The 2-D Leap-Frog: Integrability, Noise, and Digitization

Abstract

The 1-D Leap-Frog Algorithm [12] is an iterative scheme for solving a class of nonlinear optimization problems. In the present paper¹ we adapt Leap-Frog to solve an optimization problem in computer vision. The vision problem in the present paper is to recover (as far as possible) an integrable vector field (over an orthogonal grid) from a field corrupted by noise or the effects of digitization of camera images. More generally, we are dealing with integration of discrete vector fields, where every vector represents a surface normal at a grid position within a regular orthogonal grid of size N × N. Our 2-D extension of Leap-Frog is a scheme which we prove converges linearly to the optimal estimate. 1-D Leap-Frog [12] can deal with nonlinear problems such as are encountered in computer vision. In the present paper we exploit Leap-Frog’s capacity to handle large number of variables for a linear problem (in this situation Leap-Frog becomes an extension of Gauss-Seidel), and we offer a geometrical proof of convergence for the case of photometric stereo (see e.g. [10],[11]), where data is corrupted by noise or digitization. In the present paper, noise enters in an especially simple way, as Gaussian noise added to gradient estimates. So this as a first step towards more realistic (and demanding) applications, where Leap-Frog’s capacity to deal with nonlinearities is needed. The present paper also offers an alternative to other methods in photometric stereo [3], [6], [13], and [15]. The performance of 2-D Leap- Frog was demonstrated in [14] without proof of convergence: established methods are faster, but without Leap-Frog’s capacity for generalization to nonlinear problems.

This research was supported by an Australian Research Council Small Grant^a,b and an Alexander von Humboldt Research Fellowship.^b