Shape from Shading Using Near Point Light Sources

1 The Approximate Solution Curve Tracing (ASCT) Method 1.1 The correct solution space (CSS) Figure 1: The CSS represents the correct solution .space which all of the depth values on the illuminated object are included in. LB is the lower bound of the CSS. All of the depth values on the illuminated object will be not "deeper" than it. UB is the upper boumt of the CSS. All of the depth values on the illuminated object are always "deeper" than it. Before recovering the depth map of the ilhmtinated object: we can assume that there exists a range to include all of the depth values of the points on the surface, a~s shown in Figure 1. 1.2 The approximate solution difference (ASD) Now suppose we have two images I1, Is, the corresponding point light sources are (S~I,Syl,S~) and (S~> Sy2. S~2), the illuminated point is (x, y, z), and the normal vector is (p, q,-1). Then, Ii(x,y) = k. (S~1-x)p+(Su~-y)q-(S~-z) h(*, y) = k (&~-z)p + (s~-y)q-(&~-=) Substitute a value in the CSS, called z~,,, for the unknown variable z in the denominator of both equations and divide Equation (1) by Equation (2). After simplifying the coefficients, a linear equation (Equation (3)) with three unknowns is obtained. These unknown values are p, q, and z. s,-[(&~-x) ~ + (s~l-v) ~ + (&l-z~,,)2] ~ h " [(&2-X) ~ + (Sy2-y)2 + (&2-z.s)2] ~ zr = a value in the CSS