Mathematical Background of SFS Models

Many problems of mathematical physics lead to PDEs. In general, PDEs are classified in many different ways. However, in most mathematics literature, PDEs are classified on the basis of their characteristics, or curves of information propagation (see, for example, [60] and [19]). The irradiance equation (5.2) is a first-order nonlinear equation. The general format of such an equation in the two-dimensional space is given by where T is the boundary curve of the domain Œ.

In general, nonlinear PDEs are much more difficult than the linear equations, while the more the nonlinearity affects the higher derivatives, the more difficult the PDE is. The irradiance equation (5.2) with a nonlinear reflectance map (5.5) is a hyperbolic PDE of first order with severe nonlinearity. Although the nonlinearity prevents the possibility of deriving any simple method to solve the equation, there are still some techniques developed to obtain local information of the solution to a certain extent. In this section, we briefly review some basics about the irradiance equation, namely, the existence and uniqueness of

Theoretically, a compactible boundary condition should be given as

Z(x, y) = g(x, y), (x, y) e T, the solution. We also describe a technique, characteristic strip method, which leads to the solution of the equation.

5.2.1 The Uniqueness and Existence

It has been shown that surfaces with continuously varying surface orientation give rise to shaded images. The problem of shape from shading is to reconstruct the three-dimensional shape of a surface from the brightness or intensity variation in a single black-and-white photographic image of the surface. For a long time in history, the SFS model was believed ill-posed. However, it has been shown that the problem in its idealized form is actually well posed or "partially" well posed under a wide range of conditions ( [32,42]).

The standard assumptions for the idealized surface are:

• "Lambertian" reflectance—the surface is matte, rather than mirror-like and reflects light evenly in all directions,

• "Orthographic" projection—the illuminating light is from a single known direction and that the surface is distant from the camera, and

• "Nonocclusion"—all portions of the surface are visible.

If only one source of illumination is available, uniqueness can be proved. Further Saxberg [51,52] discussed conditions for existence of the solution. Olien-sis [41,42] has shown the following:

Proposition 1. For an image of a light region contained in a black background, if the reflectance map is known, as given in (5.2), then there is a unique solution for a generic surface which is smooth and non-self-occluding.

Despite various existence and uniqueness theorems for smooth solutions (see [14, 30, 34,41, 42, 51, 52, 64]), in practice the problem is unstable, which is catastrophic for general numerical algorithms [4, 18]. This is because the reflectance map is, in general, given by its sampled data rather than an analytic expression. This data may be sparse and contaminated by noise. We will not go into the detailed discussion about the uniqueness and existence issue here; the readers who are interested in this issue are referred to the excellent review paper by Hurt [32] and references [14,30,34].

5.2.2 The Characteristic Strip Method

Horn [29] established a method to find the solution of (5.2), the characteristic strip method ( [29], p. 244). This method is to generate the characteristic strip expansion for the nonlinear PDE (5.2) along a curve on the surface by solving a group of five ordinary differential equations called characteristic equations:

Z = pRp + qRq, where the dot denotes differentiation along a solution curve. The characteristic equation can be organized in a matrix format:

d dt







pRp + qRq




The solution, (x, y, Z, p, q)T, to (5.8) forms a characteristic strip along the curve. The curves traced out by the solutions of the five ordinary differential equations are called characteristic curves, and their projections in the image are called base characteristics. If an initial curve (with known derivative along this curve) is given by a parametric equation:

(n) = {x(n), y(n), Z(n)}T, then we can derive the surface by integrating the equation d Z dx dy dn dn dn

Example 2. Consider an ideal Lambertian surface illuminated by a light source close to the viewer at (p0, q0, 1) = (0, 0,1). (p0, q0) is the direction toward the light source. In this case, the image irradiance equation is

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