1 Introduction

Horn's classical approach to shape-from-shading is couched as an energy minimisation process using the apparatus of variational calculus [ 9 , 6 ]. The aim is to iteratively recover a needle-map representing local surface orientation by minimising an error-functional. This functional contains a data-closeness term, and a regularizing term that controls the smoothness of the recovered needle-map. Since the recovery of the needle-map is under-constrained, the variational equations must be augmented with boundary constraints.

Subsequent literature on the variational formulation of shape-from-shading can be viewed as focussing on one of three broad issues. The first of these is the definition of the objective function. Most of the effort here has concentrated on how to impose constraints on the recovered needle map. For instance, Legarde and Ferrie [ 4 ] have augmented the standard quadratic regularizer with a curvature consistency measure. The second aim has been to provide more realistic physical models of the reflectance process: for instance, by relaxing the requirement for a matte Lambertian surface to accommodate specularities [ 1 , ]. The final area of activity is the modelling of boundary conditions. The classical work relied almost exclusively on the use of the occluding boundary condition which demands that the needle-map is confined to the image and orthogonal to the occluding edge. If occluding boundary constraints are not available, then singular points can be used to constrain the needle-map [ 12 ]. Recently, Kimmel and Bruckstein have shown how the apparatus of level-set theory can be used to solve the reflectance equation [ 10 ] as a boundary value problem.

Despite this sustained activity, there has been little effort devoted to the statistical modelling of the process of needle map recovery. One of perennial criticisms of the conventional approach is that the smoothness model can dominate genuine features in the data. This can have the undesirable effect of blurring high-curvature surface detail. Here we focus on this source of criticism by showing how needle map recovery can by realised using the apparatus of robust statistics. In particular, we show how the use of a robust regularization term increases the capacity of the shape-from-shading process to implicitly accommodate surface discontinuities and sharp changes in orientation, whilst rejecting noise artifacts as outliers. A sensitivity study reveals that the best performance is obtained with regularizer which is derived from an influence function of the form .