On the Eikonal equation and shape from shading

Abstract

This thesis proposes and develops new techniques for the solution of the Shape from Shading (SFS) problem. For a homogeneous Lambertian surface, illuminated by a distant point light source in the direction (p, q, -1) and viewed from a large distance in the direction (0, 0, -1), the observed image intensity I(x, y) is related to the surface gradient (p, q) by the image irradiance equation where P is a constant (albedo), P = H and ? = If being the depth. When the light source is in the viewer’s direction, the image irradiance equation (1) reduces to what is known as the eikonal equation form (x, y) (2) where E(x, y) = 1 / I(x, y) and P is taken to be equal to 1. The thesis mainly dwells on the study of the eikonal equation and development of solution techniques for the same. These techniques are further extended to the solution of the general image irradiance equation (1) by suitably approximating them by equations of (eikonal) form (2). The present work is reported in seven chapters. The thesis begins with an introduction (Chapter 1) to the mathematical formulation of the shape from shading problem and then reviews various algorithms presently available. Chapter 2 of the thesis examines the ill-posedness of the SFS problem, as modelled by Horn’s image irradiance equation (1). The existing results on the mathematical properties of the SFS problem, particularly those given by Bruss [1] and Oliensis [2] are discussed. Using the elementary theory of Partial Differential Equations, the existence and uniqueness of the solution for the SFS problem, in the presence of proper boundary conditions, are analysed. Through a counterexample, it is established that the solution to (1) need not be a continuous function of input data. A new method for shape extraction from shading information proposed in Chapter 3, suggests the use of patch-wise approximation of the surfaces in the scene by discrete polynomials. The problem of solving a non-linear partial differential equation (image irradiance equation) is reduced to that of solving a finite set of non-linear algebraic equations. The polynomial approximation is studied in the context of local shading analysis of Pentland [3] and also in a global perspective to include various depth cues obtained by preprocessing the image. The method has been successfully used for the case of eikonal equation, where an explicit algebraic technique for estimation of the quadratic polynomial parameters is made available. For the general image irradiance equation an iterative technique is presented on the basis of orientation information at the singular point, and the known depth values (or slope values) at the boundary.