8 research outputs found
An algorithm for optimal transport between a simplex soup and a point cloud
We propose a numerical method to find the optimal transport map between a
measure supported on a lower-dimensional subset of R^d and a finitely supported
measure. More precisely, the source measure is assumed to be supported on a
simplex soup, i.e. on a union of simplices of arbitrary dimension between 2 and
d. As in [Aurenhammer, Hoffman, Aronov, Algorithmica 20 (1), 1998, 61--76] we
recast this optimal transport problem as the resolution of a non-linear system
where one wants to prescribe the quantity of mass in each cell of the so-called
Laguerre diagram. We prove the convergence with linear speed of a damped
Newton's algorithm to solve this non-linear system. The convergence relies on
two conditions: (i) a genericity condition on the point cloud with respect to
the simplex soup and (ii) a (strong) connectedness condition on the support of
the source measure defined on the simplex soup. Finally, we apply our algorithm
in R^3 to compute optimal transport plans between a measure supported on a
triangulation and a discrete measure. We also detail some applications such as
optimal quantization of a probability density over a surface, remeshing or
rigid point set registration on a mesh
Light in Power: A General and Parameter-free Algorithm for Caustic Design
We present in this paper a generic and parameter-free algorithm to
efficiently build a wide variety of optical components, such as mirrors or
lenses, that satisfy some light energy constraints. In all of our problems, one
is given a collimated or point light source and a desired illumination after
reflection or refraction and the goal is to design the geometry of a mirror or
lens which transports exactly the light emitted by the source onto the target.
We first propose a general framework and show that eight different optical
component design problems amount to solving a light energy conservation
equation that involves the computation of visibility diagrams. We then show
that these diagrams all have the same structure and can be obtained by
intersecting a 3D Power diagram with a planar or spherical domain. This allows
us to propose an efficient and fully generic algorithm capable to solve these
eight optical component design problems. The support of the prescribed target
illumination can be a set of directions or a set of points located at a finite
distance. Our solutions satisfy design constraints such as convexity or
concavity. We show the effectiveness of our algorithm on simulated and
fabricated examples
Transport optimal semi-discret et applications en optique anidolique
In this thesis, we are interested in solving many inverse problems arising inoptics. More precisely, we are interested in designing optical components such as mirrors andlenses that satisfy some light conservation constraints meaning that we want to control thereflected (or refracted) light in order match a prescribed intensity. This has applications incar headlight design or caustic design for example. We show that optical component designproblems can be recast as optimal transport ones for different cost functions and we explainhow this allows to study the existence and the regularity of the solutions of such problems. Wealso show how, using computational geometry, we can use an efficient numerical method namelythe damped Newton’s algorithm to solve all these problems. We will end up with a singlegeneric algorithm able to efficiently build an optical component with a prescribed reflected(or refracted) illumination. We show the convergence of the Newton’s algorithm to solve theoptimal transport problem when the source measure is supported on a finite union of simplices.We then describe the common relation between eight optical component design problemsand show that they can all be seen as discrete Monge-Ampère equations. We also apply theNewton’s method to optical component design and show numerous simulated and fabricatedexamples. Finally, we look at a problem arising in computational optimal transport namelythe choice of the initial weights. We develop three simple procedures to find “good” initialweights which can be used as a starting point in computational optimal transport algorithms.Dans cette thèse, nous nous intéressons à la résolution de nombreux problèmes d’optique anidolique. Plus précisément, il s’agit de construire des composants optiques qui satisfont des contraintes d’illumination à savoir que l’on veut que la lumière réfléchie(ou réfractée) par ce composant corresponde à une distribution fixée en avance. Comme applications, nous pouvons citer la conception de phares de voitures ou de caustiques. Nous montrons que ces problèmes de conception de composants optiques peuvent être vus comme des problèmes de transport optimal et nous expliquons en quoi cette formulation permet d’étudier l’existence et la régularité des solutions. Nous montrons aussi comment, en utilisant des outils de géométrie algorithmique, nous pouvons utiliser une méthode numérique efficace, la méthode de Newton amortie, pour résoudre tous ces problèmes. Nous obtenons un algorithme générique capable de construire efficacement un composant optique qui réfléchit (ou réfracte)une distribution de lumière prescrite. Nous montrons aussi la convergence de l’algorithme de Newton pour résoudre le problème de transport optimal dans le cas où le support de la mesure source est une union finie de simplexes. Nous décrivons également la relation commune qui existe entre huit différents problèmes de conception de composants optiques et montrons qu’ils peuvent tous être vus comme des équations de Monge-Ampère discrètes. Nous appliquons aussi la méthode de Newton à de nombreux problèmes de conception de composants optiques sur différents exemples simulés ainsi que sur des prototypes physiques. Enfin, nous nous intéressons à un problème apparaissant en transport optimal numérique à savoir le choix du point initial. Nous développons trois méthodes simples pour trouver de “bons” points initiaux qui peuvent être ensuite utilisés comme point de départ dans des algorithmes de résolution de transport optimal
Semi-discrete optimal transport and applications in non-imaging optics
Dans cette thèse, nous nous intéressons à la résolution de nombreux problèmes d’optique anidolique. Plus précisément, il s’agit de construire des composants optiques qui satisfont des contraintes d’illumination à savoir que l’on veut que la lumière réfléchie(ou réfractée) par ce composant corresponde à une distribution fixée en avance. Comme applications, nous pouvons citer la conception de phares de voitures ou de caustiques. Nous montrons que ces problèmes de conception de composants optiques peuvent être vus comme des problèmes de transport optimal et nous expliquons en quoi cette formulation permet d’étudier l’existence et la régularité des solutions. Nous montrons aussi comment, en utilisant des outils de géométrie algorithmique, nous pouvons utiliser une méthode numérique efficace, la méthode de Newton amortie, pour résoudre tous ces problèmes. Nous obtenons un algorithme générique capable de construire efficacement un composant optique qui réfléchit (ou réfracte)une distribution de lumière prescrite. Nous montrons aussi la convergence de l’algorithme de Newton pour résoudre le problème de transport optimal dans le cas où le support de la mesure source est une union finie de simplexes. Nous décrivons également la relation commune qui existe entre huit différents problèmes de conception de composants optiques et montrons qu’ils peuvent tous être vus comme des équations de Monge-Ampère discrètes. Nous appliquons aussi la méthode de Newton à de nombreux problèmes de conception de composants optiques sur différents exemples simulés ainsi que sur des prototypes physiques. Enfin, nous nous intéressons à un problème apparaissant en transport optimal numérique à savoir le choix du point initial. Nous développons trois méthodes simples pour trouver de “bons” points initiaux qui peuvent être ensuite utilisés comme point de départ dans des algorithmes de résolution de transport optimal.In this thesis, we are interested in solving many inverse problems arising inoptics. More precisely, we are interested in designing optical components such as mirrors andlenses that satisfy some light conservation constraints meaning that we want to control thereflected (or refracted) light in order match a prescribed intensity. This has applications incar headlight design or caustic design for example. We show that optical component designproblems can be recast as optimal transport ones for different cost functions and we explainhow this allows to study the existence and the regularity of the solutions of such problems. Wealso show how, using computational geometry, we can use an efficient numerical method namelythe damped Newton’s algorithm to solve all these problems. We will end up with a singlegeneric algorithm able to efficiently build an optical component with a prescribed reflected(or refracted) illumination. We show the convergence of the Newton’s algorithm to solve theoptimal transport problem when the source measure is supported on a finite union of simplices.We then describe the common relation between eight optical component design problemsand show that they can all be seen as discrete Monge-Ampère equations. We also apply theNewton’s method to optical component design and show numerous simulated and fabricatedexamples. Finally, we look at a problem arising in computational optimal transport namelythe choice of the initial weights. We develop three simple procedures to find “good” initialweights which can be used as a starting point in computational optimal transport algorithms
Initialization Procedures for Discrete and Semi-Discrete Optimal Transport
International audienc
Approximation of Digital Surfaces by a Hierarchical Set of Planar Patches
International audienceWe show that the plane-probing algorithms introduced in Lachaud et al. (J. Math. Imaging Vis., 59, 1, 23-39, 2017), which compute the normal vector of a digital plane from a starting point and a set-membership predicate, are closely related to a three-dimensional generalization of the Euclidean algorithm. In addition, we show how to associate with the steps of these algorithms generalized substitutions, i.e., rules that replace square faces by unions of square faces, to build finite sets of elements that periodically generate digital planes. This work is a first step towards the incremental computation of a hierarchy of pieces of digital plane that locally fit a digital surface
An Optimized Framework for Plane-Probing Algorithms
International audienceA plane-probing algorithm computes the normal vector of a digital plane from a starting point and a predicate "Is a point x in the digital plane?". This predicate is used to probe the digital plane as locally as possible and decide on-the-fly the next points to consider. However, several existing plane-probing algorithms return the correct normal vector only for some specific starting points and an approximation otherwise, e.g. the Hand R-algorithm proposed in Lachaud et al. (J. Math. Imaging Vis., 59, 1, 23-39, 2017). In this paper, we present a general framework for these plane-probing algorithms that provides a way of retrieving the correct normal vector from any starting point, while keeping their main features. There are O(ω log ω) calls to the predicate in the worst-case scenario, where ω is the thickness of the underlying digital plane, but far fewer calls are experimentally observed on average. In the context of digital surface analysis, the resulting algorithm is expected to be of great interest for normal estimation and shape reconstruction