Uniqueness and Approximation of a Photometric Shape-from-Shading Model

Abstract

We deal with an inverse problem where we want to determine the surface of an object using the information contained in two or more pictures which correspond to different light conditions. In particular, we will examine the case where the light source direction varies between the pictures, and we will show how this additional information allows us to obtain a uniqueness result solving the wellknown convex/concave ambiguity of the shape-from-shading problem. We will prove a uniqueness result for weak (Lipschitz continuous) solutions that improves previous results in [R. Kozera, Appl.

Direct Shape Recovery from Photometric Stereo with Shadows

“…The local ambiguity is shown to be removable in most cases by global integrability constraints. A completely different way to view PS 2 is presented in [11]. This formulation exploits the property of the photometric ratio introduced in [18].…”

“…This method emerges as one of the most readily implementable depth recovery methods and recently new ideas have been introduced in order to solve the PS problems more efficiently, see e.g. [11].…”

Perspective Photometric Stereo with Shadows

“…the entries of ∇z (namely p = ∂z ∂ξ and q = ∂z ∂η ), are computed locally. Let us start by taking into account the differential formulation (8). We exploit the linearity of the hyperbolic equations in (8) by simply summing them, resulting in the single differential equation…”

“…These domains are then going to be patched together in such a way that, across the boundaries between domains on which there are discontinuities in some derivatives, the equation (16) is satisfied. Let us recall that the points where the surface z is not differentiable are the same where the functions b w n and s w n are discontinuous (jump discontinuity) [8]. We assume the discontinuity points as the family of regular curves (γ 1 (t), .…”

“…In this context, quite a few multi-image depth recovery techniques based on inverting shading models have been addressed in the literature [4,5]. Utilizing multiple images in order to remove both the nonlinearities in the image irradiance equation and the generally unknown albedo, new ideas have been introduced in order to solve the PS problems more efficiently, see [6][7][8][9].…”