On the expected size of the 2d visibility complex

We study the expected size of the 2D visibility complex of randomly distributed objects in the plane. We prove that the asymptotic expected number of free bitangents (which correspond to 0-faces of the visibility complex) among unit discs (or polygons of bounded aspect ratio and similar size) is linear and exhibit bounds in terms of the density of the objects. We also make an experimental assessment of the size of the visibility complex for disjoint random unit discs. We provide experimental estimates of the onset of the linear behavior and of the asymptotic slope and y-intercept of the number of free bitangents in terms of the density of discs. Finally, we analyze the quality of our estimates in terms of the density of discs.

On the Degree of Standard Geometric Predicates for Line Transversals in 3D

International audienceIn this paper we study various geometric predicates for determining the existence of and categorizing the configurations of lines in 3D that are transversal to lines or segments. We compute the degrees of standard procedures of evaluating these predicates. The degrees of some of these procedures are surprisingly high (up to 168), which may explain why computing line transversals with finite-precision floating-point arithmetic is prone to error. Our results suggest the need to explore alternatives to the standard methods of computing these quantities

On the Computation of the 3D Visibility Skeleton

International audienceThe 3D visibility skeleton is a data structure that encodes the global visibility information of a set of 3D objects. While it is useful in answering global visibility queries, its large size often limits its practical use. In this paper, we address this issue by proposing a subset of the visibility skeleton, which is about $25\%$ to $50\%$ of the whole set. We show that the rest of the data structure can be recovered from the subset as needed, partially or completely. Our recovery method is efficient in the sense that it is output-dependent. We also prove that this subset is minimal for the complexity of our recovery method

Squelette de visibilité en trois dimensions: implantation et analyse

The visibility skeleton is a data structure that encodes global visibility information of a given scene in either 2D or 3D. While this data structure is in principle very useful in answering global visibility queries, its high order worst-case complexity, especially in 3D scene, appears to be prohibitive. However, previous theoretical research has indicated that the expected size of this data structure can be linear under some restricted conditions. This thesis advances the study of the size of the visibility skeleton, namely, using an experimental approach. We first show that, both theoretically and experimentally, the expected size of the visibility skeleton in 2D is linear, and present a linear asymptote that facilitates estimation of the size of the 2D visibility skeleton. We then study the 3D visibility skeleton defined by visual events, which is a subset of the full skeleton defined by Durand et al.. We first present an implementation to compute the vertices of that skeleton for convex disjoint polytopes in general position. This implementation makes it possible to carry on our empirical study in 3D. We consider input scenes that consist of disjoint convex polytopes that approximate randomly distributed unit spheres. We found that, in our setting, the size of the 3D visibility skeleton is quadratically related to the number of the input polytopes and linearly related to the expected silhouette size of the input polytopes. This estimate is much lower than the worst-case complexity, but higher than the expected linear complexity that we had initially hoped for. We also provide arguments that could explain the obtained complexity. We finally prove that, using the 3D visibility skeleton defined by visual events, we can compute the remaining vertices of the full skeleton in almost linear time in the size of their output.Le squelette de visibilité est une structure de donnée qui encode l'information de visibilité globale pour une scène donnée en 2D ou 3D. Cette structure de donnée est en principe très utile pour répondre à des requêtes de visiblité globale, mais elle est, en particulier en 3D, d'une complexité de haut degré dans le pire des cas qui semble prohibitive. Cependant, les recherches théoriques précédentes ont indiqué que l'espérance de la taille de cette structure de donnée peut être linéaire sous certaines conditions restreintes. Cette thèse approfondit l'étude de la taille du squelette de visibilité, au moyen d'une approche expérimentale. Nous montrons d'abord qu'aussi bien théoriquement qu'empiriquement, l'espérance de la taille du squelette de visibilité en 2D est linéaire, et présentons une asymptote affine qui facilite l'estimation de la taille du squelette de visibilité en 2D. Nous étudions ensuite le squelette de visibilité 3D défini par événement visuels, qui est un sous-ensemble du squelette complet défini par Durand et al. . Nous présen- tons tout d'abord une implantation calculant les sommets de ce squelette pour des polytopes convexes disjoints en position générale. Cette implantation nous permet de continuer notre étude empirique en 3D. Nous considérons des scènes données consis- tant en des polytopes convexes disjoints qui sont une approximation de sphères unités distribuées aléatoirement. Nous avons découvert que, dans ces conditions, la taille du squelette de visibilité 3D a une relation quadratique en le nombre de polytopes donnés, et linéaire en l'espérance de la taille de la silhouette des polytopes donnés. Cette estimation est bien plus basse que la complexité dans le pire des cas, mais plus haute que la complexité linéaire que nous espérions initialement. Nous présentons aussi des arguments qui pourraient expliquer la complexité obtenue. Nous prouvons finalement qu'en utilisant le squelette de visibilité 3D défini par événement visuels, nous pouvons calculer les sommets restants du squelette complet en temps presque linéaire en la taille du résultat

Rendering falling snow using an inverse Fourier transform

This thesis presents an image based falling snow rendering method which is based on spectral synthesis technique. By incorporating the natural falling snow motion property, that is, the image speed and size of the snowflakes are related to the depth, we develop a tent-like surface in frequency domain. We synthesize the power spectrum along the tent-like surface and use IFFT to bring the data function back to space-time domain, thus attain a motion parallax image sequence. Treating the motion parallax as an opacity function, we can composite it with an existing video sequence and turn it into a snowing scene. Treating the motion parallax as a stimulus for the psychophysical study, it could serve as a complex yet natural scene-like stimulus, and therefore being expected to give a new perspective to the psychophysical study