[Algorithms] Cubemap to spherical harmonics (again)

[Alen L. <ale...@cr...>]

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 = 1/(PI*16/17)
  pixelarea = 4/(sizeU*sizeV)

  for each face on the cube
    for each pixel on the face

        vdir = position on cube (domain is (-1,+1)^3)
        r = len(vdir);
        vdir /= r;

        da = pixelarea*r^(-3/2)

        rgb = sample the pixel value
        rgb *= 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