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Definition of anisotropic denoising operators via sectional curvature Stanley Durrleman

Definition of anisotropic denoising operators via sectional curvature Stanley Durrleman. September 19, 2006. The problem. crest. coast. Purpose : Denoising homogeneous areas… …without smoothing the signal at the interfaces. The problem. Autoregressive model :. The problem.

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Definition of anisotropic denoising operators via sectional curvature Stanley Durrleman

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  1. Definition of anisotropic denoising operators viasectional curvatureStanley Durrleman Air Systems Division September 19, 2006

  2. The problem crest coast Purpose : Denoising homogeneous areas… …without smoothing the signal at the interfaces Air Systems Division

  3. The problem Autoregressive model : Air Systems Division

  4. The problem Autoregressive model : Air Systems Division

  5. The problem Autoregressive model : Air Systems Division

  6. 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 : Air Systems Division

  7. The problem Images du CR 1 magnitude angle • Example : record of turbulent atmospheric clutter Air Systems Division

  8. 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 Air Systems Division

  9. Outline • Noise characterization via sectional curvature • De-noising algorithms • Results Air Systems Division

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

  11. I. Noise & Sectional Curvature • 1. Question : what is noise ?? • statistics : Bayesian filters, maximum likelihood… • geometry : which tool ? • Gradient ? • Curvature ! Air Systems Division

  12. I. Noise & Sectional Curvature • 2- Basic idea : the surface Gaussian Curvature Air Systems Division

  13. I. Noise & Sectional Curvature • 2- Basic idea : the surface Gaussian Curvature Examples : Air Systems Division

  14. I. Noise & Sectional Curvature • Noise and curvature Axiom : pixel of noise = pixel of big curvature Air Systems Division

  15. I. Noise & Sectional Curvature • How to denoise ? • By minimizing the following energy : Air Systems Division

  16. I. Noise & Sectional Curvature • 3 – Modeling • A generic ‘image’ : Air Systems Division

  17. I. Noise & Sectional Curvature • 3 – Modeling Air Systems Division

  18. I. Noise & Sectional Curvature • 3 – Modeling Curvature of a metric : Air Systems Division

  19. I. Noise & Sectional Curvature • 3 – Modelisation Curvature of a metric : That is the surface Gaussian curvature ! Air Systems Division

  20. 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 Air Systems Division

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

  22. II. De-noising algorithms • Purpose : • Minimizing : • 2 methods : • - Partial Differential Equation • - Stochastic algorithm Air Systems Division

  23. II. De-noising algorithms • 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. Air Systems Division

  24. II. De-noising algorithms • Case of gray-level images : Air Systems Division

  25. 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. Air Systems Division

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

  27. III. Results • 1 – Gray-level images (1) • Metric : • Curvature : • Flow : Air Systems Division

  28. III. Results Flow equation t = 0 t = 1 t = 10 t = 100 Air Systems Division

  29. III. Results • Transversal view : Air Systems Division

  30. III. Results Stochastic algorithm : Stochastic Stoch. + PDE Air Systems Division

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

  32. III. Results Stoch. Algo. Adaptative metric original PDE Air Systems Division

  33. III. Results median Air Systems Division

  34. III. Results • RSO Image PDE amss Stoch original Air Systems Division

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

  36. III. Results • Data : Reflection coefficients Air Systems Division

  37. III. Results Image of CR 1 : • Simulated data : magnitude angle azimut 16 Air Systems Division

  38. III. Results Image of CR 1 : • Simulated data : magnitude angle azimut 16 Air Systems Division

  39. III. Results • After de-noising: Air Systems Division

  40. III. Results • After de-noising : Air Systems Division

  41. III. Results Images du CR 1 • Real data : angle magnitude azimut 19 Air Systems Division

  42. III. Results • After de-noising : Air Systems Division

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

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