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.

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Iterative Methods for Sparse Linear Systems

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.

/pdf/active-contours-without-edges-2tt3qz9lq0.pdf

Active contours without edges

Co-Planar Stereotaxic Atlas of the Human Brain—3-Dimensional Proportional System: An Approach to Cerebral Imaging, J. Talairach, P. Tournoux. Georg Thieme Verlag, New York (1988), 122 pp., 130 figs. DM 268

Data Mining - Concepts and Techniques.

https://hal.archives-ouvertes.fr/hal-00417283/document

Independent component analysis, a new concept?

Many variational models for image denoising restoration are formulated in primal variables that are directly linked to the solution to be restored. If the total variation related semi-norm is used in the models, one consequence is that extra regularization is needed to remedy the highly non-smooth and oscillatory coefficients for effective numerical solution. The dual formulation was often used to study theoretical properties of a primal formulation. However as a model, this formulation also offers some advantages over the primal formulation in dealing with the above mentioned oscillation and non-smoothness. This paper presents some preliminary work on speeding up the Chambolle method [J. Math. Imaging Vision, 20 (2004), pp. 89–97] for solving the dual formulation. Following a convergence rate analysis of this method, we first show why the nonlinear multigrid method encounters some difficulties in achieving convergence. Then we propose a modified smoother for the multigrid method to enable it to achieve convergence in solving a regularized Chambolle formulation. Finally, we propose a linearized primaldual iterative method as an alternative stand-alone approach to solve the dual formulation without regularization. Numerical results are presented to show that the proposed methods are much faster than the Chambolle method.

https://etna.mcs.kent.edu/vol.26.2007/pp299-311.dir/abstr299-311.pdf

Iterative methods for solving the dual formulation arising from image restoration

In the past several years, domain decomposition has been a very popular topic, motivated by the ease of parallelization. However, the question of whether it is better than parallelizing some standard sequential methods has seldom been directly addressed.In this paper it is shown, with some numerical examples, that the answer is affirmative in the case of iterative solutions of elliptic problems by preconditioned conjugate gradient iteration. Specifically shown is how to construct effective incomplete factorization preconditioners based on the domain decomposition principle. In addition to having all the advantages of domain decomposition, this also results in better convergence rates than the analogous preconditioners on the whole domain.

A note on the efficiency of domain decomposed incomplete factorizations

We develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and fine grids are assumed only to be shape regular, and the domains formed by the coarse and fine grids need not be identical. In this general setting, our convergence theory leads to completely local bounds for the condition numbers of two level additive Schwarz methods, which imply that these condition numbers are optimal, or independent of fine and coarse mesh sizes and subdomain sizes if the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. In particular, we will show that additive Schwarz algorithms are still very efficient for nonselfadjoint parabolic problems with only symmetric, positive definite solvers both for local subproblems and for the global coarse problem. These conclusions for elliptic and parabolic problems improve our earlier results in [12, 15, 16]. Finally, the convergence theory is applied to multilevel additive Schwarz algorithms. Under some very weak assumptions on the fine mesh and coarser meshes, e.g., no requirements on the relation between neighboring coarse level meshes, we are able to derive a condition number bound of the orderO(ρ2
L
2), whereρ = max1≤l≤L(h
l +l− 1)/δ
l,h
l is the element size of thelth level mesh,δ
l the overlap of subdomains on thelth level mesh, andL the number of mesh levels.

A convergence theory of multilevel additive Schwarz methods on unstructured meshes

Variational method is a useful mathematical tool in various areas of research, espe- cially in image processing. Recently, solving image processing problems on general manifolds using variational techniques has become an important research topic. In this paper, we solve several image processing problems on general Riemann surfaces using variational models defined on surfaces. We propose an explicit method to solve variational problems on general Riemann surfaces, using the conformal parameterization and covariant derivatives defined on the surface. To simplify the com- putation, the surface is firstly mapped conformally to the two dimensional rectangular domains, by computing the holomorphic 1-form on the surface. It is well known that the Jacobian of a conformal map is simply the scalar multiplication of the conformal factor. Therefore with the conformal para- meterization, the covariant derivatives on the parameter domain are similar to the usual Euclidean differential operators, except for the scalar multiplication. As a result, any variational problem on the surface can be formulated to a 2D problem with a simple formula and efficiently solved by well developed numerical scheme on the 2D domain. With the proposed method, we solve various image processing problems on surfaces using different variational models, which include image segmen- tation, surface denoising, surface inpainting, texture extraction and automatic landmark tracking. Experimental results show that our method can effectively solve variational problems and tackle image processing tasks on general Riemann surfaces.

https://projecteuclid.org/download/pdf_1/euclid.maa/1254492832

Variational Method on Riemann Surfaces using Conformal Parameterization and its Applications to Image Processing

Phase diversity is a technique for obtaining estimates of both the object and the phase, by exploiting the simultaneous collection of two (or more) short-exposure optical images, one of which has been formed by further blurring the conventional image in some known fashion. This paper concerns a fast computational algorithm based upon a regularized variant of the Gauss-Newton optimization method for phase diversity-based estimation when a Gaussian likelihood fit-to-data criterion is applied. Simulation studies are provided to demonstrate that the method is remarkably robust and numerically efficient.

Tony F. Chan

Papers

Iterative methods for solving the dual formulation arising from image restoration

A note on the efficiency of domain decomposed incomplete factorizations

A convergence theory of multilevel additive Schwarz methods on unstructured meshes

Variational Method on Riemann Surfaces using Conformal Parameterization and its Applications to Image Processing

Fast algorithms for phase diversity-based blind deconvolution