Re: [Algorithms] Cubemap to spherical harmonics (again)

[<ca...@gm...>]

That seems to be right and is pretty close to what I do. The only
difference is that I compute the area element in a different way,
computing the solid angle of the texel polygon using Girard's formula:

omega =3D sum_i(alpha_i - (n-2)*pi)

where alpha_i is the dihedral angle between the planes that form the
polygon and n is the number of planes (Graphic Gems R27, R6).

You can find another derivation of the area element of the texels of
the cube map in Manne Ohrstrom thesis:

http://coblitz.codeen.org:3125/citeseer.ist.psu.edu/cache/papers/cs/30957/h=
ttp:zSzzSzwww.student.lu.sezSz~cif97mohzSzthesis.pdf/spherical-harmonics-pr=
ecomputed-radiance.pdf

I assume that r^-3/2 is the differential solid angle, where did you
get that result?

With a low number of coefficients, the results usually look weird, but
after convoling it with the cosine term and sampling it, you end up
getting the nice results.

--
Ignacio Casta=F1o
ca...@gm...


On 4/21/06, Alen Ladavac <ale...@cr...> wrote:
> Hi list,
>
> I believe that this issue, even though discussed a lot, was never
> completely clarified in one place. So as my implementation is a kind
> of a Frankenstein creation stitched from fragments of math that I
> understand and pieces that still make by head steam, I guess it can't
> hurt to run the actual result by some people that comprehend the SH
> math better than me.
>
> The reason I ask is that when I run it on a cube that has upper half
> pure red and lower half pure green, I get something that I don't quite
> expect. Upon evaluating the resulting SH in various directions, I get
> mostly yellow everywhere, with just a tiny bit of red/green in the
> up/down directions.
>
> Even with the cosine term, I'd expect the straight up/down directions
> to be pure red/green. Is my expectation wrong, or is the problem in my
> implementation?
>
> Here is the pcode:
>
>   correction =3D 1/(PI*16/17)
>   pixelarea =3D 4/(sizeU*sizeV)
>
>   for each face on the cube
>     for each pixel on the face
>
>         vdir =3D position on cube (domain is (-1,+1)^3)
>         r =3D len(vdir);
>         vdir /=3D r;
>
>         da =3D pixelarea*r^(-3/2)
>
>         rgb =3D sample the pixel value
>         rgb *=3D angle * correction;
>         addtoSHC(rgb,dir)
>
> Where "addtoSHC()" is the one normally used by the directional light.
> I.e. it uses the squared factors with extra corrections:
>
>   0.282095*0.282095 * PI*16/17 * 1,
>   0.488603*0.488603 * PI*16/17 * 2/3,
>   1.092548*1.092548 * PI*16/17 * 1/4,
>   0.315392*0.315392 * PI*16/17 * 1/4,
>   0.546274*0.546274 * PI*16/17 * 1/4
>
> I also tried using the plain set of factors for adding (without the
> squaring and the corrections), but I get even stranger results.
>
> Thanks in advance for any insight,
>
> Alen