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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‚

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