Bijective rigid motions of the 2D Cartesian grid

International audienceRigid motions are fundamental operations in image processing. While they are bijective and isometric in R^2, they lose these properties when digitized in Z^2. To investigate these defects, we first extend a combinatorial model of the local behavior of rigid motions on Z^2, initially proposed by Nouvel and Rémila for rotations on Z^2. This allows us to study bijective rigid motions on Z^2, and to propose two algorithms for verifying whether a given rigid motion restricted to a given finite subset of Z^2 is bijective

Topological alterations of 3D digital images under rigid transformations

National audienceRigid transformations in R^n are known to preserve the shape, and are often applied to digital images. However, digitized rigid transformations, defined as digital functions from Z^n to Z^n do not preserve shapes in general\string; indeed, they are almost never bijective and thus alter the topology. In order to understand the causes of such topological alterations, we first study the possible loss of voxel information and modification of voxel adjacencies induced by applications of digitized rigid transformations to 3D digital images. We then show that even very simple structured images such as digital half-spaces may not preserve their topology under these transformations. This signifies that a simple extension of the two-dimensional solution for topology preservation cannot be made in three dimensions

ConSLAM: Periodically Collected Real-World Construction Dataset for SLAM

International audienc

Déplacements sur des espaces discrets

In digital geometry, Euclidean objects are represented by their discrete approximations, e.g. subsets of the lattice of integers. Rigid motions of such sets have to be defined as maps from and onto a given discrete space. One way to design such motions is to combine continuous rigid motions defined on Euclidean space with a digitization operator. However, digitized rigid motions often no longer satisfy properties of their continuous siblings. Indeed, due to digitization, such transformations do not preserve distances, while bijectivity and point connectivity are generally lost. In the context of 2D discrete spaces, we study digitized rigid motions on the lattices of Gaussian and Eisenstein integers. We characterize bijective digitized rigid motions on the integer lattice, and bijective digitized rotations on the regular hexagonal lattice. Also, we compare the information loss induced by non-bijective digitized rigid motions defined on both lattices. Yet, for practical applications, the relevant information is not global bijectivity, but bijectivity of a digitized rigid motion restricted to a given finite subset of a lattice. We propose two algorithms testing that condition for subsets of the integer lattice, and a third algorithm providing optimal angle intervals that preserve this restricted bijectivity. We then focus on digitized rigid motions on 3D integer lattice. First, we study at a local scale geometric and topological defects induced by digitized rigid motions. Such an analysis consists of generating all the images of a finite digital set under digitized rigid motions. This problem amounts to computing an arrangement of hypersurfaces in a 6D parameter space. The dimensionality and degenerate cases make the problem practically unsolvable for state-of-the-art techniques. We propose an ad hoc solution, which mainly relies on parameter uncoupling, and an algorithm for computing sample points of 3D connected components in an arrangement of second degree polynomials. Finally, we focus on the open problem of determining whether a 3D digitized rotation is bijective or not. In our approach, we explore arithmetic properties of Lipschitz quaternions. This leads to an algorithm which answers whether a given digitized rotation—related to a Lipschitz quaternion—is bijective or notEn géométrie discrète, les objets euclidiens sont représentés par leurs approximations discrètes, telles que des sous-ensembles du réseau des points à coordonnées entières. Les déplacements de ces ensembles doivent être définis comme des applications depuis et sur un espace discret donné. Une façon de concevoir de telles transformations est de combiner des déplacements continus définis sur un espace euclidien avec un opérateur de discrétisation. Cependant, les déplacements discrétisés ne satisfont souvent plus les propriétés de leurs équivalents continus. En effet, en raison de la discrétisation, de telles transformations ne préservent pas les distances, et la bijectivité et la connexité entre les points sont généralement perdues. Dans le contexte des espaces discrets 2D, nous étudions des déplacements discrétisés sur les réseaux d'entiers de Gauss et d'Eisenstein. Nous caractérisons les déplacements discrétisés bijectifs sur le réseau carré, et les rotations bijectives discrétisées sur le réseau hexagonal régulier. En outre, nous comparons les pertes d'information induites par des déplacements discrétisés non bijectifs définis sur ces deux réseaux. Toutefois, pour des applications pratiques, l'information pertinente n'est pas la bijectivité globale, mais celle d'un déplacement discrétisé restreint à un sous-ensemble fini donné d'un réseau. Nous proposons deux algorithmes testant cette condition pour les sous-ensembles du réseau entier, ainsi qu'un troisième algorithme fournissant des intervalles d'angles optimaux qui préservent cette bijectivité restreinte. Nous nous concentrons ensuite sur les déplacements discrétisés sur le réseau cubique 3D. Tout d'abord, nous étudions à l'échelle locale des défauts géométriques et topologiques induits par des déplacements discrétisés. Une telle analyse consiste à générer toutes les images d'un ensemble du réseau fini sous des déplacements discrétisés. Un tel problème revient à calculer un arrangement d'hypersurfaces dans un espace de paramètres de dimension six. La dimensionnalité et les cas dégénérés rendent le problème insoluble, en pratique, par les techniques usuelles. Nous proposons une solution ad hoc reposant sur un découplage des paramètres, et un algorithme pour calculer des points d'échantillonnage de composantes connexes 3D dans un arrangement de polynômes du second degré. Enfin, nous nous concentrons sur le problème ouvert de déterminer si une rotation discrétisée 3D est bijective ou non. Dans notre approche, nous explorons les propriétés arithmétiques des quaternions de Lipschitz. Ceci conduit à un algorithme qui détermine si une rotation discrétisée donnée, associée à un quaternion de Lipschitz, est bijective ou no

copyme/RigidMotionsMapleTools 0.1

The first release comes after we succeeded to compute neighborhood motion maps of 18-neighborhood

Lipschitz quaternions in the range [−10, 10]^4, which induce bijective 3D digitized rotations

<p>The file contains Lipschitz quaternions in the range [−10, 10]^4, such that they induce bijective 3D digitized rotations. It is a comma-separated values file format such that each line contains a different quaternion.</p

CataEx - Export Code for Google Earth Engine

&lt;p&gt;Please read &lt;strong&gt;CataEx_readme.txt&lt;/strong&gt; includedd in the ZIP-file.&lt;/p&gt;This work is part of the project "Global Assessment of Glacier-Landslide Interactions and Associated Geo-Hazards" (2021/42/E/ST10/00186), funded by the Polish National Science Center (NCN)

Bijective digitized rigid motions on subsets of the plane

International audienceRigid motions in $\mathbb{R}^2$ are fundamental operations in 2D image processing. They satisfy many properties: in particular, they are isometric and therefore bijective. Digitized rigid motions, however, lose these two properties. To investigate the lack of injectivity or surjectivity and more generally their local behavior, we extend the framework initially proposed by Nouvel and R\'emila to the case of digitized rigid motions. Yet, for practical applications, the relevant information is not global bijectivity, which is seldom achieved, but bijectivity of the motion restricted to a given finite subset of $\mathbb{Z}^2$. We propose two algorithms testing that condition. Finally, because rotation angles are rarely given with infinite precision, we propose a third algorithm providing optimal angle intervals that preserve this restricted bijectivity

Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image

International audienceRigid motions are fundamental operations in image processing. While bijective and isometric in $\mathbb{R}^3$, they lose these properties when digitized in $\mathbb{Z}^3$. To understand how the digitization of 3D rigid motions affects the topology and geometry of a chosen image patch, we classify the rigid motions according to their effect on the image patch. This classification can be described by an arrangement of hypersurfaces in the parameter space of 3D rigid motions of dimension six. However, its high dimensionality and the existence of degenerate cases make a direct application of classical techniques, such as cylindrical algebraic decomposition or critical point method, difficult. We show that this problem can be first reduced to computing sample points in an arrangement of quadrics in the 3D parameter space of rotations. Then we recover information about remaining three parameters of translation. We implemented an ad-hoc variant of state-of-the-art algorithms and applied it to an image patch of cardinality $7$. This leads to an arrangement of 81 quadrics and we recovered the classification in less than one hour on a machine equipped with 40 cores

Bijectivity certification of 3D digitized rotations

International audienceEuclidean rotations in R^n are bijective and isometric maps. Nevertheless, they lose these properties when digitized in Z^n. For n=2, the subset of bijective digitized rotations has been described explicitly by Nouvel and R\émila and more recently by Roussillon and Coeurjolly. In the case of 3D digitized rotations, the same characterization has remained an open problem. In this article, we propose an algorithm for certifying the bijectivity of 3D digitized rational rotations using the arithmetic properties of the Lipschitz quaternions