A mathematical and algorithmic study of the Lambertian SFS problem for orthographic and pinhole cameras

This report proposes a mathematical and algorithmic study of the Lambertian SFS problem for orthographic and pinhole cameras. Our approach is based upon the notion of viscosity solutions of Hamilton-Jacobi equations. This approach provides a mathematical framework in which we can prove the well-posedness of the problem (proof of the existence of a solution and characterization of all solutions). This mathematical approach allows also to prove the correctness of our methods. In particular, we describe a simple monotonous stability condition for the studied decentered schemes and we prove the convergence of their solutions toward the viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Also, we show that this theory naturally applies to the SFS problems. Our work extends previous work in the SFS area in three directions. First, it models the camera both as orthographic and as perspective (pinhole), i.e whereas most authors assume an orthographic projection (see for a panorama of the SFS problem up to 1989 and for more recent surveys); thus we extend the applicability of shape from shading methods to more realistic acquisition models. In particular it extends the work of and . Also, by introducing a «generic» Hamiltonian, we work in a general framework allowing to deal with both models, thereby simplifying the formalization of the problem. Second, it gives some novel mathematical formulations of this problem yielding new partial differential equations. Results about the existence and uniqueness of their solution are also obtained. Third, it allows us to come up with two new generic algorithms for computing numerical approximations of the "continuous" solution (of the «generic SFS problem») as well as a proof of their convergence toward that solution. Moreover, our two generic algorithms are able to deal with discontinuous images as well as images containing black shadows. Also, one of the algorithms we propose in this report, seems to be the most effective iterative algorithm of the SFS literature.    From a more general viewpoint, our numerical results follow from a new method for solving Hamilton-Jacobi-Bellman equations. We propose two decentered finite difference schemes. We detail the proofs of the stability and the consistency of these schemes, and the proof of the convergence of their associated algorithms

Rôle clé de la Modélisation en Shape From Shading

National audienceLe problème du "Shape From Shading" (SFS) consiste à reconstruire la forme tri-dimensionnelle d'une surface à partir d'une unique image noir et blanc de cette surface. Le "Shape From Shading" est connu comme étant un problème mal posé. Dans cet article, nous montrons que si nous modélisons le problème de manière différente de celle qui est habituellement proposée (plus précisément en prenant en compte l'atténuation de l?éclairage due à la distance), le "Shape From Shading" devient complètement bien posé. Ainsi l'information d'ombrage permet, à elle seule, de reconstruire (presque) n'importe quelle surface à partir d'une unique image (de cette surface). Aucune donnée additionnelle telle que la hauteur de la solution aux "minima" locaux (contrairement à [7, 28, 12, 31, 15]) et aucune hypothèse de régularité (contrairement à [20, 13], par exemple) ne sont nécessaires. Plus précisément, nous reformulons le problème sous la forme d?une nouvelle Equation aux Dérivées Partielles (EDP), nous développons une étude mathématique complète de cette équation, enfin nous proposons une nouvelle méthode numérique permettant de résoudre cette équation. Par ailleurs, nous prouvons la convergence de notre méthode. Nous démontrons aussi expérimentalement la pertinence de notre nouvelle méthode en l'appliquant avec succès à diverses images synthétiques et réelles

Perspective shape from shading and viscosity solutions

International audienceThis article proposes a solution of the Lambertian shape from shading (SFS) problem in the case of a pinhole camera model (performing a perspective projection). Our approach is based upon the notion of viscosity solutions of Hamilton-Jacobi equations. This approach allows us to naturally deal with nonsmooth solutions and provides a mathematical framework for proving correctness of our algorithms. Our work extends previous work in the area in three aspects. First, it models the camera as a pinhole whereas most authors assume an orthographic projection, thereby extending the applicability of shape from shading methods to more realistic images. Second, by adapting the brightness equation to the perspective problem, we obtain a new partial differential equation (PDE). Results about the existence and uniqueness of its solution are also obtained. Third, it allows us to come up with a new approximation scheme and a new algorithm for computing numerical approximations of the ?continuous? solution as well as a proof of their convergence toward that solution

Shape from Shading: a well-posed problem?

International audienceShape From Shading is known to be an ill-posed problem. Contrary to the previous work, we show here that if we model the problem in a more realistic way than it is usually done (we take into account the 1/r2 attenuation term of the lighting), Shape From Shading can be completely well-posed. Thus the shading allows to recover (almost) any surface from only one image (of this surface), without any additional data (in particular, without regularity assumptions and without the knowledge of the heights of the solution at the local "minima". More precisely, in this report we formulate the problem as that of solving a new PDE, we develop a complete mathematical study of this equation (existence and uniqueness of the solution) and we design a new provably convergent numerical method. Finally, we test our new SFS method on various synthetic images and on our database of real images of faces, with success

Reconstruction photogramétrique des formes 3D; nouveaux résultats théoriques et nouveaux algorithmes pour des projections orthographique et en perspective

National audienceCet article propose une solution au problème de ``Shape from Shading'' (SFS) en modélisant l'appareil photographique par une projection orthographique et une projection en perspective. Notre approche est basée sur la notion de solution de viscosité des équations de Hamilton-Jacobi. Cette approche fournit un cadre mathématique permettant de bien poser le problème et de prouver la pertinence de nos algorithmes. Nos travaux étendent les précédents dans le domaine sous trois aspects : 1) ils modélisent l'appareil photographique par une projection en perspective alors que la plupart des auteurs considèrent une projection orthographique (voir [Horn-Brooks:89,Zhang-Tsai-etal:99,Kozera:98] pour un panorama complet sur le ``Shape from Shading'') nos algorithmes sont ainsi plus efficaces sur des images plus réalistes. 2) La formulation ``en perspective'' aboutit, comme avec la formulation orthographique, à une équation aux dérivées partielles (EDP). Ainsi, des résultats sur l'existence et l'unicité de la solution sont obtenus. 3) Notre approche nous permet d'obtenir deux nouveaux algorithmes (un pour chaque modélisation) permettant de calculer des approximations numériques de la solution. Nous prouvons enfin la convergence de nos approximations vers la solution

A rigorous and realistic Shape From Shading method and some of its applications

This article proposes a rigorous and realistic solution of the Lambertian Shape From Shading (SFS) problem. The power of our approach is threefolds. First, our work is based on a rigorous mathematical method: we define a new notion of weak solutions (in the viscosity sense) which does not necessarily requires boundary data (contrary to the work of [rouy-tourin:92,prados-faugeras-etal:02,prados-faugeras:03,camilli-falcone:96,falcone-sagona-etal:01]) and which allows to define a solution as soon as the image is (Lipschitz) continuous (contrary to the work of [oliensis:91,dupuis-oliensis:94]). We prove the existence and uniqueness of this (new) solution and we approximate it by using a provably convergent algorithm. Second, it improves the applicability of the SFS to real images: we complete the realistic work of [prados-faugeras:03,tankus-sochen-etal:03], by modeling the problem with a pinhole camera and with a single point light source located at the optical center. This new modelization appears very relevant for applications. Moreover, our algorithm can deal with images containing discontinuities and black shadows. It is very robust to pixel noise and to errors on parameters. It is also generic: i.e. we propose a unique algorithm which can compute numerical solutions of the various perspective and orthographic SFS models. Finally, our algorithm seems to be the most efficient iterative algorithm of the SFS literature. Third, we propose three applications (in three different areas) based on our SFS method

A unifying and rigorous Shape From Shading method adapted to realistic data and applications

International audienceWe propose a new method for the Lambertian Shape From Shading (SFS) problem based on the notion of Crandall-Lions viscosity solution. This method has the advantage of requiring the knowledge of the solution (the surface to be reconstructed) only on some part of the boundary and/or of the singular set (the set of the points at maximal intensity). Moreover it unifies in an unique mathematical formulation the works of Rouy and Tourin, Falcone et al., Prados and Faugeras, based on the notion of viscosity solutions and the work of Dupuis and Oliensis dealing with classical solutions and value functions. Also, it allows to generalize their results to the "perspective SFS" problem

Mathematical formulation of REDEM algorithm for National soft landing CO2 trajectories under global carbon budgets

Global warming may be one of the greatest threats facing the human civilization. It is now widely shared that it is necessary to reduce quickly and significantly the greenhouse gas emissions to avoid uncontrolled and irreversible evolutions of climate. It has now become urgent to develop a legal instrument addressing the post-2020 period and to achieve a successful outcome in the international climate negotiations. In this paper we propose a new computational tool which provides elements of benchmarking for the climate negotiations. The model and algorithm we propose is designed on rationale elaborated by energy and climate policy experts. We detail how to estimate the parameters of the model and how this benchmarking tool could be used.Le changement climatique est probablement l'une des plus grandes menaces à laquelle la civilisation humaine doit faire face. Il est désormais largement partagé qu'il est nécessaire de réduire rapidement et de manière très importante les émissions de gaz à effet de serre afin d'éviter un emballement irréversible du système climatique. Il est aujourd'hui urgent de développer les instruments politiques et juridiques pour la période post-2020 et de faire aboutir les négociations internationales. Dans ce rapport, nous proposons un nouvel outil permettant de calculer des courbes de références des émissions nationales de gaz à effet de serre. L'algorithme et le modèle proposés se basent sur une logique développée par des économistes spécialistes de l'énergie et du changement climatique. Nous montrons comment les paramètres du modèles peuvent être estimés et comment ces outils peuvent être utilisés

National Soft Landing CO2 trajectories under global carbon budgets

One of the most important outcomes during the last Conferences of the Parties was the Durban Platform for Enhanced Action, which can be seen not only as a window of opportunity, but as a necessity to act. Our concern in the present paper is with the identification of an appropriate international climate-policy architecture in order to foster climate governance. The paper attempts to raise awareness towards the Soft landing commitment scheme which is proposed as an element of answer to the climate-policy dilemma. The scheme (REDEM for REDuction of Emissions) proposes a pathway to stabilize the emissions with a timing and a level of commitment, which are differentiated on the basis of capability (per capita income levels) and responsibility (per capita emissions). We present a couple of simulation examples which show different ways to conceive the shapes of the emissions and their rates of variation. The REDEM algorithm is designed as a tool for the benchmarking of supposed national emission reduction trajectories. The tool shows a practical way to guide the potential national trajectories, through a convergence mechanism into a comprehensible framework

A viscosity method for Shape-from-Shading without boundary data

This report proposes a solution of the Lambertian Shape From Shading (SFS) problem by designing a new mathematical framework based on the notion of viscosity solutions. The power of our approach is twofolds: 1) it defines a notion of weak solutions (in the viscosity sense) which does not necessarily require boundary data. Note that, in the previous SFS work of Rouy et al., Falcone et al. , Prados et al., the characterization of a viscosity solution and its computation require the knowledge of its values on the boundary of the image. This was quite unrealistic because in practice such values are not known. 2) it unifies the work of Rouy et al., Falcone et al., Prados et al., based on the notion of viscosity solutions and the work of Dupuis and Oliensis dealing with classical ($C^1$) solutions and value functions. Also, we generalize their work to the ``perspective SFS'' problem recently introduced by Prados and Faugeras. The notion of viscosity solutions described in this paper is obtained by slightly modifying the notion of singular viscosity solutions developped by Camilli and Siconolfi. We demonstrate the existence and the uniqueness of the solution for a class of Hamilton-Jacobi equations H(x,\nablau)=0, in a bounded open domain. Some stability results are proved. Moreover, we show that this framework allows to characterize the classical discontinuous viscosity solutions by their ``minimums''. In this report, we also propose some algorithms which provide numerical approximations of these new solutions. These provably convergent algorithms are quite robust and do not necessarily require boundary data. Finally, we have successfully applied our SFS method to real images and we have suggested a number of real-life applications