| dc.description.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. | |
