Evaluation of Algorithms for Linear Shape from Shading

We analyse different sequential algorithms for the recovery of object shape from a single shading pattern generated under the assumption of a linear reflectance map. The algorithms are based on the finite difference approximation of the derivatives. They operate on a rectangular discrete image (or part of it) and use the height of the sought-after surface along a curve in the image (image boundary) as initial data. The evaluation of different numerical schemes is achieved by comparing stability, convergence, and domains of influence of each scheme in question. The relative difficulty of handling a linear case indicates that the case of non-linear reflectance maps is far from being trivial. * The University of Western Australia, Department of Computer Science, Nedlands, WA 6907 Australia ** The University of Auckland, Tamaki Campus, Computing and Information Technology Research, Computer Vision Unit, Auckland, New Zealand 1 Introduction In this paper, we present some results concerning the shape-from-shading problem in which the re ectance map is linear. Such a special case arises e.g. in the study of the maria of the moon (see [1, Subsections 10.9 and 11.1.2]). If a small portion of a surface, described by the graph of a function u, having re ectivity properties approximated by a linear re ectance map, is illuminated by a distant point source of unit power in direction (a1; a2; 1), then the corresponding image E(x1; x2) satis es a linear image irradiance equation of the following form a1 @u @x1 (x1; x2) + a2 @u @x2 (x1; x2) + 1 (a 1 + a 2 + 1) 1=2 = E(x1; x2); (1) over = f(x; y) 2 IR : E(x1; x2) > 0g. Letting E(x1; x2) = E(x1; x2)(a 2 1 + a 2 + 1) 1, one can rewrite (1) as a transformed linear image irradiance equation a1 @u @x1 (x1; x2) + a2 @u @x2 (x1; x2) = E(x1; x2): (2) In this paper we evaluate di erent nite di erence algorithms for a direct shape recovery modelled by the equation (2). The original idea of this work is an extension of Kozera work in [4, 5], where the convergence analysis of the nite di erence scheme based on central di erence approximation of the derivatives has been discussed. We continue to investigate here the issue of the stability and the convergence of di erent algorithms based on the combination of the forward and backward derivative approximations. Convergence, stability, and domain of in uence, will be considered here as algorithmic features and used in this paper for evaluating shape reconstruction algorithms based on nite di erence schemes. Critical to our approach is the assumption that u is given along some (not necessarily smooth) initial curve in the image (image boundary). The algorithms provide the numerical solution of the following Cauchy problem (for u 2 C( ) \ C( )) considered over a rectangle : L(u(x1; x2)) = E(x1; x2) (3a) u(x1; 0) = f(x1) 0 x1 a; for sgn(a1a2) 0; (3b) u(x1; b) = f(x1) 0 x1 a; for sgn(a1a2) < 0; (3c) u(0; x2) = g(x2) 0 x2 b; (3d) here Lu = a1ux1 + a2ux2 , and functions f 2 C([0; a]) \ C ((0; a)) and g 2 C([0; b]) \ C((0; b)) satisfy f(0) = g(0), E 2 C( ), and a1 and a2 are constants such that (a1; a2) 6= (0; 0). To simplify