Showing papers by "Tony F. Chan published in 2002"

A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model

Abstract: We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141–151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266–277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies, including triple junctionss in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.

Mathematical models for local nontexture inpaintings

Abstract: Dedicated to Stanley Osher on the occasion of his 60th birthday. Abstract. Inspired by the recent work of Bertalmio et al. on digital inpaintings (SIGGRAPH 2000), we develop general mathematical models for local inpaintings of nontexture images. On smooth regions, inpaintings are connected to the harmonic and biharmonic extensions, and inpainting orders are analyzed. For inpaintings involving the recovery of edges, we study a variational model that is closely connected to the classical total variation (TV) denoising model of Rudin, Osher, and Fatemi (Phys. D, 60 (1992), pp. 259-268). Other models are also discussed based on the Mumford-Shah regularity (Comm. Pure Appl. Math., XLII (1989), pp. 577-685) and curvature driven diffusions (CDD) of Chan and Shen (J. Visual Comm. Image Rep., 12 (2001)). The broad applications of the inpainting models are demonstrated through restoring scratched old photos, disocclusion in vision analysis, text removal, digital zooming, and edge-based image coding.

Wavelet Algorithms for High-Resolution Image Reconstruction

Abstract: High-resolution image reconstruction refers to the reconstruction of high-resolution images from multiple low-resolution, shifted, degraded samples of a true image. In this paper, we analyze this problem from the wavelet point of view. By expressing the true image as a function in ${\cal L}({\Bbb R}^2)$, we derive iterative algorithms which recover the function completely in the ${\cal L}$ sense from the given low-resolution functions. These algorithms decompose the function obtained from the previous iteration into different frequency components in the wavelet transform domain and add them into the new iterate to improve the approximation. We apply wavelet (packet) thresholding methods to denoise the function obtained in the previous step before adding it into the new iterate. Our numerical results show that the reconstructed images from our wavelet algorithms are better than that from the Tikhonov least-squares approach. Extension to super-resolution image reconstruction, where some of the low-resolution images are missing, is also considered.

Variational PDE models in image processing

Abstract: : Image processing, a traditionally engineering field, has attracted the attention of many mathematicians during the past two decades. From the vision and cognitive science point of view, image processing is a basic tool used to reconstruct the relative order, geometry, topology, patterns, and dynamics of the 3-D world from 2-D images. Therefore, it cannot be merely a historic coincidence that mathematics must meet image processing in this digital technology era. The role of mathematics is also determined by the broad range of applications of image processing in contemporary science and technology. These include astronomy and aerospace exploration, medical imaging, molecular imaging, computer graphics, human and machine vision, telecommunication, auto-piloting, surveillance video, and biometric security identification (such as fingerprints and face identification), etc. All these highly diversified disciplines have made it necessary to develop the common mathematical foundation and frameworks for image analysis and processing. Mathematics at all levels must be introduced to meet the crucial qualities demanded by this new era-genericity, well-posedness, accuracy, and computational efficiency, just to name a few. In return, image processing has created tremendous opportunities for mathematical modeling, analysis, and computation. In this article, we intend to give a broad picture of mathematical image processing through one of the most recent and very successful approaches - the variational PDE method. We first discuss two crucial ingredients for image processing: image modeling or representation, and processor modeling. We then focus on the variational PDE method. The backbone of the article consists of two major problems in image processing - inpainting and segmentation, which we have personally worked on, but by no means do we intend to have a comprehensive review of the entire field of image processing.

Active Contour and Segmentation Models using Geometric PDE’s for Medical Imaging

Abstract: This paper is devoted to the analysis and the extraction of information from bio-medical images. The proposed technique is based on object and contour detection, curve evolution and segmentation. We present a particular active contour model for 2D and 3D images, formulated using the level set method, and based on a 2-phase piecewise-constant segmentation. We then show how this model can be generalized to segmentation of images with more than two segments. The techniques used are based on the Mumford-Shah [21] model. By the proposed models, we can extract in addition measurements of the detected objects, such as average intensity, perimeter, area, or volume. Such informations are useful when in particular a time evolution of the subject is known, or when we need to make comparisons between different subjects, for instance between a normal subject and an abnormal one. Finally, all these will give more informations about the dynamic of a disease, or about how the human body growths. We illustrate the efficiency of the proposed models by calculations on two-dimensional and three-dimensional bio-medical images.

Multiscale and multiresolution methods : theory and applications

Abstract: Invited Papers: A. Brandt: Multiscale Scientific Computation: Review 2001.- B. Engquist, O. Runborg: Wavelet-Based Numerical Homogenization with Applications.- D.L. Donoho, X. Huo: Beamlets and Multiscale Image Analysis.- C. Schwab, A.-M. Matache: Generalized FEM for Homogenization Problems.- J.-L. Starck: Nonlinear Multiscale Transforms. Contributed Papers: F. Arandiga, G. Chiavassa, R. Donat: Application of Harten's Framework for Multiresolution: From Conservation Laws to Image Compression.- F. Fairag: A Two Level Finite Element Technique for Pressure Recovery from the Stream Function Formulation of the Navier-Stokes Equations.- I.K. Fodor, C. Kamath: The Role of Multiresolution in Mining Massive Image Datasets.- J. Hoffman: Dynamic Subgrid Modeling for Scalar Convection-Diffusion-Reaction Equations with Fractal Coefficients.- C.R. Johnson, M. Mohr, U. Rude, A. Samsonov, K. Zyp: Multilevel Methods for Inverse Bioelectric Field Problems.- O.E. Livne, A. Brandt: Multiscale Eigenbasis Calculations: N Eigenfunctions in O(N log N).- G. Schmidlin, C. Schwab: Wavelet Galerkin BEM on Unstructures Meshes by Aggregation. Appendix: Collected Color Figures

ENO-Wavelet Transforms for Piecewise Smooth Functions

Abstract: We have designed an adaptive essentially nonoscillatory (ENO)-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENO-wavelet transforms we adaptively change the function and use the same uniform stencils. The ENO-wavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate.

Modular solvers for image restoration problems using the discrepancy principle

Abstract: Many problems in image restoration can be formulated as either an unconstrained non-linear minimization problem, usually with a Tikhonov-like regularization, where the regularization parameter has to be determined; or as a fully constrained problem, where an estimate of the noise level, either the variance or the signal-to-noise ratio, is available. The formulations are mathematically equivalent. However, in practice, it is much easier to develop algorithms for the unconstrained problem, and not always obvious how to adapt such methods to solve the corresponding constrained problem. In this paper, we present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of non-linear equations, which couples the constraint and the regularized problem, but it does not require knowledge of the Jacobian of the irregularity functional. The existing solver is only used as a black box solver, which for a fixed regularization parameter returns an improved solution to the unconstrained minimization problem given an initial guess. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identifies the regularization parameter, during the iterative solution process. We present some numerical results. The results indicate that even in the worst case the constrained solver requires only about twice as much work as the unconstrained one, and in some instances the constrained solver can be even faster. Copyright © 2002 John Wiley & Sons, Ltd.

Inpainting from multiple views

Abstract: Inpainting refers to the task of filling in missing or damaged regions of an image. In this paper we are interested in the inpainting problem where the missing regions are so large that local inpainting methods fail. As an alternative to the local principle, we make use of other images with related global information to enable a reasonable inpainting. Our method has roughly three phases: landmark matching, interpolation, and copying. The experimental results obtained are promising.

Subspace correction multi-level methods for elliptic eigenvalue problems

Abstract: In this work, we apply the ideas of domain decomposition and multi-grid methods to PDE-based eigenvalue problems represented in two equivalent variational formulations. To find the lowest eigenpair, we use a “subspace correction” framework for deriving the multiplicative algorithm for minimizing the Rayleigh quotient of the current iteration. By considering an equivalent minimization formulation proposed by Mathew and Reddy, we can use the theory of multiplicative Schwarz algorithms for non-linear optimization developed by Tai and Espedal to analyse the convergence properties of the proposed algorithm. We discuss the application of the multiplicative algorithm to the problem of simultaneous computation of several eigenfunctions also formulated in a variational form. Numerical results are presented. Copyright © 2001 John Wiley & Sons, Ltd.

On Two Variants of an Algebraic Wavelet Preconditioner

Abstract: A recursive method of constructing preconditioning matrices for the nonsymmetric stiffness matrix in a wavelet basis is proposed for solving a class of integral and differential equations. It is based on a level-by-level application of the wavelet scales decoupling the different wavelet levels in a matrix form just as in the well-known nonstandard form. The result is a powerful iterative method with built-in preconditioning leading to two specific algebraic multilevel iteration algorithms: one with an exact Schur preconditioning and the other with an approximate Schur preconditioning. Numerical examples are presented to illustrate the efficiency of the new algorithms.

Bayesian Inpainting Based on Geometric Image Models

Abstract: Image inpainting is an image restoration problem, with wide applications in image processing, vision analysis, and the movie industry This paper surveys and summarizes all the recent inpainting models based on the Bayesian and variational principle A unified view is developed around the central topic of geometric image models We also discuss their associated Euler-Lagrange PDE’s and numerical implementation A few open problems are proposed