing, 18(12):2661–2672.

website: http://perso.telecom-paristech.fr/~ deledall


[Buades et al., 2005] Buades, A., Coll, B., and Morel, J. (2005).
A Non-Local Algorithm for Image Denoising.
Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society
Conference on, 2.

[Deledalle et al., 2009] Deledalle, C., Denis, L., and Tupin, F. (2009).
Iterative Weighted Maximum Likelihood Denoising with Probabilistic Patch-Based Weights.
IEEE Transactions on Image Processing, 18(12):2661–2672.

[Lee, 1981] Lee, J. (1981).
Reﬁned ﬁltering of image noise using local statistics.
Computer graphics and image processing, 15(4):380–389.

[Polzehl and Spokoiny, 2006] Polzehl, J. and Spokoiny, V. (2006).
Propagation-separation approach for local likelihood estimation.
Probability Theory and Related Fields, 135(3):335–362.

[Vasile et al., 2006] Vasile, G., Trouv´, E., Lee, J., and Buzuloiu, V. (2006).
e
Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR
parameters estimation.
IEEE Transactions on Geoscience and Remote Sensing, 44(6):1609–1621.

[Yaroslavsky, 1985] Yaroslavsky, L. (1985).
Digital Picture Processing.
Springer-Verlag New York, Inc. Secaucus, NJ, USA.

Results of NL-PolSAR (Simulated data)

(a) Noise-free (b) Noisy (c) Boxcar 7 × 7

(d) Lee (e) IDAN (f) NL-PolSAR

|1 − α/π|, |1 − H|, |HH| + |VV | + |HV |
| {z } | {z } | {z }
H S V

Results of NL-PolSAR (Dresden, L-Band E-SAR data c DLR)

(a) Noisy (b) NL-PolSAR

|1 − α/π|, |1 − H|, |HH| + |VV | + |HV |
| {z } | {z } | {z }
H S V


Results of NL-PolSAR (Dresden, L-Band E-SAR data c DLR)



|1 − α/π|, |1 − H|, |HH| + |VV | + |HV |
| {z } | {z } | {z }
H S V

WE4.L09 - POLARIMETRIC SAR ESTIMATION BASED ON NON-LOCAL MEANS

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