2010 IEEE International Geoscience and Remote Sensing Symposium
                                                  Honolulu...
Motivation

   Why non-local methods ?




                                         (a) Noisy image                 (b) No...
Motivation

   Why non-local methods ?




                                         (a) Noisy image                 (b) No...
Table of contents




   1    Non-local means (NL means)


   2    Non-local estimation for PolSAR data
          Weighted...
Table of contents




   1    Non-local means (NL means)


   2    Non-local estimation for PolSAR data
          Weighted...
Non-local means (NL means) (1/2)

   Background
           Local filters (e.g. boxcar, Gaussian filter...) ⇒ resolution loss...
Non-local means (NL means) (1/2)

   Background
           Local filters (e.g. boxcar, Gaussian filter...) ⇒ resolution loss...
Non-local means (NL means) (1/2)

   Background
           Local filters (e.g. boxcar, Gaussian filter...) ⇒ resolution loss...
Non-local means (NL means) (1/2)

   Background
           Local filters (e.g. boxcar, Gaussian filter...) ⇒ resolution loss...
Non-local means (NL means) (2/2)

   Similarity criterion [Buades et al., 2005]
           Euclidean distance:            ...
Non-local means (NL means) (2/2)

   Similarity criterion [Buades et al., 2005]
           Euclidean distance:            ...
Table of contents




   1    Non-local means (NL means)


   2    Non-local estimation for PolSAR data
          Weighted...
Weighted maximum likelihood for PolSAR data

   Context and notations
           Searched data:                           ...
Weighted maximum likelihood for PolSAR data

   Context and notations
           Searched data:                           ...
Weighted maximum likelihood for PolSAR data

   Context and notations
           Searched data:                           ...
Weighted maximum likelihood for PolSAR data

   Context and notations
           Searched data:                           ...
Weighted maximum likelihood for PolSAR data

   Context and notations
           Searched data:                           ...
Setting of the weights for PolSAR data (1/3)


           We search weights w (s, t) such that:
                   w (s, t...
Setting of the weights for PolSAR data (1/3)


           We search weights w (s, t) such that:
                   w (s, t...
Setting of the weights for PolSAR data (2/3)

   Bayesian decomposition
                            p(T1 = T2 |k1 , k2 ) ∝...
Setting of the weights for PolSAR data (2/3)

   Bayesian decomposition
                            p(T1 = T2 |k1 , k2 ) ∝...
Setting of the weights for PolSAR data (2/3)

   Bayesian decomposition
                            p(T1 = T2 |k1 , k2 ) ∝...
Setting of the weights for PolSAR data (2/3)

   Bayesian decomposition
                            p(T1 = T2 |k1 , k2 ) ∝...
Setting of the weights for PolSAR data (2/3)

   Bayesian decomposition
                            p(T1 = T2 |k1 , k2 ) ∝...
Setting of the weights for PolSAR data (3/3)

   Iterative scheme
           The refined similarity involves an iterative s...
Table of contents




   1    Non-local means (NL means)


   2    Non-local estimation for PolSAR data
          Weighted...
Results of NL-PolSAR (Simulations)




                              (a) Noise-free                     (b) Noisy         ...
Results of NL-PolSAR (Dresden, -Band E-SAR data c DLR)




                                  (a) Noisy HH                 ...
Results of NL-PolSAR (Dresden, -Band E-SAR data c DLR)




                              (a) Noisy HH                    (...
Results of NL-PolSAR (Dresden, -Band E-SAR data c DLR)




                              (a) Noisy HH             (b) Boxc...
Conclusion

           We proposed an efficient estimator of local covariance matrices for PolSAR data
           based on n...
Conclusion

           We proposed an efficient estimator of local covariance matrices for PolSAR data
           based on n...
Conclusion

           We proposed an efficient estimator of local covariance matrices for PolSAR data
           based on n...
[Buades et al., 2005] Buades, A., Coll, B., and Morel, J. (2005).
     A Non-Local Algorithm for Image Denoising.
     Com...
Results of NL-PolSAR (Simulated data)




                              (a) Noise-free                     (b) Noisy      ...
Results of NL-PolSAR (Dresden, L-Band E-SAR data c DLR)




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




                              (a) Noisy HH                  (b...
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WE4.L09 - POLARIMETRIC SAR ESTIMATION BASED ON NON-LOCAL MEANS

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

  1. 1. 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
  2. 2. Motivation Why non-local methods ? (a) Noisy image (b) Non-local means Noise reduction + resolution preservation C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 2 / 18
  3. 3. 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, C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 2 / 18
  4. 4. 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 C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 3 / 18
  5. 5. 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 C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 4 / 18
  6. 6. 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] C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 4 / 18
  7. 7. 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] 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. C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 4 / 18
  8. 8. 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] 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). C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 4 / 18
  9. 9. 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] Algorithm of NL means C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 4 / 18
  10. 10. Non-local means (NL means) (2/2) 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 C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 5 / 18
  11. 11. Non-local means (NL means) (2/2) 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. C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 5 / 18
  12. 12. 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 C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 6 / 18
  13. 13. 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 . C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 6 / 18
  14. 14. 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 . 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. C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 6 / 18
  15. 15. 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 . Example (Refined Lee - oriented neighborhood [Lee, 1981]) Redundant pixels are located in one of these eight neighborhoods: image extracted from [Lee, 1981] C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 7 / 18
  16. 16. 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 . 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] C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 7 / 18
  17. 17. 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 . Example (NL means - non-local neighborhood) Redundant pixels are located anywhere in the image: image extracted from [Buades et al., 2005] C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 7 / 18
  18. 18. 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 , C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 8 / 18
  19. 19. 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 , 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. C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 8 / 18
  20. 20. Setting of the weights for PolSAR data (2/3) Bayesian decomposition p(T1 = T2 |k1 , k2 ) ∝ p(k1 , k2 |T1 = T2 ) × p(T1 = T2 ) | {z } | {z } similarity likelihood a priori similarity C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 10 / 18
  21. 21. Setting of the weights for PolSAR data (2/3) Bayesian decomposition p(T1 = T2 |k1 , k2 ) ∝ p(k1 , k2 |T1 = T2 ) × p(T1 = T2 ) | {z } | {z } similarity likelihood a priori similarity 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 ) C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 11 / 18
  22. 22. Setting of the weights for PolSAR data (2/3) Bayesian decomposition p(T1 = T2 |k1 , k2 ) ∝ p(k1 , k2 |T1 = T2 ) × p(T1 = T2 ) | {z } | {z } similarity likelihood a priori similarity 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 ) 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 | C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 12 / 18
  23. 23. Setting of the weights for PolSAR data (2/3) Bayesian decomposition p(T1 = T2 |k1 , k2 ) ∝ p(k1 , k2 |T1 = T2 ) × p(T1 = T2 ) | {z } | {z } similarity likelihood a priori similarity 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 ) 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. C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 13 / 18
  24. 24. Setting of the weights for PolSAR data (2/3) Bayesian decomposition p(T1 = T2 |k1 , k2 ) ∝ p(k1 , k2 |T1 = T2 ) × p(T1 = T2 ) | {z } | {z } similarity likelihood a priori similarity 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] C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 14 / 18
  25. 25. Setting of the weights for PolSAR data (3/3) 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 ˆi w (s, t)kt k† t Ts ← P . w (s, t) The procedure converges in about ten iterations. Scheme of the iterative filtering process C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 15 / 18
  26. 26. 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 C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 16 / 18
  27. 27. 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 C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 16 / 18
  28. 28. Results of NL-PolSAR (Dresden, -Band E-SAR data c DLR) (a) Noisy HH (b) NL-PolSAR |HH + VV |, |HH − VV |, |HV | | {z } RVB C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 17 / 18
  29. 29. Results of NL-PolSAR (Dresden, -Band E-SAR data c DLR) (a) Noisy HH (b) Boxcar 7 × 7 (c) Lee (e) IDAN (f) NL-PolSAR |HH + VV |, |HH − VV |, |HV | | {z } RVB C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 17 / 18
  30. 30. Results of NL-PolSAR (Dresden, -Band E-SAR data c DLR) (a) Noisy HH (b) Boxcar 7 × 7 (c) Lee (e) IDAN (f) NL-PolSAR Entropy H C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 17 / 18
  31. 31. 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. C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 18 / 18
  32. 32. 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. 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. C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 18 / 18
  33. 33. 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. 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 C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 18 / 18
  34. 34. [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. C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 19 / 18
  35. 35. 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 C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 20 / 18
  36. 36. 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 C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 21 / 18
  37. 37. Results of NL-PolSAR (Dresden, L-Band E-SAR data c DLR) (a) Noisy HH (b) Boxcar 7 × 7 (c) Lee (e) IDAN (f) NL-PolSAR |1 − α/π|, |1 − H|, |HH| + |VV | + |HV | | {z } | {z } | {z } H S V C. Deledalle, L. Denis, F. Tupin (Telecom Paristech) NL-PolSAR July 28, 2010 21 / 18

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