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.

/pdf/iterative-methods-for-sparse-linear-systems-34ozhpwq89.pdf

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?

A new agglomeration multigrid method is proposed in this paper for general unstructured grids. By a proper local agglomeration of finite elements, a nested sequence of finite dimensional subspaces are obtained by taking appropriate linear combinations of the basis functions from previous level of space. Our algorithm seems to be able to solve, for example, the Poisson equation discretized on any shape-regular finite element grids with nearly optimal complexity. In this paper, we discuss a multilevel method applied to problems on general unstructured grids. We will describe an approach for designing a multilevel method for the solution of large systems of linear algebraic equations, arising from finite element discretizations on unstructured grids. Our interest will be focused on the performance of an agglomeration multigrid method for unstructured grids. One approach of constructing coarse spaces is based on generating node-nested coarse grids, which are created by selecting subsets of a vertex set, retriangulating the subset, and using piecewise linear interpolation between the grids (see [8, 5]). This still provides an automatic way of generating coarse grids and faster implementations of the interpolation in O(n) time. The drawback is that in three dimensions, retetrahedralization can be problematic. Another effective coarsening strategy has been proposed by Bank and Xu [1]. It uses the geometrical coordinates of the fine grid and the derefinement algorithm is based on the specified patterns of fine grid elements. The interpolation between grids is done by interpolating each fine grid node using only 2 coarse grid nodes. As a consequence of that the fill-in in the coarse grid matrices is reasonably small. The hierarchy of spaces is defined by interpolating the basis. Recently a new approach, known as auxiliary space method, was proposed by Xu [16]. In this method only one non-nested (auxiliary) grid is created and then all consecutive grids are nested. This can be done by using as auxiliary grid a uniform

An agglomeration multigrid method for unstructured grids

Circulant preconditioners for elliptic problems

The active contour segmentation model of Chan and Vese has been widely used and generalized in different contexts in the literature. One possible modification is to employ Euler's elastica as the regularization of active contour. In this paper, we study the new effects of this modification and validate them numerically using the augmented Lagrangian method.

Image Segmentation Using Euler's Elastica as the Regularization

We investigate how TV regularization naturally recognizes the scale of individual features of an image, and we show how this perception of scale depends on the amount of regularization applied to the image. We give an automatic method driven by the geometry of the image for finding the minimum value of the regularization parameter needed to remove all features below a user-chosen threshold. We explain the relation of Meyer's G norm to the perception of scale, which provides a more intuitive understanding of this norm. We consider other applications of this ability to recognize scale, including the multiscale effects of TV regularization and the rate of loss of image features of various scales as a function of increasing amounts of regularization. Several numerical results are given.

Scale Recognition, Regularization Parameter Selection, and Meyer's G Norm in Total Variation Regularization

The success of some recent texture synthesis methods, see [8, 17], suggests that there exists an underlying formulation explaining their performance and paving the way to more involved modeling Based on their ideas, we formalize a low-level global deterministic solution for image inpainting A correspondence map is defined as linking each blank or missing pixel to the pixel where its value is taken from, in the seed image The above-mentioned algorithms are seen as descent procedures to minimize a functional of this correspondence map, the inpainting energy We discuss why they should not be seen as procedures to sample a probability distribution on the correspondence maps We therefore question the claims that probability is anywhere involved at this explanatory level The algorithm we use is mostly taken from [17] The latter however suffers from a strong directional bias, the direction in which texture is grown We restore rotationinvariance at the level of both the target function and the algorithm Our encouraging numerical results could not have been obtained by a directional texture-growing algorithm

/pdf/image-inpainting-by-correspondence-maps-a-deterministic-2erz8oi4cl.pdf

Tony F. Chan

Papers

An agglomeration multigrid method for unstructured grids

Circulant preconditioners for elliptic problems

Image Segmentation Using Euler's Elastica as the Regularization

Scale Recognition, Regularization Parameter Selection, and Meyer's G Norm in Total Variation Regularization

Image inpainting by correspondence maps: A deterministic approach