Spherical Harmonics Diffuse CubeMap. How to get coefficients?

Question

• envmap.pdf | An Efficient Representation for Irradiance Environment Maps
• Spherical Harmonics and applications in real time graphics

I am using Spherical Harmonics to turn reflective(mirror like) cube map to diffuse light cube map.

From what I understand for an image, I calculate 9 coefficient(for red green blue) that describes the color of the cube map.

The nine coefficients are this...

And when you visualize them they become...

Then I do something with these coefficients to get color of the cube map.

The part that I am not understanding is using an image to get that 9 coefficients. All the papers kinda assume that you somehow got that coefficient calculated because it is "easy".

I don't know how to do that calculation.

---

Update

So while I were looking at Spherical Harmonic Lighting | Simon's Tech Blog

I found this equation...

So now I know that the mysterious yellow part of the equation was "solid angle related". I do know how to do solid angle but I still don't understand how to do it.

Answer 1

So at start you have samples from your cube map. Each sample has color and normal (dir) at which you sampled that color. This is how I do it. I use coeffs from this paper (the same you linked), there are values for the first 9 of them.

So for constructing for each sample you:

• Compute your SH basis using normal from sample

float Y00     = 0.282095;
float Y11     = 0.488603 * normal.x;
float Y10     = 0.488603 * normal.z;
float Y1_1    = 0.488603 * normal.y;
float Y21     = 1.092548 * normal.x*normal.z;
float Y2_1    = 1.092548 * normal.y*normal.z;
float Y2_2    = 1.092548 * normal.y*normal.x;
float Y20     = 0.946176 * normal.z * normal.z - 0.315392;
float Y22     = 0.546274 * (normal.x*normal.x - normal.y*normal.y);

• Then multiply these coeffs by your color (L)

vec3 L00   = L * Y00;
vec3 L11   = L * Y11;
vec3 L10   = L * Y10;
vec3 L1_1  = L * Y1_1;
vec3 L21   = L * Y21;
vec3 L2_1  = L * Y2_1;
vec3 L2_2  = L * Y2_2;
vec3 L20   = L * Y20;
vec3 L22   = L * Y22;

And thats it, you have your coeffs. Now, you can store them in some data structure. You accumulate each of them like:

someArray[0] += L00;
someArray[1] += L11;
etc...

And you divide these accumulated coeffs by a number of samples;

Then when reconstructing :

• compute your SH basis one more time for normal at point of the scene.
• for approximating diffuse irradiance at given point of scene, convolution with cosine lobe is performed using normal from that point. Convolution means, for each SH order, multiplication by $ \hat{A}=[\pi, \frac{2}{3}\pi, \frac{\pi}{4}]$

$\hat{A_0} = 3.141593$, $\hat{A_1} = 2.094395$, $\hat{A_2} = 0.785398$

• L coeffs are sampled from data structure.

vec3 color = A0*Y00*L00  
             + A1*Y1_1*L1_1 + A1*Y10*L10 + A1*Y11*L11 
             + A2*Y2_2*L2_2 + A2*Y2_1*L2_1 + A2*Y20*L20 + A2*Y21*L21 + A2*Y22*L22;

Disclaimer: This basically works (for me) but I am not an expert and something might be missing or not perfect because this IS confusing topic...

Answer 2

To start I'll assume you have a list of (x,y,z) points each with a r, g, b color that are the samples you want to approximate with the spherical harmonics.

To get the coefficients you make a matrix n x 9 $A$ where each row is the sequence $y_0(x,y,z), y_1^{-1}(x,y,z), y_1^{0}(x,y,z), y_1^{1}(x,y,z), y_2^{-2}(x,y,z), y_2^{-1}(x,y,z), y_2^{0}(x,y,z), y_2^{1}(x,y,z), y_2^{2}(x,y,z)$ of a point and also a n x 3 matrix $b$ with each corresponding row the color at that point.

Then you take a 9x3 matrix $x$ of the unknown coefficients (9 for each color). Then $Ax=b$ is the equation to solve. This equation will be overdetermined so you'll need to approximate the solution.

This means doing a linear regression to find the coefficients that give the smallest error. This is well described in linear algebra.