Results 1 - 10 of 35

Geodesic entropic graphs for dimension and entropy estimation in manifold learning

by Jose A. Costa, Alfred O. Hero - IEEE TRANS. ON SIGNAL PROCESSING , 2004
"... In the manifold learning problem, one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional linear vector space, based on a set of measured sample points on the surface. In this paper, we consider the closely related problem of estimating the manifold ..."
Abstract - Cited by 99 (5 self) - Add to MetaCart
In the manifold learning problem, one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional linear vector space, based on a set of measured sample points on the surface. In this paper, we consider the closely related problem of estimating the manifold’s intrinsic dimension and the intrinsic entropy of the sample points. Specifically, we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold. We introduce a novel geometric approach based on entropic graph methods. Although the theory presented applies to this general class of graphs, we focus on the geodesic-minimal-spanning-tree (GMST) to obtaining asymptotically consistent estimates of the manifold dimension and the Rényi-entropy of the sample density on the manifold. The GMST approach is striking in its simplicity and does not require reconstruction of the manifold or estimation of the multivariate density of the samples. The GMST method simply constructs a minimal spanning tree (MST) sequence using a geodesic edge matrix and uses the overall lengths of the MSTs to simultaneously estimate manifold dimension and entropy. We illustrate the GMST approach on standard synthetic manifolds as well as on real data sets consisting of images of faces.

Mapping cortical change in Alzheimer’s disease, brain development, and schizophrenia

by Paul M. Thompson, et al. , 2004
"... ..."
Abstract - Cited by 64 (16 self) - Add to MetaCart
Abstract not found

O.: Regularizing flows for constrained matrix-valued images

by D. Tschumperlé, R. Deriche, O. Faugeras - J. Math. Imaging Vision , 2004
"... Abstract. Nonlinear diffusion equations are now widely used to restore and enhance images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a differential-geometric framework to define PDEs acting on some manifol ..."
Abstract - Cited by 47 (8 self) - Add to MetaCart
Abstract. Nonlinear diffusion equations are now widely used to restore and enhance images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a differential-geometric framework to define PDEs acting on some manifold constrained datasets. We consider the case of images taking value into matrix manifolds defined by orthogonal and spectral constraints. We directly incorporate the geometry and natural metric of the underlying configuration space (viewed as a Lie group or a homogeneous space) in the design of the corresponding flows. Our numerical implementation relies on structure-preserving integrators that respect intrinsically the constraints geometry. The efficiency and versatility of this approach are illustrated through the anisotropic smoothing of diffusion tensor volumes in medical imaging. Note: This is the draft
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Numerical Methods for p-Harmonic Flows and Applications to Image Processing

by Luminita A. Vese, Stanley J. Osher - SIAM J. NUMER. ANAL , 2002
"... We propose in this paper an alternative approach for computing p-harmonic maps and flows: instead of solving a constrained minimization problem on S N- i, we solve an unconstrained minimization problem on the entire space of functions. This is possible, using the projection on the sphere of any arbi ..."
Abstract - Cited by 43 (6 self) - Add to MetaCart
We propose in this paper an alternative approach for computing p-harmonic maps and flows: instead of solving a constrained minimization problem on S N- i, we solve an unconstrained minimization problem on the entire space of functions. This is possible, using the projection on the sphere of any arbitrary function. Then we show how this formulation can be used in practice, for problems with both isotropic and anisotropic diffusion, with applications to image processing, using a new finite difference scheme.

Fourth order partial differential equations on general geometries

by John B. Greer, Andrea L. Bertozzi, Guillermo Sapiro - UNIVERSITY OF CALIFORNIA LOS ANGELES , 2005
"... We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces (Bertalmío, Cheng, Osher, and Sapiro 2001) to fourth order PDEs including the Cahn-Hilliard equation and a lubrication model for curved surfaces. By representing a surface in ¡ N as the ..."
Abstract - Cited by 26 (4 self) - Add to MetaCart
We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces (Bertalmío, Cheng, Osher, and Sapiro 2001) to fourth order PDEs including the Cahn-Hilliard equation and a lubrication model for curved surfaces. By representing a surface in ¡ N as the level set of a smooth function, φ ¢ we compute the PDE using only finite differences on a standard Cartesian mesh in ¡ N. The higher order equations introduce a number of challenges that are of small concern when applying this method to first and second order PDEs. Many of these problems, such as time-stepping restrictions and large stencil sizes, are shared by standard fourth order equations in Euclidean domains, but others are caused by the extreme degeneracy of the PDEs that result from this method and the general geometry. We approach these difficulties by applying convexity splitting methods, ADI schemes, and iterative solvers. We discuss in detail the differences between computing these fourth order equations and computing the first and second order PDEs considered in earlier work. We explicitly derive schemes for the linear fourth order diffusion, the Cahn-Hilliard equation for phase transition in a binary alloy, and surface tension driven flows on complex geometries. Numerical examples validating our methods are presented for these flows for data on general surfaces.
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Local solvability of a constrained gradient system of total variation

by Yoshikazu Giga, Yohei Kashima, Noriaki Yamazaki - Abstr. Appl. Anal
"... A 1−harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in RN is formulated by use of subd-ifferentials of a singular energy- the total variation. An abstract convergence result is established to show that solutions of appro ..."
Abstract - Cited by 12 (4 self) - Add to MetaCart
A 1−harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in RN is formulated by use of subd-ifferentials of a singular energy- the total variation. An abstract convergence result is established to show that solutions of approximate problem converge to a solution of the limit problem. As an application of our convergence result a local-in-time solution of 1−harmonic map flow equation is constructed as a limit of the solutions of p−harmonic (p> 1) map flow equation, when the initial data is smooth with small total variation under periodic boundary condition.
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Maintaining the Point Correspondence in the Level Set Framework

by J.-P. Pons , G. Hermosillo , R. Keriven , O. Faugeras , 2006
"... In this paper, we propose a completely Eulerian approach to maintain a point correspondence during a level set evolution. Our work is in the spirit of some recent methods (Adalsteinsson & Sethian, J. Comp. Phys. 2003; Xu & Zhao, J. Sci. Comp. 2003) for handling interfacial data on moving lev ..."
Abstract - Cited by 12 (4 self) - Add to MetaCart
In this paper, we propose a completely Eulerian approach to maintain a point correspondence during a level set evolution. Our work is in the spirit of some recent methods (Adalsteinsson & Sethian, J. Comp. Phys. 2003; Xu & Zhao, J. Sci. Comp. 2003) for handling interfacial data on moving level set interfaces. Our approach maintains an explicit backward correspondence from the evolving interface to the initial one, by advecting the initial point coordinates with the same velocity as the level set function. It leads to a system of coupled Eulerian partial differential equations. We describe in detail a robust numerical implementation of our approach, in accordance with the narrow band methodology. We show in a variety of numerical experiments that it can handle both normal and tangential velocities, large deformations, shocks, rarefactions and topological changes. The possible applications of our approach include scientific visualization, computer graphics and image processing.

A short-time Beltrami kernel for smoothing images and manifolds

by Alon Spira, Ron Kimmel, Nir Sochen - IEEE TRANS. IMAGE PROCESSING , 2007
"... We introduce a short-time kernel for the Beltrami image enhancing flow. The flow is implemented by “convolving” the image with a space dependent kernel in a similar fashion to the solution of the heat equation by a convolution with a Gaussian kernel. The kernel is appropriate for smoothing regular ( ..."
Abstract - Cited by 12 (2 self) - Add to MetaCart
We introduce a short-time kernel for the Beltrami image enhancing flow. The flow is implemented by “convolving” the image with a space dependent kernel in a similar fashion to the solution of the heat equation by a convolution with a Gaussian kernel. The kernel is appropriate for smoothing regular (flat) 2-D images, for smoothing images painted on manifolds, and for simultaneously smoothing images and the manifolds they are painted on. The kernel combines the geometry of the image and that of the manifold into one metric tensor, thus enabling a natural unified approach for the manipulation of both. Additionally, the derivation of the kernel gives a better geometrical understanding of the Beltrami flow and shows that the bilateral filter is a Euclidean approximation of it. On a practical level, the use of the kernel allows arbitrarily large time steps as opposed to the existing explicit numerical schemes for the Beltrami flow. In addition, the kernel works with equal ease on regular 2-D images and on images painted on parametric or triangulated manifolds. We demonstrate the denoising properties of the kernel by applying it to various types of images and manifolds.

A New Riemannian Setting for Surface Registration

by Martin Bauer, Martins Bruveris , 2011
"... Abstract. We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In this article we compare our approach with the diffeomorphic matching framework. In the latte ..."
Abstract - Cited by 8 (7 self) - Add to MetaCart
Abstract. We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In this article we compare our approach with the diffeomorphic matching framework. In the latter approach a deformation is prescribed on the ambient space, which then drags along an embedded surface. In contrast our metric is defined directly on the deformation vector field and can therefore be called an inner metric. We also show how to discretize the corresponding geodesic equation and compute the gradient of the cost functional using finite elements.
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A CURVILINEAR SEARCH METHOD FOR P-HARMONIC FLOWS ON SPHERES

by Donald Goldfarb, Zaiwen Wen, Wotao Yin , 2008
"... The problem of finding p-harmonic flows arises in a wide range of applications including color image (chromaticity) denoising, micromagnetics, liquid crystal theory, and directional diffusion. In this paper, we propose an innovative curvilinear search method for minimizing p-harmonic energies over ..."
Abstract - Cited by 6 (0 self) - Add to MetaCart
The problem of finding p-harmonic flows arises in a wide range of applications including color image (chromaticity) denoising, micromagnetics, liquid crystal theory, and directional diffusion. In this paper, we propose an innovative curvilinear search method for minimizing p-harmonic energies over spheres. Starting from a flow (map) on the unit sphere, our method searches along a curve that lies on the sphere in a manner similar to a standard inexact line search descent method. We show that our method is globally convergent if the step length satisfies the Armijo-Wolfe conditions. Computational tests are presented to demonstrate the efficiency of the proposed method and a variant of it that uses Barzilai-Borwein steps.
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