An optimal control formulation and related numerical methods for a problem in shape reconstruction (Q1333378)

The main problem considered in this paper is the construction of numerical methods for the problem of ``shape from shading'', which is a central problem in computer vision. This problem consists in reconstructing the three-dimensional shape of a surface from the brightness or intensity variation in a black-and-white photographic image of the surface. Within an idealized framework, the authors study the determination of the height function \(z: {\mathcal D}\to \mathbb{R}\). In order to identify \(z\), they formulate a pair of optimal control problems. The dynamics of the control problems is the simplest one: \(\dot\phi(t)= u(t)\). In fact, they can be considered as calculus of variations problems. The value function of the control problem allows to reconstruct easily the height function \(z\). Then the dynamic programming approach is followed. Numerical schemes are constructed. The proof of the convergence is based on a representation of the approximation to the height as a functional of a controlled Markov chain.