FLEXIBLE APPEARANCE MODEL - 3D Face Processing: Modeling, Analysis and Synthesis - page 64

Given a diffuse sphere of constant reflectance coefficient let denote the
distant lighting distribution. The irradiance on the sphere is then a function of
normal given by an integral over the upper hemisphere

The intensity of the sphere is

is the radiance environment map. Notice that radiance environment
map depends on both the lighting and the reflectance coefficient The inten-

sity of the sphere is usually warped to a reference plane to extract

2D image map where is the coordinate system of the 2D

reference plane.
We can derive similar formulas for face appearance if we assume faces are
Lambertian‚ and ignore cast shadows which is a common assumption for en-
vironment map based techniques. Let denote the normal and
albedo of a face surface point at texture plane Suppose the face is in the
same lighting environment as The irradiance on the face is same as
which can be re-written using coordinate system as:

The intensity of the neutral face point at is

We can observe that for sphere and face points with the same normal‚ the pixels
intensity can be computed with the same lighting The only difference
is the different albedos for the two surfaces‚ which we can handle using the
ratio-image technique described in Section 2.2.

2.1.2 Approximating a radiance environment map using spherical
harmonics
A radiance environment map can be captured by taking photographs of a
sphere painted with a constant diffuse material. It usually requires multiple
views and they need to be stitched together. However‚ spherical harmonics
technique [Ramamoorthi and Hanrahan‚ 2001a] provides a way to approximate
a radiance environment map from one or more images of a sphere or other types
of surfaces.