A minimalistic approach for fast computation of geodesic distances on triangular meshes

The computation of geodesic distances is an important research topic in Geometry Processing and 3D Shape Analysis as it is a basic component of many methods used in these areas. In this work, we present a minimalistic parallel algorithm based on front propagation to compute approximate geodesic distances on meshes. Our method is practical and simple to implement and does not require any heavy pre-processing. The convergence of our algorithm depends on the number of discrete level sets around the source points from which distance information propagates. To appropriately implement our method on GPUs taking into account memory coalescence problems, we take advantage of a graph representation based on a breadth-first search traversal that works harmoniously with our parallel front propagation approach. We report experiments that show how our method scales with the size of the problem. We compare the mean error and processing time obtained by our method with such measures computed using other methods. Our method produces results in competitive times with almost the same accuracy, especially for large meshes. We also demonstrate its use for solving two classical geometry processing problems: the regular sampling problem and the Voronoi tessellation on meshes.Comment: Preprint submitted to Computers & Graphic

A triangulation-invariant method for anisotropic geodesic map computation on surface meshes

pre-printThis paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner

An Iterative Parallel Algorithm for Computing Geodesic Distances on Triangular Meshes

Analiza el cálculo de distancias geodésicas, este es un importante tópico de investigación en geometría computacional y análisis de formas, porque muchos métodos tienen como componte el cálculo de geodésicas. Este trabajo propone y desarrolla un algoritmo iterativo, depende del número de anillos alrededor de los puntos de origen, a partir de los cuales la información de la distancia se propaga. Así, este método es particularmente eficiente para la computación de distancias geodésicas de múltiples fuentes. En los experimentos, se muestra cómo el método escala con el tamaño del problema y comparamos su error promedio y los tiempos de procesamiento con los de otros métodos encontrados en la literatura. También demostramos su uso para resolver dos problemas comunes de procesamientoTesi

Trends in Mathematical Imaging and Surface Processing

Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments

Laplacian regularized eikonal equation with Soner boundary condition on polyhedral meshes

In this paper, we propose a numerical algorithm based on a cell-centered finite volume method to compute a distance from given objects on a three-dimensional computational domain discretized by polyhedral cells. Inspired by the vanishing viscosity method, a Laplacian regularized eikonal equation is solved and the Soner boundary condition is applied to the boundary of the domain to avoid a non-viscosity solution. As the regularization parameter depending on a characteristic length of the discretized domain is reduced, a corresponding numerical solution is calculated. A convergence to the viscosity solution is verified numerically as the characteristic length becomes smaller and the regularization parameter accordingly becomes smaller. From the numerical experiments, the second experimental order of convergence in the $L^1$ norm error is confirmed for smooth solutions. Compared to solve a time-dependent form of eikonal equation, the Laplacian regularized eikonal equation has the advantage of reducing computational cost dramatically when a more significant number of cells is used or a region of interest is far away from the given objects. Moreover, the implementation of parallel computing using domain decomposition with $1$-ring face neighborhood structure can be done straightforwardly by a standard cell-centered finite volume code

Efficient Algorithms for Image and High Dimensional Data Processing Using Eikonal Equation on Graphs

International audienceIn this paper we propose an adaptation of the static eikonal equation over weighted graphs of arbitrary structure using a framework of discrete operators. Based on this formulation, we provide explicit solu- tions for the L1,L2 and L∞ norms. Efficient algorithms to compute the explicit solution of the eikonal equation on graphs are also described. We then present several applications of our methodology for image processing such as superpixels decomposition, region based segmentation or patch- based segmentation using non-local configurations. By working on graphs, our formulation provides an unified approach for the processing of any data that can be represented by a graph such as high-dimensional data

Automatic segmentation of wall structures from cardiac images

One important topic in medical image analysis is segmenting wall structures from different cardiac medical imaging modalities such as computed tomography (CT) and magnetic resonance imaging (MRI). This task is typically done by radiologists either manually or semi-automatically, which is a very time-consuming process. To reduce the laborious human efforts, automatic methods have become popular in this research. In this thesis, features insensitive to data variations are explored to segment the ventricles from CT images and extract the left atrium from MR images. As applications, the segmentation results are used to facilitate cardiac disease analysis. Specifically, 1. An automatic method is proposed to extract the ventricles from CT images by integrating surface decomposition with contour evolution techniques. In particular, the ventricles are first identified on a surface extracted from patient-specific image data. Then, the contour evolution is employed to refine the identified ventricles. The proposed method is robust to variations of ventricle shapes, volume coverages, and image quality. 2. A variational region-growing method is proposed to segment the left atrium from MR images. Because of the localized property of this formulation, the proposed method is insensitive to data variabilities that are hard to handle by globalized methods. 3. In applications, a geometrical computational framework is proposed to estimate the myocardial mass at risk caused by stenoses. In addition, the segmentation of the left atrium is used to identify scars for MR images of post-ablation.PhDCommittee Chair: Yezzi, Anthony; Committee Co-Chair: Tannenbaum, Allen; Committee Member: Egerstedt, Magnus ; Committee Member: Fedele, Francesco ; Committee Member: Stillman, Arthur; Committee Member: Vela,Patrici