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

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

Regularization: Analytical vs Statistical 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

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

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

The Image Restoration Problem A given Observed image z Related to True Image u Through Blur K And Noise n 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 !

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

Comparison of different methods for signal denoising & reconstruction

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

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

TV Inpaintings: disocclusion

Examples of TV Inpaintings Where is the Inpainting Region?

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

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

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:

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)

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.

Application: “active contour” Initial Curve Evolutions Detected Objects

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)

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

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”

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-curve C

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

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

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

Experimental Results Automatically detects interior contours! 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

A plane in a noisy environment Europe nightlights

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

Example: two level set functions and four phases

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

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