Invariant geometric evolutions of surfaces and volumetric smoothing

Includes bibliographical references (p. 21-25).Supported by the National Science Foundation. DMS-9116672 DMS-9204192 DMS-8811084 ECS-9122106 Supported by the Air Force Office of Scientific Research. AFOSR-90-0024 F49620-94-1-00S8DEF Supported by the Army Research Office. DAAL03-92-G-0115 DAAL03-91-G-0019 DAAH04-93-G-0332 DAAH04-94-G-0054 Supported by the Rothschild Foundation-Yad Hanadiv and by Image Evolutions, Ltd.Peter J. Olver, Guillermo Sapiro, Allen Tannenbaum

Optical Flow on Moving Manifolds

Optical flow is a powerful tool for the study and analysis of motion in a sequence of images. In this article we study a Horn-Schunck type spatio-temporal regularization functional for image sequences that have a non-Euclidean, time varying image domain. To that end we construct a Riemannian metric that describes the deformation and structure of this evolving surface. The resulting functional can be seen as natural geometric generalization of previous work by Weickert and Schn\"orr (2001) and Lef\`evre and Baillet (2008) for static image domains. In this work we show the existence and wellposedness of the corresponding optical flow problem and derive necessary and sufficient optimality conditions. We demonstrate the functionality of our approach in a series of experiments using both synthetic and real data.Comment: 26 pages, 6 figure

A Geometric Snake Model for Segmentation of Medical Imagery

Abstract-In this note, we employ the new geometric active contour models formulated in [25] and [26] for edge detection and segmentation of magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound medical imagery. Our method is based on defining feature-based metrics on a given image which in turn leads to a novel snake paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus, the snake is attracted very quickly and efficiently to the desired feature

Segmentation and polyp detection in virtual colonoscopy : a complete system for computer aided diagnosis

El cancer colorectal es una de las mayores causas de muerte por cancer en el mundo. La deteccion temprana de polipos es fundamental para su tratamiento, permitiendo alcanzar tasas del 90% de curabilidad. La tecnica habitual para la deteccion de polipos, debido a su elevada performance, es la colonoscopia optica (tecnica invasiva y extremadamente cara). A mediados de los '90 surge la tecnica denominada colonoscopia virtual. Esta tecnica consiste en la reconstruccion 3D del colon a partir de cortes de tomografia computada. Es por ende una tecnica no invasiva, y relativamente barata, pero la cantidad de falsos positivos y falsos negativos producida por estos metodos esta muy por encima de los maximos aceptados en la practica medica. Los avances recientes en las tecnicas de imagenologia parecerian hacer posible la reduccion de estas tasas. Como consecuencia de esto, estamos asistiendo a un nuevo interes por la colonoscopia virtual. En este trabajo se presenta un sistema completo de diagnostico asistido por computadora. La primera etapa del sistema es la segmentacion, que consiste en la reconstruccion 3D de la superficie del colon a partir del volumen tomografico. El aporte principal en este paso es el suavizado de la imagen. A partir de la superficie, se detectan aquellas zonas candidatas de ser polipos mediante una estrategia multi-escala que permite delinear con precision la zona. Luego para cada candidato se extraen caracteristicas geometricas y de textura, que son calculadas tambien en el tejido que rodea la zona a efectos de compararlas. Finalmente las zonas candidatas se clasifican utilizando SVM. Los resultados obtenidos son prometedores, permitiendo detectar un 100% de los polipos mayoresColorectal cancer is the second leading cause of cancer-related death in the United States, and the third cause worldwide. The early detection of polyps is fundamental, allowing to reduce mortality rates up to 90%. Nowadays, optical colonoscopy is the most used detection method due in part to its relative high performance. Virtual Colonoscopy is a promising alternative technique that emerged in the 90's. It uses volumetric Computed Tomographic data of the cleansed and air-distended colon, and the examination is made by a specialist from the images in a computer. Therefore, this technique is less invasive and less expensive than optical colonoscopy, but up to now the false positive and false negative rates are above the accepted medical limits. Recent advances in imaging techniques have the potential to reduce these rates; consequently, we are currently re-experiencing an increasing interest in Virtual Colonoscopy. In this work we propose a complete pipeline for a Computer-Aided Detection algorithm. The system starts with a novel and simple segmentation step. We then introduce geometrical and textural features that take into account not only the candidate polyp region, but the surrounding area at multiple scales as well. This way, our proposed CAD algorithm is able to accurately detect candidate polyps by measuring local variations of these features. Candidate patches are then classi ed using SVM. The whole algorithm is completely automatic and produces state-of-the-art results, achieving 100% sensitivity for polyps greater than 6mm in size with less than one false positive per case, and 100% sensitivity for polyps greater than 3mm in size with 2:2 false positives per case

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

Non-rigid registration by geometry-constrained diffusion.

. Assume that only partial knowledge about a non-rigid registration is given so that certain points, curves, or surfaces in one 3D image map to certain certain points, curves, or surfaces in another 3D image. We are facing the aperture problem because along the curves and surfaces, point correspondences are not given. We will advocate the viewpoint that the aperture and the 3D interpolation problem may be solved simultaneously by finding the simplest displacement field. This is obtained by a geometry-constrained diffusion which yields the simplest displacement field in a precise sense. The point registration obtained may be used for growth modelling, shape statistics, or kinematic interpolation. The algorithm applies to geometrical objects of any dimensionality. We may thus keep any number of fiducial points, curves, and/or surfaces fixed while finding the simplest registration. Examples of inferred point correspondences in a longitudinal growth study of the mandible are g..

A Summary of Geometric Level-Set Analogues for a General Class of Parametric Active Contour and Surface Models

© 2001 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.DOI: 10.1109/VLSM.2001.938888Geometric active contours (GACs) and surfaces (GASs) implemented via level set techniques enjoy many advantages over parametric active contours (PACs) and surfaces (PASs), such as computational stability and the ability to change topology during deformation. While many capabilities of earlier PACs and PASs have been reproduced by various GACs and GASs, and while relationships have been discussed for a variety of specific cases, a comprehensive accounting of the connections between these two worlds (particularly regarding rigid forces) has not been consolidated thus far. In this paper we present the precise mathematical relationships between the two for an extensive family of both active contour and surface models, encompassing spatially-varying coefficients, both tension and rigidity, and both conservative and non-conservative external forces. The result is a very general geometric formulation for which the intuitive design principles of PACs and PASs can be applied. We also point out which type of PAC and PAS methodologies cannot be adapted to the geometric level set framework. We conclude by demonstrating several geometric adaptations of specfic PACs and PASs in several simulations