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?

This paper proposes a novel approach for extracting two intrinsic feature curves on hippocampal (HC) surfaces. The hippocampus is a key target of study in medical imaging, as it degenerates in conditions such as epilepsy and Alzheimer's disease (AD), but its structure is complex. To facilitate HC morphometry, we generate two intrinsic feature curves that describe their global geometries. For example, the separation of them captures thickness changes in HC surfaces, which can be used to effectively measure HC atrophy found in patients with AD. They also separate HC surfaces into upper and lower surface patches where intrinsic shape analysis using conformal modules can be carried out. Based on these curves, we further propose a parameterization of HC surfaces called the eigen-harmonic parameterization (EHP). EHP maps each HC surface onto a parameter domain and imposes longitudinal and azimuthal coordinates on each surface, which follow the gradient and level sets of its first nontrivial Laplace--Beltrami ei...

Intrinsic Feature Extraction on Hippocampal Surfaces and Its Applications

We propose an effective framework for multi-phase image segmentation and semi-supervised data clustering by introducing a novel region force term into the Potts model. Assume the probability that a pixel or a data point belongs to each class is known a priori. We show that the corresponding indicator function obeys the Bernoulli distribution and the new region force function can be computed as the negative log-likelihood function under the Bernoulli distribution. We solve the Potts model by the primal-dual hybrid gradient method and the augmented Lagrangian method, which are based on two different dual problems of the same primal problem. Empirical evaluations of the Potts model with the new region force function on benchmark problems show that it is competitive with existing variational methods in both image segmentation and semi-supervised data clustering.

New region force for variational models in image segmentation and high dimensional data clustering

Following logic synthesis and circuit design, an integrated circuit is represented as a collection of interconnected rectangular modules of fixed dimensions Timing constraints on signal propagation paths along sequences of connections are also specified The task of circuit placement is to arrange the modules inside a prescribed rectangular region such that no two modules overlap, timing constraints are satisfied, and the estimated total wirelength needed to implement the connections is minimized Thus, an algorithm for placement derives a suitable spatial characterization of a given circuit from a logical-temporal one

Multilevel Circuit Placement

We propose a variational formulation for dimension reduction on Riemannian manifolds. The algorithm is developed based on the level set method together with a recently developed principal flow algorithm. The original principal flow algorithm is a Lagrangian technique which extends the principal component analysis (PCA) to dimension reduction on Riemannian manifolds. We propose to incorporate the level set method to obtain a fully implicit formulation so that the overall algorithm can naturally handle various topological changes in the curve evolution. The variational formulation consists of two terms which try to balance the contributions from both the dataset itself and the principal direction by the PCA. We will demonstrate that the method is insensitive to the initial guess and is robust enough for noisy data.

A Level Set Based Variational Principal Flow Method for Nonparametric Dimension Reduction on Riemannian Manifolds

In this work, we focus on using the intrinsic geometric method to study variational problems and Laplace-Beltrami eigen-geometry on 3D triangulated surfaces and their applications to computational brain anatomy. Two classes of problems will be discussed in this dissertation. In the first part, we study how to tackle image processing problems on surfaces by using variational approaches. Starting from the proof for the suitability of total variation for image processing problems on surfaces, we generalize the well-known total variation related imaging models to study imaging problems on surfaces by using differential geometry techniques. As an advantage of this intrinsic method, popular algorithms for solving the total variation related problems can be adapted to solve the generalized models on surfaces. We also demonstrate that this intrinsic method provides us a robust and efficient way to solve imaging problems on surfaces. In the second part, we focus on studying surfaces' own geometry. Specifically, we will study how to detect local and global surface geometry and its applications to computational brain anatomy. The main tool we use is the Laplace-Beltrami (LB) operator and its eigen-systems, which provide us an intrinsic and robust tool to study surface geometry. We first propose to use LB nodal count sequences as a surface signature to characterize surface and demonstrate its applications to isospectral surfaces resolving and surface classification. Then, we provide a novel approach of computing skeletons of simply connected surfaces by constructing Reeb graphs from the eigenftmctions of an anisotropie Laplace-Beltrami operator. In the last topic about the LB eigen-geometry, we propose a general framework to define a mathematically rigorous distance between surfaces by using the eigen-system of the LB operator, and then we demonstrate one of its applications to tackle the challenging sulci region identification problem in computational brain anatomy.

Tony F. Chan

Papers

Intrinsic Feature Extraction on Hippocampal Surfaces and Its Applications

New region force for variational models in image segmentation and high dimensional data clustering

Multilevel Circuit Placement

A Level Set Based Variational Principal Flow Method for Nonparametric Dimension Reduction on Riemannian Manifolds

Computational differential geometry and intrinsic surface processing