Total Variation and Geometric Regularization for Inverse Problems

1. Total Variation and Geometric Regularization for Inverse Problems Regularization in Statistics September 7-11, 2003 BIRS, Banff, Canada Tony ChanDepartment of Mathematics, UCLA

2. Outline • TV & Geometric Regularization (related concepts) • PDE and Functional/Analytic based • Geometric Regularization via Level Sets Techniques • Applications (this talk): • Image restoration • Image segmentation • Elliptic Inverse problems • Medical tomography: PET, EIT

3. Regularization: Analytical vs Statistical • Analytical: • Controls “smoothness” of continuous functions • Function spaces (e.g. Sobolov, Besov, BV) • Variational models -> PDE algorithms • Statistical: • Data driven priors • Stochastic/probabilistic frameworks • Variational models -> EM, Monte Carlo

4. Taking the Best from Each? • Concepts are fundamentally related: • e.g. Brownian motion  Diffusion Equation • Statistical frameworks advantages: • General models • Adapt to specific data • Analytical frameworks advantages: • Direct control on smoothness/discontinuities, geometry • Fast algorithms when applicable

5. Total Variation Regularization • Measures “variation” of u, w/o penalizing discontinuities. • |.| similar to Huber function in robust statistics. • 1D: If u is monotonic in [a,b], then TV(u) = |u(b) – u(a)|, regardless of whether u is discontinuous or not. • nD: If u(D) = char fcn of D, then TV(u) = “surface area” of D. • (Coarea formula) • Thus TV controls both size of jumps and geometry of boundaries. • Extensions to vector-valued functions • Color TV: Blomgren-C 98; Ringach-Sapiro, Kimmel-Sochen

6. The Image Restoration Problem A given Observed imagez Related to True Imageu Through BlurK And Noisen Initial Blur Blur+Noise Inverse Problem: restore u, given K and statistics for n. Keeping edges sharp and in the correct location is a key problem !

7. Total Variation Restoration Regularization: Variational Model: * First proposed by Rudin-Osher-Fatemi ’92. * Allows for edge capturing (discontinuities along curves). * TVD schemes popular for shock capturing. Gradient flow: anisotropic diffusion data fidelity

8. Comparison of different methods for signal denoising & reconstruction

9. Image Inpainting (Masnou-Morel; Sapiro et al 99) Disocclusion Graffiti Removal

10. Unified TV Restoration & Inpainting model (C- J. Shen 2000)

11. TV Inpaintings: disocclusion

12. Examples of TV Inpaintings Where is the Inpainting Region?

13. TV Zoom-in Inpaint Region: high-res points that are not low-res pts

14. Edge Inpainting edge tube T No extra data are needed. Just inpaint! Inpaint region: points away from Edge Tubes

15. Extensions • Color (S.H. Kang thesis 02) • “Euler’s Elastica” Inpainting (C-Kang-Shen 01) • Minimizing TV + Boundary Curvature • “Mumford-Shah” Inpainting (Esedoglu-Shen 01) • Minimizing boundary + interior smoothness:

18. Geometric Regularization • Minimizing surface area of boundaries and/or volume of objects • Well-studied in differential geometry: curvature-driven flows • Crucial: representation of surface & volume • Need to allow merging and pinching-off of surfaces • Powerful technique: level set methodology (Osher/Sethian 86)

19. Level Set Representation (S. Osher - J. Sethian ‘87) Inside C Outside C Outside C C C= boundary of an open domain • Example: mean curvature motion • * Allows automatic topology changes, cusps, merging and breaking. • Originally developed for tracking fluid interfaces.

20. Application: “active contour” Initial Curve Evolutions Detected Objects

21. Basic idea in classical active contours Curve evolution and deformation (internal forces): Min Length(C)+Area(inside(C)) Boundary detection: stopping edge-function (external forces) Example: Snake model (Kass, Witkin, Terzopoulos 88) Geodesic model (Caselles, Kimmel, Sapiro 95)

22. Limitations - detects only objects with sharp edges defined by gradients - the curve can pass through the edge - smoothing may miss edges in presence of noise - not all can handle automatic change of topology Examples

23. A fitting term “without edges” where Fit > 0 Fit > 0 Fit > 0 Fit ~ 0 Minimize: (Fitting +Regularization) Fitting not depending on gradient detects “contours without gradient”

24. An active contour model “without edges” (C. + Vese 98) Fitting + Regularization terms (length, area) C = boundary of an open and bounded domain |C|= the length of the boundary-curveC

25. Mumford-Shah Segmentation 89 S=“edges” MS reg: min boundary + interior smoothness CV model = p.w. constant MS

26. Variational Formulations and Level Sets (Following Zhao, Chan, Merriman and Osher ’96) The Heaviside function The level set formulation of the active contour model

27. The Euler-Lagrange equations Using smooth approximations for the Heaviside and Delta functions

28. Experimental Results Advantages Automatically detects interior contours! Works very well for concave objects Robust w.r.t. noise Detects blurred contours The initial curve can be placed anywhere! Allows for automatical change of topolgy

29. A plane in a noisy environment Europe nightlights

30. Multiphase level set representations and partitions allows for triple junctions, with no vacuum and no overlap of phases 4-phase segmentation 2 level set functions 2-phase segmentation 1 level set function

31. Example: two level set functions and four phases

32. An MRI brain image Phase 11 Phase 10 Phase 01 Phase 00 mean(11)=45 mean(10)=159 mean(01)=9 mean(00)=103

34. References for PDE & Level Sets in Imaging • * IEEE Tran. Image Proc. 3/98, Special Issue on PDE Imaging • * J. Weickert 98: Anisotropic Diffusion in Image Processing • * G. Sapiro 01: Geometric PDE’s in Image Processing • Aubert-Kornprost 02: Mathematical Aspects of Imaging Processing • Osher & Fedkiw 02: “Bible on Level Sets” • Chan, Shen & Vese Jan 03, Notices of AMS