Maurizio Falcone

Numerical methods for shape-from-shading: A new survey with benchmarks

Many algorithms have been suggested for the shape-from-shading problem, and some years have passed since the publication of the survey paper by Zhang et al. [R. Zhang, P.-S. Tsai, J.E. Cryer, M. Shah, Shape from shading: a survey, IEEE Transactions on Pattern Analysis and Machine Intelligence 21 (8) (1999) 690-706]. In this new survey paper, we try to update their presentation including some recent methods which seem to be particularly representative of three classes of methods: methods based on partial differential equations, methods using optimization and methods approximating the image irradiance equation. One of the goals of this paper is to set the comparison of these methods on a firm basis. To this end, we provide a brief description of each method, highlighting its basic assumptions and mathematical properties. Moreover, we propose some numerical benchmarks in order to compare the methods in terms of their efficiency and accuracy in the reconstruction of surfaces corresponding to synthetic, as well as to real images.

Convergence Analysis for a Class of High-Order Semi-Lagrangian Advection Schemes

The convergence properties of a class of high-order semi-Lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions. We review the general convergence results for discrete-time approximation schemes belonging to that class and we prove some a priori estimates in

Semi-Lagrangian approximation schemes for linear and Hamilton-Jacobi equations

L^\infty

An approximation scheme for the minimum time function

and L2 for the rate of convergence of fully discrete schemes. We prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions. Several examples of schemes and numerical tests are presented.

Numerical Methods for Pursuit-Evasion Games via Viscosity Solutions

This largely self-contained book provides a unified framework of semi-Lagrangian strategy for the approximation of hyperbolic PDEs, with a special focus on Hamilton-Jacobi equations. The authors provide a rigorous discussion of the theory of viscosity solutions and the concepts underlying the construction and analysis of difference schemes; they then proceed to high-order semi-Lagrangian schemes and their applications to problems in fluid dynamics, front propagation, optimal control, and image processing. The developments covered in the text and the references come from a wide range of literature.

Fast Semi-Lagrangian Schemes for the Eikonal Equation and Applications

This paper presents an approximation scheme for the nonlinear minimum time problem with compact target. The scheme is derived from a discrete dynamic programming principle and the main convergence result is obtained by applying techniques related to discontinuous viscosity solutions for Hamilton–Jacobi equations. The convergence is proved under general controllability assumptions on both the continuous-time and the discrete-time systems. An explicit sufficient condition on the system and the target ensuring the desired controllability is given. This condition is shown to be necessary and sufficient for the Lipschitz continuity of the minimum time function if the target is smooth. An extension to the case of a point-shaped target is given.

Discrete Dynamic Programming and Viscosity Solutions of the Bellman Equation

We present a class of numerical schemes for the Isaacs equation of pursuit-evasion games. We consider continuous value functions, where the solution is interpreted in the viscosity sense, as well as discontinuous value functions, where the notion of viscosity envelope-solution is needed. The convergence of the approximation scheme to the value function of the game is proved in both cases. A priori estimates of the convergence in L∞ are established when the value function is Holder continuous. We also treat problems with state constraints and discuss several issues concerning the implementation of the approximation scheme, the synthesis of approximate feedback controls, and the approximation of optimal trajectories. The efficiency of the algorithm is illustrated by a number of numerical tests, either in the case of one player (i.e., minimum time problem) or for some 2-players games.

Level sets of viscosity solutions: some applications to fronts and rendez-vous problems

We introduce and analyze a fast version of the semi-Lagrangian algorithm for front propagation originally proposed in [M. Falcone, “The minimum time problem and its applications to front propagation,” in Motion by Mean Curvature and Related Topics, A. Visintin and G. Buttazzo, eds., de Gruyter, Berlin, 1994, pp. 70-88]. The new algorithm is obtained using the local definition of the approximate solution typical of semi-Lagrangian schemes and redefining the set of “neighboring nodes” necessary for fast marching schemes. A new proof of convergence is needed since that definition produces a new narrow band centered at the interphase which is larger than the one used in fast marching methods based on finite differences. We show that the new algorithm converges to the viscosity solution of the problem and that its complexity is

NUMERICAL METHODS FOR DIFFERENTIAL GAMES BASED ON PARTIAL DIFFERENTIAL EQUATIONS

O(N \log N_{nb})

Fully Discrete Schemes for the Value Function of Pursuit-Evasion Games

, as it is for the fast marching method based on finite difference (