Re: [Algorithms] PRT equations confussion.

[<Pau...@sc...>]

gda...@li... wrote on 27/09/2005 
13:21:31:

> Hi,
> 
> hmm, this means I got the transfer vector right, which is what i thought 

> really. Then maybe this means that my L' coefficients vector is wrong.
> 
> From the PRT equations it looks like that is calculated by doing a 
> projection of the light environment on the spherical function (SH in my 
> case)
> 
> L(s) = Sum( li Yi(s)) [ where i = 0 to order^2 ]
> 
> Which is a bit confusing because it does not tell me how to actually 
obtain 
> the li coefficients. So what I do instead is follow the Ramamoorhi and 
> Hanrahan paper (An Efficient Representation of Irradiance Environment 
Maps) 
> and project my env map using a weighted lookup. So I get
> 
> L = Int (L(s) Y(s) )ds
> 
> where L(s) is the light incomming from direction s and Y(s) is a 
spherical 
> harmonics function evaluated in the direction s. (this is a double 
integral 
> on theta and phi and there is a sin(theta) in there as well, but I just 
> presented it like that for simplicity).
> 
> Now in my case L (calculated above) is just a vector Nx3 which is what 
li 
> coefficients put together are.
> 
> So my question is: is it correct to use my calculated L coefficients (as 
of 
> above) in to the PRT equations? Are they the same as the li coefficients 
(or 
> L' in matrix/vector form) in the PRT equations? If not, how else do I 
> calculate the li coefficients needed for diffuse PRT?
> 
> Thanks in advance,

You have to make sure the maths is the same in both cases - i.e. the way 
you build the SH for your environment map  and they way you build your SH 
for each 'normal' potential incoming radiance:

Project both into SH using the same maths... from 

http://www.research.scea.com/gdc2003/spherical-harmonic-lighting.pdf

a = 0.5sqrt(1/PI)
b = 0.5sqrt(3/PI)
c = 0.5sqrt(15/PI)
d = 0.25sqrt(5/PI)

Li[0] = a*1
Li[1] = b*y
Li[2] = b*z
Li[3] = b*x
Li[4] = c*y*x
Li[5] = c*y*z
Li[6] = d*2*z^2-x^2-y^2
Li[7] = c*z*x
Li[8] = c*x^2-y^2

So vector components x, y and z (the environment map and your bent 
'normals') go though this function to generate the SH coefficients. If you 
have R, G and B you will have three such sets.

Then at runtime, you just dot these two vectors (i.e. environment SH and 
per vertex SH) together to yield the intensity. You may have to do some 
other normalisation which i can't recall right now...

If you haven't read that paper i linked to, it might shed some more light 
(um, another pun, horrah) on the subject.

Cheers, Paul.

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