Finsler Active Contours

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/TPAMI.2007.70713In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest (typically edges) is small. The conformal factor that is chosen depends only upon position and is in this sense isotropic. Although directional information has been studied previously for other segmentation frameworks, here, we show that if one desires to add directionality in the conformal active contour framework, then one gets a well-defined minimization problem in the case that the factor defines a Finsler metric. Optimal curves may be obtained using the calculus of variations or dynamic programming-based schemes. Finally, we demonstrate the technique by extracting roads from aerial imagery, blood vessels from medical angiograms, and neural tracts from diffusion-weighted magnetic resonance imagery

Piecewise rigid curve deformation via a Finsler steepest descent

This paper introduces a novel steepest descent flow in Banach spaces. This extends previous works on generalized gradient descent, notably the work of Charpiat et al., to the setting of Finsler metrics. Such a generalized gradient allows one to take into account a prior on deformations (e.g., piecewise rigid) in order to favor some specific evolutions. We define a Finsler gradient descent method to minimize a functional defined on a Banach space and we prove a convergence theorem for such a method. In particular, we show that the use of non-Hilbertian norms on Banach spaces is useful to study non-convex optimization problems where the geometry of the space might play a crucial role to avoid poor local minima. We show some applications to the curve matching problem. In particular, we characterize piecewise rigid deformations on the space of curves and we study several models to perform piecewise rigid evolution of curves

Tubular Surface Evolution for Segmentation of the Cingulum Bundle From DW-MRI

Presented at the 2nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy: Geometrical and Statistical Methods for Biological Shape Variability Modeling, September 6th, 2008, Kimmel Center, New York, USA.This work provides a framework for modeling and extracting the Cingulum Bundle (CB) from Diffusion-Weighted Imagery (DW-MRI) of the brain. The CB is a tube-like structure in the brain that is of potentially of tremendous importance to clinicians since it may be helpful in diagnosing Schizophrenia. This structure consists of a collection of fibers in the brain that have locally similar diffusion patterns, but vary globally. Standard region-based segmentation techniques adapted to DW-MRI are not suitable here because the diffusion pattern of the CB cannot be described by a global set of simple statistics. Active surface models extended to DW-MRI are not suitable since they allow for arbitrary deformations that give rise to unlikely shapes, which do not respect the tubular geometry of the CB. In this work, we explicitly model the CB as a tube-like surface and construct a general class of energies defined on tube-like surfaces. An example energy of our framework is optimized by a tube that encloses a region that has locally similar diffusion patterns, which differ from the diffusion patterns immediately outside. Modeling the CB as a tube-like surface is a natural shape prior. Since a tube is characterized by a center-line and a radius function, the method is reduced to a 4D (center-line plus radius) curve evolution that is computationally much less costly than an arbitrary surface evolution. The method also provides the center-line of CB, which is potentially of clinical significance

Geodesic tractography segmentation for directional medical image analysis

Acknowledgements page removed per author's request, 01/06/2014.Geodesic Tractography Segmentation is the two component approach presented in this thesis for the analysis of imagery in oriented domains, with emphasis on the application to diffusion-weighted magnetic resonance imagery (DW-MRI). The computeraided analysis of DW-MRI data presents a new set of problems and opportunities for the application of mathematical and computer vision techniques. The goal is to develop a set of tools that enable clinicians to better understand DW-MRI data and ultimately shed new light on biological processes. This thesis presents a few techniques and tools which may be used to automatically find and segment major neural fiber bundles from DW-MRI data. For each technique, we provide a brief overview of the advantages and limitations of our approach relative to other available approaches.Ph.D.Committee Chair: Tannenbaum, Allen; Committee Member: Barnes, Christopher F.; Committee Member: Niethammer, Marc; Committee Member: Shamma, Jeff; Committee Member: Vela, Patrici

Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups $SE(2)$ and $SE(3)$. In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate $SE(n)$, and which are obtained via the exponential and logarithmic map on $SE(n)$. In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on $SE(n)$. The quantitative experiments further confirm the accuracy of the approximations. Quantitative results are obtained by evaluating perceptual grouping performance of retinal blood vessels in 2D images and curves in challenging 3D synthetic volumes. The results show that 1) sub-Riemannian geometry is essential in achieving top performance and 2) that grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on $\mathbb{R}^n$ and $SE(n)$.Comment: 18 pages, 9 figures, 3 tables, in review at JMI