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**Total Variation and Geometric Regularization for Inverse Problems**

Regularization in Statistics September 7-11, 2003 BIRS, Banff, Canada Tony Chan Department of Mathematics, UCLA

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**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

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**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

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**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

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**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

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**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 !

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**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

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**Comparison of different methods for signal denoising & reconstruction**

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**Image Inpainting (Masnou-Morel; Sapiro et al 99)**

Disocclusion Graffiti Removal

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**Unified TV Restoration & Inpainting model**

(C- J. Shen 2000)

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**TV Inpaintings: disocclusion**

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**Examples of TV Inpaintings**

Where is the Inpainting Region?

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

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**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

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**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:

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**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)

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**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.

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**Application: “active contour”**

Initial Curve Evolutions Detected Objects

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**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)

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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

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**A fitting term “without edges”**

where Fit > Fit > Fit > Fit ~ 0 Minimize: (Fitting +Regularization) Fitting not depending on gradient detects “contours without gradient”

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**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

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**Mumford-Shah Segmentation 89**

S=“edges” MS reg: min boundary + interior smoothness CV model = p.w. constant MS

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**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

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**Using smooth approximations for the Heaviside and Delta functions**

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

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**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

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**A plane in a noisy environment**

Europe nightlights

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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

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**Example: two level set functions and four phases**

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**An MRI brain image Phase 11 Phase 10 Phase 01 Phase 00**

mean(11)= mean(10)= mean(01)= mean(00)=103

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**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

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