Author

# Tony F. Chan

Other affiliations: Kent State University, University of California, National Science Foundation ...read more

Bio: Tony F. Chan is an academic researcher from Hong Kong University of Science and Technology. The author has contributed to research in topics: Domain decomposition methods & Image restoration. The author has an hindex of 82, co-authored 437 publications receiving 48083 citations. Previous affiliations of Tony F. Chan include Kent State University & University of California.

##### Papers published on a yearly basis

##### Papers

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TL;DR: A new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets is proposed, which can detect objects whose boundaries are not necessarily defined by the gradient.

Abstract: We propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by the gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We give a numerical algorithm using finite differences. Finally, we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.

10,404 citations

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TL;DR: A new version of ENO (essentially non-oscillatory) shock-capturing schemes which is called weighted ENO, where, instead of choosing the "smoothest" stencil to pick one interpolating polynomial for the ENO reconstruction, a convex combination of all candidates is used.

3,023 citations

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TL;DR: A new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations, and validated by numerical results for signal and image denoising and segmentation.

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.

2,649 citations

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TL;DR: A blind deconvolution algorithm based on the total variational (TV) minimization method proposed is presented, and it is remarked that psf's without sharp edges, e.g., Gaussian blur, can also be identified through the TV approach.

Abstract: We present a blind deconvolution algorithm based on the total variational (TV) minimization method proposed by Acar and Vogel (1994). The motivation for regularizing with the TV norm is that it is extremely effective for recovering edges of images as well as some blurring functions, e.g., motion blur and out-of-focus blur. An alternating minimization (AM) implicit iterative scheme is devised to recover the image and simultaneously identify the point spread function (PSF). Numerical results indicate that the iterative scheme is quite robust, converges very fast (especially for discontinuous blur), and both the image and the PSF can be recovered under the presence of high noise level. Finally, we remark that PSFs without sharp edges, e.g., Gaussian blur, can also be identified through the TV approach.

1,220 citations

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TL;DR: 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.

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.

1,189 citations

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01 Apr 2003

TL;DR: This chapter discusses methods related to the normal equations of linear algebra, and some of the techniques used in this chapter were derived from previous chapters of this book.

Abstract: Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

13,484 citations

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TL;DR: A new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets is proposed, which can detect objects whose boundaries are not necessarily defined by the gradient.

Abstract: We propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by the gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We give a numerical algorithm using finite differences. Finally, we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.

10,404 citations

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9,362 citations

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TL;DR: An efficient algorithm is proposed, which allows the computation of the ICA of a data matrix within a polynomial time and may actually be seen as an extension of the principal component analysis (PCA).

8,522 citations