1. Air Systems Division Definition of anisotropic denoising operators via sectional curvature Stanley Durrleman September 19, 2006

2. 1 MIA’06 - September 2006 Air Systems Division The problem coast crest Purpose : Denoising homogeneous areas… …without smoothing the signal at the interfaces

3. 2 MIA’06 - September 2006 Air Systems Division The problem Autoregressive model :

4. 3 MIA’06 - September 2006 Air Systems Division The problem Autoregressive model :

5. 4 MIA’06 - September 2006 Air Systems Division The problem Autoregressive model :

6. 5 MIA’06 - September 2006 Air Systems Division The problem Autoregressive model : Burg algorithm enables : -better estimation in case of short sample signals -fewer interference peaks -recursive computation : real time algorithm -estimation of the spectral density function :

7. 6 MIA’06 - September 2006 Air Systems Division The problem Example : record of turbulent atmospheric clutter Images du CR 1 magnitude angle

8. 7 MIA’06 - September 2006 Air Systems Division What’s in the image proceesing toolbox ? -Statistical models of noise Bayesian models, Markov fields… : - good model of noise - how to take the geometry into account ? -Geometrical models : Linear filters (Gaussian,…) : do not preserve the discontinuities Non-linear filters : - Curvature motion & morphologic filters (AMSS, mean curvature motion, median filter) : - noise = level set of small areas - specific for gray-level images - Geometric filters : (Kimmel, Sochen, Barbaresco) : - model data as a sub-manifold - depend on the way data are parametrized (mean curvature flow) - model of noise ?  Our goal : define anisotropic operators that can denoise data…  of any dimension (gray-level images, radar signal…)  independently of the data parametrization  and restore piecewise constant data

9. 8 MIA’06 - September 2006 Air Systems Division Outline  Noise characterization via sectional curvature  De-noising algorithms  Results

10. Air Systems Division I. Noise characterization through sectional curvature MIA – September 19, 2006

11. 10 MIA’06 - September 2006 Air Systems Division I. Noise & Sectional Curvature 1. Question : what is noise ?? statistics : Bayesian filters, maximum likelihood… geometry : which tool ? Gradient ? Curvature !

12. 11 MIA’06 - September 2006 Air Systems Division I. Noise & Sectional Curvature 2- Basic idea : the surface Gaussian Curvature

13. 12 MIA’06 - September 2006 Air Systems Division I. Noise & Sectional Curvature 2- Basic idea : the surface Gaussian Curvature Examples :

14. 13 MIA’06 - September 2006 Air Systems Division I. Noise & Sectional Curvature Noise and curvature Axiom : pixel of noise = pixel of big curvature

15. 14 MIA’06 - September 2006 Air Systems Division How to denoise ? By minimizing the following energy : I. Noise & Sectional Curvature

16. 15 MIA’06 - September 2006 Air Systems Division I. Noise & Sectional Curvature 3 – Modeling A generic ‘image’ :

17. 16 MIA’06 - September 2006 Air Systems Division I. Noise & Sectional Curvature 3 – Modeling

18. 17 MIA’06 - September 2006 Air Systems Division I. Noise & Sectional Curvature 3 – Modeling Curvature of a metric :

19. 18 MIA’06 - September 2006 Air Systems Division I. Noise & Sectional Curvature 3 – Modelisation Curvature of a metric : That is the surface Gaussian curvature !

20. 19 MIA’06 - September 2006 Air Systems Division I. Noise & Sectional Curvature Summary : 1/ One defines : h metric on the data space e metric on the acquisition space => a ‘mixed’ metric : g 2/ One computes the sectional curvature: K 3/ One defines the energy : E

21. Air Systems Division II. De-noising algorithms MIA’06 - September 19, 2006

22. 21 MIA’06 - September 2006 Air Systems Division II. De-noising algorithms Purpose : Minimizing : 2 methods : - Partial Differential Equation - Stochastic algorithm

23. 22 MIA’06 - September 2006 Air Systems Division 1.Descent gradient scheme : 1/ initialise with the given noisy image 2/ Evolve towards a minimum of : using the gradient : Hence, the evolution equation : implemented with a finite difference scheme. II. De-noising algorithms

24. 23 MIA’06 - September 2006 Air Systems Division II. De-noising algorithms Case of gray-level images :

25. 24 MIA’06 - September 2006 Air Systems Division II. De-noising algorithms 2. Stochastic method : - One picks randomly a pixel in the (noisy) image. - One adds a small random Gaussian variable to the pixel’s value. - If the energy decreases : one keeps the change Else : the change is rejected.

26. Air Systems Division III. Results MIA’06 - September 19, 2006

27. 26 MIA’06 - September 2006 Air Systems Division III. Results 1 – Gray-level images (1) Metric : Curvature : Flow :

28. 27 MIA’06 - September 2006 Air Systems Division III. Results t = 0t = 1t = 10t = 100 Flow equation

29. 28 MIA’06 - September 2006 Air Systems Division III. Results Transversal view :

30. 29 MIA’06 - September 2006 Air Systems Division III. Results Stochastic Stoch. + PDE Stochastic algorithm :

31. 30 MIA’06 - September 2006 Air Systems Division III. Results 2 – Gray level images (2) Adaptative metric : ‘Dilate’ geodesics in D far from the minimum

32. 31 MIA’06 - September 2006 Air Systems Division III. Results originalPDE Stoch. Algo.Adaptative metric

33. 32 MIA’06 - September 2006 Air Systems Division III. Results median

34. 33 MIA’06 - September 2006 Air Systems Division III. Results RSO Image original amss PDE Stoch

35. 34 MIA’06 - September 2006 Air Systems Division III. Results 2. Radar signal Reminder : complex auto-regressive analysis Parametrization thanks to complex auto-regressive analysis 8 complex magnitudes 7 reflection coefficients Doppler spectrum Burg Algo.

36. 35 MIA’06 - September 2006 Air Systems Division III. Results Data : Reflection coefficients

37. 36 MIA’06 - September 2006 Air Systems Division III. Results Simulated data : Image of CR 1 : magnitude angle azimut 16

38. 37 MIA’06 - September 2006 Air Systems Division III. Results Simulated data : Image of CR 1 : magnitude angle azimut 16

39. 38 MIA’06 - September 2006 Air Systems Division III. Results After de-noising:

40. 39 MIA’06 - September 2006 Air Systems Division III. Results After de-noising :

41. 40 MIA’06 - September 2006 Air Systems Division III. Results Real data : Images du CR 1 magnitude angle azimut 19

42. 41 MIA’06 - September 2006 Air Systems Division III. Results After de-noising :

43. Air Systems Division Thank you for your kind attention MIA’06 - September 19, 2006

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