Optimal Separable Algorithms to Compute the Reverse Euclidean Distance Transformation and Discrete Medial Axis in Arbitrary Dimension

In binary images, the distance transformation (DT) and the geometrical skeleton extraction are classic tools for shape analysis. In this paper, we present time optimal algorithms to solve the reverse Euclidean distance transformation and the reversible medial axis extraction problems for $d$-dimensional images. We also present a $d$-dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape

Efficient Distance Transformation for Path-based Metrics

In many applications, separable algorithms have demonstrated their efficiency to perform high performance volumetric processing of shape, such as distance transformation or medial axis extraction. In the literature, several authors have discussed about conditions on the metric to be considered in a separable approach. In this article, we present generic separable algorithms to efficiently compute Voronoi maps and distance transformations for a large class of metrics. Focusing on path-based norms (chamfer masks, neighborhood sequences...), we propose efficient algorithms to compute such volumetric transformation in dimension $n$. We describe a new $O(n\cdot N^n\cdot\log{N}\cdot(n+\log f))$ algorithm for shapes in a $N^n$ domain for chamfer norms with a rational ball of $f$ facets (compared to $O(f^{\lfloor\frac{n}{2}\rfloor}\cdot N^n)$ with previous approaches). Last we further investigate an even more elaborate algorithm with the same worst-case complexity, but reaching a complexity of $O(n\cdot N^n\cdot\log{f}\cdot(n+\log f))$ experimentally, under assumption of regularity distribution of the mask vectors

Discrete bisector function and Euclidean skeleton in 2D and 3D

International audienceWe propose a new definition and an exact algorithm for the discrete bisector function, which is an important tool for analyzing and filtering Euclidean skeletons. We also introduce a new thinning algorithm which produces homotopic discrete Euclidean skeletons. These algorithms, which are valid both in 2D and 3D, are integrated in a skeletonization method which is based on exact transformations, allows the filtering of skeletons, and is computationally efficient

Medial Axis Approximation with Constrained Centroidal Voronoi Diagrams On Discrete Data

International audienceIn this paper, we present a novel method for me-dial axis approximation based on Constrained Centroidal Voronoi Diagram of discrete data (image, volume). The proposed approach is based on the shape boundary subsampling by a clustering approach which generates a Voronoi Diagram well suited for Medial Axis extraction. The resulting Voronoi Diagram is further filtered so as to capture the correct topology of the medial axis. The resulting medial axis appears largely invariant with respect to typical noise conditions in the discrete data. The method is tested on various synthetic as well as real images. We also show an application of the approximate medial axis to the sizing field for triangular and tetrahedral meshing

Décomposition volumique d'images pour l'étude de la microstructure de la neige

Les avalanches de neige sont des phénomènes naturels complexes dont l'occurrence s'explique principalement par la structure et les propriétés du manteau neigeux. Afin de mieux comprendre les évolutions de ces propriétés au cours du temps, il est important de pouvoir caractériser la microstructure de la neige, notamment en termes de grains et de ponts de glace les reliant. Dans ce contexte, l'objectif de cette thèse est la décomposition d'échantillons de neige en grains individuels à partir d'images 3-D de neige obtenues par microtomographie X. Nous présentons ici deux méthodes de décomposition utilisant des algorithmes de géométrie discrète. Sur la base des résultats de ces segmentations, certains paramètres, comme la surface spécifique et la surface spécifique de contact entre grains sont ensuite estimés sur des échantillons de neiges variées. Ces méthodes de segmentation ouvrent de nouvelles perspectives pour la caractérisation de la microstructure de la neige, de ses propriétés, ainsi que de leur évolution au cours du temps.Snow avalanches are complex natural phenomena whose occurrence is mainly due to the structure and properties of the snowpack. To better understand the evolution of these properties over time, it is important to characterize the microstructure of snow, especially in terms of grains and ice necks that connect them. In this context, the objective of this thesis is the decomposition of snow samples into individual grains from 3-D images of snow obtained by X-ray microtomography. We present two decomposition methods using algorithms of discrete geometry. Based on the results of these segmentations, some parameters such as the specific surface area and the specific contact area between grains are then estimated from samples of several snow types. These segmentation methods offer new outlooks for the characterization of the microstructure of snow, its properties, and its time evolution

Medial Axis Lookup Table and Test Neighborhood Computation for 3D Chamfer Norms

International audienceChamfer distances are discrete distances based on the propagation of local distances, or weights defined in a mask. The medial axis, i.e. the centers of the maximal disks (disks which are not contained in any other disk), is a powerful tool for shape representation and analysis. The extraction of maximal disks is performed in the general case with comparison tests involving look-up tables representing the covering relation of disks in a local neighborhood. Although look-up table values can be computed efficiently, the computation of the look-up table neighborhood tend to be very time-consuming. By using polytope descriptions of the chamfer disks, the necessary operations to extract the look-up tables are greatly reduced

Fast and Memory-Efficient Voronoi Diagram Construction on Triangle Meshes

© 2017 The Author(s) Computer Graphics Forum © 2017 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. Geodesic based Voronoi diagrams play an important role in many applications of computer graphics. Constructing such Voronoi diagrams usually resorts to exact geodesics. However, exact geodesic computation always consumes lots of time and memory, which has become the bottleneck of constructing geodesic based Voronoi diagrams. In this paper, we propose the window-VTP algorithm, which can effectively reduce redundant computation and save memory. As a result, constructing Voronoi diagrams using the proposed window-VTP algorithm runs 3–8 times faster than Liu et al.'s method [LCT11] , 1.2 times faster than its FWP-MMP variant and more importantly uses 10–70 times less memory than both of them

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor