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  • 1. 2011 IEEE International Geoscience and Remote Sensing Symposium A NEW ALGORITHM FOR NOISE REDUCTION ANDQUALITY IMPROVEMENT IN SAR INTERFEROGRAMS USING INPAINTING AND DIFFUSION Silvia Liberata Ullo, Maurizio di Bisceglie, Carmela Galdi1 Universit` degli Studi del Sannio a Benevento, ITALY July 28, 2011 1 This work is supported by Centro Euro-Mediterraneo per i Cambiamenti Climaticiwithin a framework project by Italian Ministry of Environment.
  • 2. Introduction Main objective of SAR interferometry is the generation of high–quality and high–resolution digital elevation maps (DEM’s). Accuracy of DEM’s is strongly related to the quality of the generated interferograms.2 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 3. In this context • We first present an inpainting–diffusion algorithm recently introduced 2 to improve interferograms quality and therefore final quality of DEM’s • Secondly we go through the Complex Ginzburg–Landau (CGL) equation successfully applied to image restoration 3 , and in our work applied to the inpainting scheme for restoration of SAR interferograms • At the end we propose a modified version of starting algorithm and evaluate the efficiency of two algorithms also in the presence of noise 2 A. Borz` M. di Bisceglie, C. Galdi, L. Pallotta, and S. L. Ullo, Phase retrieval in ı, SAR interferograms using diffusion and inpainting, proceedings of IEEE Transactions on Geoscience and Remote Sensing Symposium, Honolulu, Hawaii, July 2010 3 A. Borz` H. Grossauer, and O. Scherzer, Analysis of iterative methods for solving ı, a Ginzburg–Landau equation, International Journal of Computer Vision, vol.64, pp.203–219, September 20053 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 4. Preliminary Considerations • We start from an interferogram, unwrapped via the Minimum Cost Flow (MCF) approach, centered on Ariano Irpino (AV), Italy, produced through a couple of SAR images acquired with ERS–1 and ERS–2 on the 13th and the 14th of July 1995. • A selected area of the interferogram and its corresponding coherence map are shown as follows4 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 5. • Based on the coherence map and fixing an appropriate threshold, a new interferogram is produced by discarding phase values whose correlation coefficients are lower than the threshold. • Dark pixels in the figure represent the discarded phase values.5 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 6. Threshold Setting The choice for the threshold is subject to some considerations: • a correlation coefficient of 0.25 is sufficient to guarantee height errors of 7.5 meters or less (thus approaching a suitable error for topographic mapping) 4 • for coherence values under 0.05 the corresponding unwrapped phases are very noisy • coherence values are regarded as low if they vary between 0.05 and 0.20 Therefore we make the threshold vary from 0.05 to 0.25 in our experiments. 4 H. A. Zebker, C. L. Werner, P.A. Rosen, and S. Hensley, Accuracy of topographic maps derived from ERS–1 interferomtric radar, IEEE Transactions on Geoscience and Remote Sensing, vol.32, no. 4, pp.823–836, July 19946 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 7. Inpainting is a well known method for restoration of missing or damaged portions of images or paintings. Complex Ginzburg–Landau inpainting is a technique that can be conveniently considered to fill fragmentary areas.7 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 8. The Complex Ginzburg–Landau inpainting scheme The basic element of the algorithm is the CGL differential diffusion equation stated in the space–time coordinates in the form ∂u 1 = ∆u + 2 1 − |u|2 u = 0 (1) ∂t ε where • an inpainting domain Ω is defined where there are the phase values to be restored • the function u : D → C is the complex valued solution of the equation such that Ω ⊂ D • ε is a suitably chosen parameter8 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 9. • It has been proved [3] that the solution u of the equation lies on the unit radius sphere • The real part of u, u, includes the interferogram values scaled and normalized, such that u may assume any value in the interval [−1, +1] • The imaginary part (u) is computed as 2 (u) = 1 − (u) (2) such that |u| = 1 • a sequence of images are generated where the information along the borders is smoothly propagated inside the region of missing data • this is achieved by moving mostly along the level line directions to preserve edges • the steady state is achieved when the smoothness of the image is nearly constant along the level lines ( ∂u = 0) ∂t9 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 10. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 11. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 12. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 13. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 14. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 15. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 16. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 17. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 18. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 19. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 20. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 21. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 22. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 23. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 24. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 25. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 26. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 27. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 28. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 29. Application of the algorithm10 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 30. Evaluation of the algorithm To get a measure of performance for the proposed algorithm we used the Signal–to–Noise Ratio (SNR) as expressed by the equation: Nt SN R = 20 log (3) Nr where Nt is the number of total pixels in the interferogram and Nr is the number of residuals, where the residuals are those points around which a close integral of the phase differences gives a non–zero result 5 . 5 U. Wegmuller, C. Werner, T. Strozzi, and A. Wiesmann, Phase unwrapping with GAMMA ISP, Technical Report, Gamma Report Sensing, May 200211 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 31. • SNR=27.3483 dB with the MCF algorithm Table: SNR for the CGL inpainting algorithm. Threshold SNR Number of Number of [dB] iterations residuals [Nr ] 0.05 27.3549 21 10636 0.10 27.3860 47 10598 0.15 27.4287 185 10546 0.20 27.6009 639 10339 0.25 27.9436 3509 9939 Results show that the proposed algorithm works better than the MCF based algorithm.12 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 32. Modified CGL based algorithm • The CGL inpainting equation works by filling the regions where there are missing values with zeros at the first iteration and later on through the reaction diffusion and inpainting procedure explained before. • In a new version of the algorithm we modify the inpainting equation (1) and the low–coherence phase values are not discarded but used as initial conditions. The CGL equation is forced to use these values at the first iteration to drive the inpainting scheme. The idea is that even if these phase values have low–coherence they may contain a part of useful information in any case.13 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 33. Results for the modified and the previous algorithm are shown in the table Table: SNR for the new and the old CGL inpainting algorithm. Threshold SN R1 SN R0 [Nr1 ] [Nr0 ] ∆ [Nr ] [dB] [dB] 0.05 27.3565 27.3549 10634 10636 -2 0.10 27.3863 27.3860 10596 10598 -2 0.15 27.4584 27.4287 10510 10546 -36 0.20 27.6480 27.6009 10283 10339 -56 0.25 28.0199 27.9436 9852 9939 -13 0.30 28.6393 28.6336 9174 9180 -6 Modified algorithm performs even better and reaches its best performance for a threshold of 0.20 This appears to be reasonable. With the modified version of the algorithm we re–used as initial conditions the discarded pixels. Obviously as the threshold increases besides a certain value all this work sounds to be worthless.14 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 34. Inpainting in the presence of noise In another set of experiments the algorithm is tested in the presence of noise. The idea is to introduce noise only in some parts of the interferogram, in a controlled manner. We could add noise to the interferogram in a random way, but we have preferred to use the following method. • first a region of the interferogram, with high–coherence phase values only, is selected • then a mask is created from a different part of the interferogram, where also low–coherence phase values are present: the mask is composed by all these low-coherence pixels and keeps their shape • the high–coherence region is marked using this mask to make a footprint • at the end noise is added only over the footprint: practically through the mask, some high–coherence phase values are identified first and corrupted with noise later on. Since we know exactly the true values before adding the noise, after implementing our algorithm, we can use them for comparison.15 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 35. In figure the red rectangle includes a part of interferogram where high–coherence phase values only are present; the black circles on the right and on the left of red rectangle are regions where also low–coherence phase values are present. These last two regions will be used as masks to establish where, in the good interferogram, noise will be added.16 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 36. The phase noise distribution For the generation of the random additive term necessary to degrade the interferometric phase, an interesting theoretical model has been used 6-7 . √ The phase noise ∆φ is the differential phase of the product S(g1 , g2 ), often referred in the literature as the compound–Gaussian model, where g1 and g2 are complex Gaussian variables, independent of S, with zero mean and assigned covariance matrix K. Interestingly, since S is a real–values random variable, the pdf of ∆φ depends on the covariance matrix of the Gaussian components only. 6 M. di Bisceglie, C. Galdi, and R. Lanari, Statistical characterization of the phase process in interferometric SAR images, proceedings of IEEE Transactions on Geoscience and Remote Sensing Symposium, Lincoln, NE, USA, May 1996 7 D. Just, R. Bambler, Phase Statistics of Interferograms with Applications to Synthetic Aperture Radar, Applied Optics, Vol.33, No.20; July 1994, pp.4361-436817 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 37. Generation of the additive phase noise Two complex vectors c1 = (X1 + jY1 ) and c2 =(X2 + jY2 ) of independent Gaussian variables with zero mean and unitary variance are generated whose length must cover the total number of pixels we want to corrupt with noise. Since each scatterer is observed at two slightly different viewpoints, some degree of correlation is expected between the two vectors. To take into account such a correlation, the vectors c1 and c2 ,are returned into vectors g1 and g2 through the covariance matrix K given by   1 0 ρc ρs K = 0 1 −ρs ρc    ρc −ρs  (4) 1 0 ρs ρc 0 1 where ρc is the correlation coefficient of the components (X1n , X2n ) and (Y1n , Y2n ); ρs is the cross–correlation coefficient of Xin , Yjn , for i, j = 1, 2 ∗ and i = j. We get the noise to be added as the phase of the product g1 g2 .18 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 38. Validation in the presence of noise In sequence (left-right, top-bottom) : 1) high–coherence interferogram represented before in the red rectangle; 2) the same region marked with the mask in the circle on the right; 3) the same region after noise is added; 4) the interferogram restored with the CGL algorithm.19 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 39. Results and some considerations In this case with a threshold of 0.20 the SNR is equal to 59.92 dB, very high, both in the case of MCF and in the case of CGL algorithm. This can be justified because first we start from a high–coherence part of the interferogram, and moreover the mask used to mark the ” good ” region is small. Consequently the quantity of phase values corrupted with noise has been little. The final amount of residuals is actually only 13. If we use also the Mean Square Error (MSE) to appreciate the difference between the two interferograms we get a MSE = 5 × 10−7 if the CGL algorithm is applied without noise and a MSE = 1.8 × 10−6 if noise is added. Results appear to be pretty good and show the algorithm works well in reconstructing the missing values even in the presence of noise.20 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 40. If instead we use as mask the one on the left of the red rectangle SNR = 27.09 dB when noise is added This can be explained because if the noise is added over several parts of the interferogram, we should consider a covariance matrix K not equal for the whole region, but varying. Moreover, the more pixels are marked to add the noise, the more corrupted regions are considered and the greater the number of these regions that take a part into the algorithm, resulting in a lower performance. We remark also that with respect to the modified algorithm that took in consideration the true values, even if with low–coherence coefficients, in this case the algorithm starts from noisy values that are generated in accordance with a theoretical model but are not real values.21 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 41. Conclusions and future work • An inpainting scheme based on Complex Ginzburg–Landau equation has been successfully applied to image restoration of SAR interferograms • A new algorithm is used to restore interferograms by using the low coherence–phase values as initial conditions • Results appear to be very good especially in the medium–coherence area • The algorithm shows to work well in reconstructing the interferogram even in presence of noise if the region is restrained • It is under analysis the possibility to adapt the algorithm even when the noise is added over a region spatially distributed • Future verifications can be done also through comparison of final DEM’s to test algorithm efficiency22 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium
  • 42. THANK YOU FOR YOUR ATTENTION!23 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium