In the early 2000's, people like Ravi Ramamoorthi and Peter Pike Sloane properly introduced a new powerful and amazing tool to the Computer Graphics industry: Spherical Harmonics (or SH).
Although Spherical Harmonics have always been around for quite some time, for example in the excellent "Predicting Reflectance Functions from Complex Surfaces" 1992 paper from Westin et al., they only piqued the public interest since their introduction as an efficient way of realistic indirect lighting via Pre-computed Radiance Transfer (PRT).
Contents
What are Spherical Harmonics?
According to wikipedia, SH are special functions defined on the surface of a sphere.
In computer graphics, we're using them as a tool to quickly and easily encode or decode a directional information. We use a specific set of spherical harmonics, denoted <math>Y^m_l(\theta,\phi)</math> called Laplace's spherical harmonics.
SH have interesting properties regarding their orthogonality, parity, symmetry and rotation that I will not cover here (cf. the wikipedia page for more info) as this page only is an overview. An excellent source of information is Spherical Harmonics Lighting: the Gritty Details by Robin Green that actually covers the practical use of SH for Computer Graphics, it's a well-explained extension of the original work done by Peter Pike Sloane about Pre-computed Radiance Transfer (PRT).
I will rather quickly talk about how to construct the SH coefficients and how to encode/decode/convolve signals using SH.
Generalities about SH orders
As can be seen in the inset on the right, Spherical Harmonics are functions defined over the sphere. The SH functions are denoted <math>Y^m_l(\theta,\phi)</math> where <math>l</math> is the order of the function. Since SH define harmonic series, there are an infinite amount of possible orders.
For a given order <math>l</math> you have <math>2l+1</math> coefficients selected by the superscript <math>m\in[-l,+l]</math>. It ensues that the indices and amount of coefficients for each order is then:
0 Order 0 - Total = 1 = 1²
1 2 3 Order 1 - Total = 1 + 3 = 4 = 2²
4 5 6 7 8 Order 2 - Total = 1 + 3 + 5 = 9 = 3²
9 10 11 12 13 14 15 Order 3 - Total = 1 + 3 + 5 + 9 = 16 = 4²
<math>\dots</math>
We quickly see the amount of coefficients to properly represent each order grows quadratically.
Intuitively, we also can notice that the more coefficients we have, the more "directionalities" we can cover. Order 0 is a constant so it represents the "ambient term" — the average response of a signal — and has no specific direction. Coefficients for order 1 represent the response to a signal aligned with the X, Y and Z axes. At order 2, we start covering some diagonal directions and higher frequencies.
In fact, if we used an infinite sum of SH coefficients, we could encode or decode a signal perfectly. A great advantage of using SH is that we can recover a partial signal (i.e. a band-limited signal) and we will obtain a low-frequency representation of that signal, as opposed to wavelets that attempt to reconstruct all frequencies of a signal and can show high-frequency noise, a partial SH-encoded signal will always be smooth.
It is common in real-time computer graphics to use only order 2 (9 coefficients) or order 3 (16 coefficients) at most, each coefficient being a RGB float3 triplet. SH are often used to encode irradiance, which is generally spatially varying pretty smoothly. Although it varies at higher frequency than irradiance, it is also possible to encode the radiance into SH.
Actually, both radiance and irradiance are related in terms of SH, as explained by this great 2001 paper from Ramamoorthi and Hanrahan.
It was shortly followed by another seminal paper called "An Efficient Representation for Irradiance Environment Maps" that extends on the first paper and shows that order 2 SH (9 coefficients) is often enough to properly represent the irradiance field surrounding an object since additional order disappear very quickly. We will come back to these papers later as I will show an extension on this irradiance estimate to introduce the Ambient Occlusion term.
NOTE: It is sometimes desirable to express a SH coefficient <math>Y^m_l(\theta,\phi)</math> by a single index <math>Y_i(\theta,\phi)</math> where <math>i=l(l+1)+m</math>. It can be noted that <math>l(l+1)</math> gives us the "central index" (If you refer to the pyramid of coefficient indices above, these are the bold indices 0, 2, 6, 12, 20, 30, etc.) for any given order <math>l</math>, then adding <math>m</math> will address the "left" or "right" coefficients depending on the sign of <math>m</math>.
Inversely, when we want to retrieve <math>l, m</math> from <math>i</math> then <math>l=\lfloor\sqrt{i}\rfloor</math> and <math>m=i - l(l+1)</math>.
Constructing the SH Coefficients
We're working with a Z-Up orientation:
The expression of vector <math>v</math> in cartesian coordinates is <math>v(\theta,\phi) = (\sin(\theta)\cos(\phi), \sin(\theta)\sin(\phi), \cos(\theta))</math>.
The general spherical harmonics coefficients <math>Y^m_l(\theta,\phi)</math> are expressed as:
- <math>Y^m_l(\theta,\phi) = K^m_l P^m_l(cos(\theta)) e^{im\phi}</math>
Where <math>K^m_l = \sqrt{ \frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}</math> is a normalization factor and <math>P^m_l(\theta,\phi)</math> is the associated Legendre polynomial (shown on the right inset figure).
We're interested in the real form of spherical harmonics that is given by:
- <math>Y^m_l(\theta,\phi) =
\begin{cases}
\sqrt{2} K^{-m}_l P^{-m}_l(cos(\theta)) \sin(-m\phi) & \quad \text{if } m < 0\\
K^0_l P^0_l(cos(\theta)) & \quad \text{if } m = 0\\
\sqrt{2} K^m_l P^{| m |}_l(cos(\theta)) cos(m\phi) & \quad \text{if } m > 0\\
\end{cases} </math>
Here is a code snippet (source Robin Green's gritty details) about how to build the SH functions:
// Renormalisation constant for SH function
double K( int l, int m ) {
double temp = ((2.0*l+1.0)*factorial(l-m)) / (4.0*PI*factorial(l+m)); // Here, you can use a precomputed table for factorials
return sqrt(temp);
}
// Evaluate an Associated Legendre Polynomial P(l,m,x) at x
// For more, see “Numerical Methods in C: The Art of Scientific Computing”, Cambridge University Press, 1992, pp 252-254
double P( int l, int m,double x ) {
double pmm = 1.0;
if ( m > 0 ) {
double somx2 = sqrt((1.0-x)*(1.0+x));
double fact = 1.0;
for ( int i=1; i<=m; i++ ) {
pmm *= (-fact) * somx2;
fact += 2.0;
}
}
if( l == m )
return pmm;
double pmmp1 = x * (2.0*m+1.0) * pmm;
if ( l == m+1 )
return pmmp1;
double pll = 0.0;
for ( int ll=m+2; ll<=l; ++ll ) {
pll = ( (2.0*ll-1.0)*x*pmmp1-(ll+m-1.0)*pmm ) / (ll-m);
pmm = pmmp1;
pmmp1 = pll;
}
return pll;
}
// Returns a point sample of a Spherical Harmonic basis function
// l is the band, range [0..N]
// m in the range [-l..l]
// theta in the range [0..Pi]
// phi in the range [0..2*Pi]
double SH( int l, int m, double theta, double phi ) {
const double sqrt2 = sqrt(2.0);
if( m == 0 ) return K(l,0)*P(l,m,cos(theta));
else if( m > 0 ) return sqrt2*K(l,m)*cos(m*phi)*P(l,m,cos(theta));
else return sqrt2*K(l,-m)*sin(-m*phi)*P(l,-m,cos(theta));
}
Analytical Expressions for the first SH Coefficients
As indicated in the beginning of this page, most real-time applications only use a limited amount of SH orders (2, 3 or more rarely 4 orders) so we're usually dealing with at most 9 or 16 SH coefficients.
These coefficients have analytical expressions that we are now listing below (we dropped the <math>\theta,\phi</math> for conciseness):
- <math>
\begin{cases} \text{order }l=0\\
Y^0_0 = \frac{1}{2}\sqrt{\frac{1}{\pi}}\\
\text{order }l=1\\
Y^{-1}_1 = \frac{1}{2}\sqrt{\frac{3}{\pi}}y & \quad m = -1\\
Y^{0}_1 = \frac{1}{2}\sqrt{\frac{3}{\pi}}z & \quad m = 0\\
Y^{1}_1 = \frac{1}{2}\sqrt{\frac{3}{\pi}}x & \quad m = 1\\
\text{order }l=2\\
Y^{-2}_2 = \frac{1}{2}\sqrt{\frac{15}{\pi}}xy & \quad m = -2\\
Y^{-1}_2 = \frac{1}{2}\sqrt{\frac{15}{\pi}}yz & \quad m = -1\\
Y^{0}_2 = \frac{1}{4}\sqrt{\frac{5}{\pi}}(3z^2-1) & \quad m = 0\\
Y^{1}_2 = \frac{1}{2}\sqrt{\frac{15}{\pi}}xz & \quad m = 1\\
Y^{2}_2 = \frac{1}{4}\sqrt{\frac{15}{\pi}}(x^2-y^2) & \quad m = 2\\
\end{cases} </math>
Where <math>
\begin{cases}
x = \sin(\theta)\cos(\phi)\\ y = \sin(\theta)\sin(\phi)\\ z = \cos(\theta)\\
\end{cases} </math>
Common Usages for SH
Signal Encoding
A signal <math>f(\theta,\phi)</math> whose value is available for all directions can be encoded (or rather, projected) into SH coefficients by estimating the following integral:
- <math>C_i = \int\limits_0^{2\pi}{\int\limits_0^{\pi} {f(\theta,\phi)Y_i(\theta,\phi)\sin(\theta)} \; \mathrm{d}\theta \; \mathrm{d}\phi}</math>
To implement this integration numerically we can use the naïve pseudo-code:
float EncodeSHCoeff( int l, int m ) {
const int STEPS_PHI = 200;
const int STEPS_THETA = 100;
const float dPhi = 2*PI/STEPS_PHI;
const float dTheta = PI/STEPS_THETA;
float coeff = 0.0f;
for ( int i=0; i < STEPS_PHI; i++ ) {
float phi = i * dPhi;
for ( int j=0; j < STEPS_THETA; j++ ) {
float theta = (0.5f+j) * dTheta;
float value = EstimateFunction( phi, theta );
float SHvalue = EstimateSH( l, m, phi, theta )
coeff += value * SHValue * sin( theta ) * dPhi * dTheta;
}
}
return coeff;
}
A better (and faster!) way to do that would be to use importance sampled Monte Carlo sampling. Please refer to Robin Green's paper for details about the technique.
We first need to generate an arbitrary set of samples on the sphere. We choose the samples so that they are uniformly distributed on the sphere, thus ensuring that each sample carries the same weight that will be <math>\frac{4\pi}{N} \text{ where } N \text{ is the amount of samples}</math>:
struct sample_t {
float3 direction;
float theta, phi;
float* Ylm;
};
// Fills an N*N*2 array with uniformly distributed samples across the sphere using jittered stratification
void PreComputeSamples( int sqrt_n_samples, int n_bands, sample_t samples[], float ) {
int i = 0; // array index
double oneoverN = 1.0 / sqrt_n_samples;
for ( int a=0; a < sqrt_n_samples; a++ ) {
for ( int b=0; b < sqrt_n_samples; b++ ) {
// Generate unbiased distribution of spherical coords
double x = (a + random()) * oneoverN; // Do not reuse results
double y = (b + random()) * oneoverN; // Each sample must be random!
double theta = 2.0 * acos( sqrt( 1.0 - x ) ); // Uniform sampling on theta
double phi = 2.0 * PI * y; // Uniform sampling on phi
// Convert spherical coords to unit vector
samples[i].direction = float3( sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta) );
samples[i].theta = theta;
samples[i].phi = phi;
// precompute all SH coefficients for this sample
for ( int l=0; l < n_bands; ++l ) {
for ( int m=-l; m<=l; ++m ) {
int index = l*(l+1)+m;
samples[i].Ylm[index] = EstimateSH( l, m, theta, phi );
}
}
++i;
}
}
}
We keep that nice array around because we will re-use it for as many encodings as we want by applying that now very straightforward pseudo-code:
typedef double (*EstimateFunction)( double theta, double phi );
// Here, n_coeffs = n_bands*n_bands and n_samples = sqrt_n_samples*sqrt_n_samples
void SHProject( EstimateFunction estimator, int n_samples, int n_coeffs, const sample_t samples[], double result[] ) {
for ( i=0; i < n_coeff; ++i ) {
result[i] = 0.0;
// For each sample
for ( int i=0; i < n_samples; ++i ) {
double theta = samples[i].theta;
double phi = samples[i].phi;
for ( int n=0; n < n_coeff; ++n ) {
result[n] += estimator( theta, phi ) * samples[i].Ylm[n];
}
}
// Divide the result by weight and number of samples
double factor = 4.0*PI / n_samples;
for ( i=0; i < n_coeff; ++i ) {
result[i] = result[i] * factor;
}
}
NOTE: Projecting a function into SH coefficients is obviously not something we want to be doing every frame! The goal is to project once, and re-use the SH coefficients all around. The memory gain is enormous: for example imagine you want to encode a distant environment cube map into an order 2 SH to use it as the diffuse radiance source, then assuming you're dealing with a 256x256x6xRGB32F cube map, you start from a 256*256*6*16 = 6 solid Mega Bytes down to a mere 9*12 = 108 Bytes (9 float3 SH coefficients).
Of course, encoding with so little coefficients is sufficient enough for diffuse irradiance estimates and barely anything else, don't expect a great many deal of details out of that!
Signal Decoding
To reconstruct the band-limited version of the original signal from a bunch of SH coefficients, you simply need to apply the inverse transform:
<math>f(\theta,\phi)=\displaystyle\sum_{l=0}^{N} \displaystyle\sum_{m=-l}^{l} C_l^m Y_l^m(\theta,\phi)\sin(\theta) \text{ where } N \text{ is the SH order}</math>
Signal Product
This property is certainly the most important regarding SH.
In computer graphics it's often that we need to compute that kind of integral:
- <math>\int\limits_\Omega{A(\boldsymbol\omega_i) B(\boldsymbol\omega_i) \; \mathrm{d\omega_i}}</math>
Where:
- <math>
\Omega \text{ represents the set of all directions on the sphere}\\
\boldsymbol{\omega_i} \text{ is the unit integration vector (i.e. the incoming direction)}\\
\mathrm{d\omega_i} \text{ is the solid angle (a scalar!) in the incoming direction}\\
</math>
Well, assuming you managed to encode <math>A(\boldsymbol\omega_i)</math> into SH coefficients <math>a_l^m</math> and similarly you obtained the <math>b_l^m</math> SH coefficients corresponding to the <math>B(\boldsymbol\omega_i)</math> signal then the integral in terms of SH simply is:
<math>\displaystyle\sum_{l=0}^{N} \displaystyle\sum_{m=-l}^{l} a_l^m \, b_l^m</math>
Examples
For example, the integration of radiance into irradiance over the hemisphere is:
- <math>E(\mathbf n) = \int\limits_\Omega{L(\boldsymbol{\omega_i}) \lfloor (\boldsymbol\omega_i\cdot\mathbf n )\rfloor \; \mathrm{d\omega_i}}</math>
Where:
- <math>
L(\mathbf n,\boldsymbol{\omega_i}) \text{ is the radiance arriving in incoming direction } \boldsymbol{\omega_i}\\
E(\mathbf n) \text{ is the irradiance perceived in direction } \mathbf{n}\\
\mathbf n \text{ is the surface normal direction}\\
\lfloor (\mathbf\omega_i\cdot\mathbf n)\rfloor \text{ represents the dot product clamped over the upper hemisphere}
</math>
Another example is the surface radiance estimate:
- <math>L(\boldsymbol{\omega_o}) = \int\limits_{\Omega^+}{L(\boldsymbol{\omega_i}) \, f_r( \boldsymbol{\omega_o}, \boldsymbol{\omega_i} ) \, (\boldsymbol\omega_i\cdot\mathbf n ) \; \mathrm{d\omega_i}}</math>
Where:
- <math>
L(\boldsymbol{\omega_i}) \text{ is the radiance arriving in incoming direction } \boldsymbol{\omega_i}\\
L(\boldsymbol{\omega_o}) \text{ is the radiance leaving in outgoing direction } \boldsymbol{\omega_o}
</math>
- <math> f_r( \boldsymbol{\omega_o}, \boldsymbol{\omega_i} )</math> is the surface's BRDF
- <math> \Omega^+ \text{ represents the upper hemisphere of directions (so there is no need for the clamped dot product here!)}\\</math>
Signal Triple Product
@TODO:
Triple Product of Zonal and Spherical Harmonics
@TODO:
About Distant Radiance and Irradiance Environment
As mentioned
Archives
You can find an old (2009) explanation about SH Environment Maps.