FLEXIBLE APPEARANCE MODEL - 3D Face Processing: Modeling, Analysis and Synthesis - page 65

Using the notation of [Ramamoorthi and Hanrahan‚ 2001a]‚ the irradiance
can be represented as a linear combination of spherical harmonic basis func-
tions:

Ramamoorthi and Hanrahan [Ramamoorthi and Hanrahan‚ 2001a] showed that
for diffuse reflectance‚ only 9 coefficients are needed to approximate the irra-
diance function. Therefore‚ given an image of a diffuse surface with constant
albedo its reflected radiance at a point with normal can be approximated

as

If we treat as a single variable for each and we can solve for these
9 variables using a least square procedure‚ thus obtaining the full radiance
environment map. This approximation gives a very compact representation of
the radiance environment map‚ using only 9 coefficients per color channel.
An important extension is the type of surface whose albedo‚ though not
constant‚ does not have low-frequency components (except the constant com-
ponent). To justify this‚ we define a function

such that equals to the

average albedo of surface points whose normal is We expand using

spherical harmonics as:

where is the constant component and contains other higher order
components. Together with equation (6.6)‚ we have

If does not have first four order components‚ the second
term of the righthand side in equation (6.7) contains components with orders
equal to or higher than 3 (see Appendix A for the explanation). Therefore‚ if we
define SpharmonicProj(I) to the be function which projects the face image
I into the 9 dimensional spherical harmonic space‚ we have

Therefore‚ the 9 coefficients of order estimated from with a lin-

ear least square procedure are where Hence‚