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Computational Algebraic Problems in

Computational Algebraic Problems in Variational PDE Image Processing. Tony F. Chan Department of Mathematics, UCLA International Summer School in Numerical Linear Algebra

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Computational Algebraic Problems in

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  1. Computational Algebraic Problems in Variational PDE Image Processing Tony F. Chan Department of Mathematics, UCLA International Summer School in Numerical Linear Algebra Chinese University of Hong Kong Reports:www.math.ucla.edu/~imagers Supported by ONR, NIH and NSF

  2. Collaborators Peter Blomgren (Stanford) Raymond Chan (CUHK) Gene Golub (Stanford) Pep Mulet (Valencia) Jackie Shen (Minnesota) Luminita Vese (UCLA) Justin Wan (Waterloo) C.K. Wong Jamylle Carter (MSRI) Berta Sandberg (TechFinity) Ke Chen (Liverpool) Xue-Cheng Tai (Bergen, Norway) Jean-Francois Aujol (Cachan) Selim Esedoglu (Michigan) Fred Park (UCLA -> Michigan)

  3. Outline of 4 Lectures • Introduction to PDE Image Models & Algorithms: • Denoising, deblurring, active contours, segmentation • 2. Algebraic Problems from Denoising & Deblurring • Linear preconditioning techniques • Nonlinear + Optimization (duality) techniques • Multigrid techniques • Algebraic Problems from Active Contours/Segmentation • Curve evolution techniques • Direct optimization techniques

  4. Goal of Lectures • Broad overview rather than latest techniques • Details on only a few topics --- see papers + reports + web • No comprehensive referencing • Limit to PDE aspects (there are non-PDE approaches; see forthcoming SIAM book by Hansen, Nagy, O’Leary) • Please ask questions (English, Cantonese, Mandarin)

  5. TYPICAL IMAGE PROCESSING TASKS * Denoising/Inpaint * Object Detection/Identification * Deblurring * Object/Pattern Recognition * Enhancement * Gray-Scale vs Vector-Valued * Compression * Still vs Video * Segmentation * Registration Related fields: Computer Graphics, Computer Vision.

  6. IP And Applied Math • Important applications: • Medical, astronomy, Comp. Vision/Comp. Graphics, • Math Models: • standard or create your own • Math Tools: • harmonic analysis, PDEs, Baysean statistics, differential geometry, CFD, multiscale, optimization,… • Analysis of Models: • existence, uniqueness, properties • Challenging Computations • 3D+time+multi-components, nonlinearity, non-smooth

  7. Examples of PDE Image Models Denoising and Inpainting

  8. The 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. Ill-posed: needs proper regularization. Keeping edges sharp and in the correct location is a key problem !

  9. 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

  10. 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

  11. Comparison of different methods for signal denoising & reconstruction

  12. Denoising: TV vs H1

  13. Inpaintings: Generalized Restoration Models Disocclusion Scratch Removal Graffiti Removal

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

  15. Unified TV Restoration & Inpainting model

  16. Examples of PDE Image Models Blind (unknown blur) and Non-blind Deburring

  17. Deblurring Original Image Out of focus blur Blurred Image - Blurring operator K usually ill-conditioned - Need to solve systems with differential-convolution operator:

  18. Debluring: TV vs H1

  19. TV Blind Deconvolution C. + Wong (98) * Variational Model: * Alternating minimization algorithm: • * Algorithm gives 1-parameter family of solutions, • determined by SNR.

  20. TV Blind deconvolution Recovered image Recovered Blurring Function 0 1e-7 1e-5 1e-4

  21. Original image Out of focus blurred blind non-blind Gaussianblurred blind non-blind

  22. Examples of PDE Image Models Active Contours & Segmentation

  23. Active Contour w/o Edges (C.-Vese 99) Features: 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 Objects Found

  24. MRI brain image Europe nightlights The model detects contours without gradient (cognitive contours)

  25. Motion Segmentation (Moelich, Chan 2004) Motion determined by logical AND & OR on frame differences over 3 consecutive frames. Designed for low frame rate videos. Another extension of Chan-Vese 2001 model. Implemented via level sets. Westwood Blvd Olympic Blvd, LA (2 frames per sec) UCLA (1 frame per sec)

  26. Other Related PDE Image Models * Geometry driven diffusion (see book by Bart M. ter Haar Romeny 1994) * Anisotropic diffusion (Perona-Malik 87) * Fundamental IP PDE (Alvarez-Guichard-Lions-Morel 92) Affine invariant flow (Sapiro-Tanenbaum 93) * TV + Textures (Meyer 2001, Osher-Vese 02, Osher-Sole-Vese 02, Osher-Sapiro-Vese 02, Chambolle 03)

  27. Different Frameworks for Image Processing Statistical/Stochastic Models: Maximum Likelihood Estimation with uncertain data Transform-Based Models: Fourier/Wavelets --- process features of images (e.g. noise) in transform space (e.g. thresholding) Variational PDE Models: Evolve image according to local derivative/geometric info, e.g. denoising  diffusion Concepts are related mathematically: Brownian motion – Fourier Analysis --- Diffusion Equation

  28. Features & Advantages of PDE Imaging Models * Use PDE concepts: gradients, diffusion, curvature, level sets * Exploit sophisticated PDE and CFD (e.g. shock capturing) techniques Restoration: - sharper edges, less edge artifacts, often morphological Segmentation: - scale adaptivity, geometry-based, controlled regularity of boundaries, segments can have complex topologies Newer, less well developed/accepted. Combining PDE with other techniques.

  29. Computational challenges * size: large # of pixels, color and multi-channel, 3D, videos. * TV(u)non-differentiable if need numerical regularization: * High nonlinearity of * Ill-conditioning: * Highly varying coefficients: * Need to precondition differential + convolution operators.

  30. Some Books/Surveys for PDE Imaging • Morel-Solimini 94: Variational Meths in Image Segmentation • Romeny 94: Geometry Driven Diffusion in Computer Vision • Alvarez-Morel 94: Acta Numerica (review article) • IEEE Tran. Image Proc. 3/98, Special Issue • J. Weickert 98: Anisotropic Diffusion in Image Processing • G. Sapiro 2000: Geometric PDE & Imaging • Aubert-Kornprost 2002: Math Aspects of Image Processing • Osher-Fedkiw 2003: “Level Set Bible” • Chan, Shen & Vese Jan 03, Notices of AMS (review) • Paragios, Chen, Faugeras 2005: Collection of articles • Chan-Shen 2005: Image Processing & Analysis

  31. Available since Sept 05 (www.siam.org)

  32. Outline of 4 Lectures • Introduction to PDE Image Models & Algorithms: • Denoising, deblurring, active contours, segmentation • 2. Algebraic Problems from Denoising & Deblurring • Linear preconditioning techniques • Nonlinear + Optimization (duality) techniques • Multigrid techniques • Algebraic Problems from Active Contours/Segmentation • Curve evolution techniques • Direct optimization techniques

  33. Restoration Problem in Discrete Algebraic Form  where denotes the discrete gradient , divergence. Gradient Flow: 

  34. Morphological Diagonal Scaling (Marquina, Osher 2000) • Two different but related motivations: • morphological evolution of level sets --- moves in direction of normal with speed proportional to curvature, independent of contrast. • diagonally scaled Richardson stationary iteration. • Advantages: • cost per time step similar to time marching • much faster convergence to steady state

  35. Unconstrained Modular Solver for Discrepancy Principle Blomgren-C. 96 Discrepancy Principle Constrained Problem: min R(u) subject to u Tikhonov Unconstrained Problem: Modular Solver: efficient solver for fixed (e.g. Time Marching, Fixed Point, Primal-Dual.) - Can make use of S to solve constrained problem efficiently. - Based on block-elimination + Newton’s method for constrained problem via calls to S for computing directional derivatives.

  36. A modular Newton’s Method System of nonlinear equations: G(u, ) = 0; N(u, ) = 0. Newton’s Method: Gu G u - G = Nu N - N Block elimination: Define w = - Gu-1G; v = Gu-1G  = - (N – Nu v)-1 (Nu w + N) Main idea: Replace computation of w & v by calls to S(u,)

  37. Robustness of Modular Solver

  38. Efficiency of Modular Solver for Constrained Problem

  39. Outline of 4 Lectures • Introduction to PDE Image Models & Algorithms: • Denoising, deblurring, active contours, segmentation • 2. Algebraic Problems from Denoising & Deblurring • Linear preconditioning techniques • Nonlinear + Optimization (duality) techniques • Multigrid techniques • Algebraic Problems from Active Contours/Segmentation • Curve evolution techniques • Direct optimization techniques

  40. Difficulties with Primal TV • TV Norm Non Differentiable ) Regularize Functional: • Primal: Gradient Flow Also Needs Regularization : • Problem Becomes difficult for Small  • But edges smeared for large  • Artificial Time Marching at best Linearly Convergent • Nonlinear relaxation (e.g. GS) non-convergent w/o  • Ill-conditioning due to spatial scales (CFL; MG)

  41. Towards Quadratically Convergent Methods Newton’s Method [R. Chan, T. Chan, Zhou, ‘95] • As ! 0 ) Size of Domain of Convergence ! 0 • Tied Continuation on . • But Efficient Continuation Not Easy to Obtain!

  42. Primal-Dual Method:(Chan, Golub, and Mulet ‘95) Instead of u system: Introduce Auxiliary (Dual Variable) w:=ru / |ru| , Note |w|=1. Linearize the (w,u)-system: • Similar to primal-dual (Conn & Overton ’95, Anderson ’95)

  43. Why Linearization Works Well • w =ru/|ru| Normal to the Level Sets of u • r ¢ w =  Curvature of Level Sets • Time Marching Regarded as Curvature Driven Flow • Better Global Convergence Of Newton Meth for (w,u)- system: (w,u) system more “globally linear” than u-system

  44. Linearized Systems • Linearization of u-system • Linearization of (w,u)-system • Similar Structure and Cost for Both Systems

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