Spherical Harmonic Lighting

Spherical Harmonic Lighting: The Gritty Details

Spherical Harmonic Lighting A efficient method for capturing and displaying Global Illumination solutions across the surface of an object.

Background

Light Modeling

For a point $x$ at the surface, the reflected light intensity from viewing direction $ \mathbf d_0$ is modelled as

\[\begin{align*} L(x, \mathbf d_0) &= L_e(x, \mathbf d_0) + \int_S f_r(x, \mathbf d_i - \mathbf d_0) L_r(x', \mathbf d_i) G(x,x') \mathbf I(x\text{ see } x')d\: \mathbf d_i\\ \end{align*}\]

where $S$ is the unit sphere centered at $x$ and
$L_e(x, \mathbf d_0)$ is the direct emission light
$f_r(x, \mathbf d_i - \mathbf d_0)$ is the BRDF term, transforming incoming light $ \mathbf d_i$ to $\mathbf d_0$
$L_r(x,x')$ the the light reflected from $x'$ to $x$
$G(x, x')$ is the geometric relationship between $x$ and $x'$

Of course, this computation is intractable and we do lots compromises (given up some terms, simplify the model).

Monte Carlo Integration

Known that the expectation is defined as

\[E(f(x)) = \int f(x)p(x) dx\]

for any function $f$, thus if we have that $f' = f/p$, we have

\[E(\frac{f(x)}{p(x)}) = \int \frac{f(x)}{p(x)}p(x) dx = \int f(x)dx \approx \hat E(\frac{f(x)}{p(x)}) = \frac{\sum^N f(x_i)/{p(x_i)}}{N}\]

More over, if we can uniformally sample $N$ points on the sphere surface, then we have $p(x_i) = 1/4\pi$ is constant. so that the equation is reduced to summing samples.

Note that uniform sampling from polar coordinate $(\theta\in [0,2\pi), \phi\in[-\pi/2,\pi/2))$ is not a uniform sampling over the sphere, since we are sampling over the circles with different radius $\sin\phi$.

Instead, we transforms from unit square $(u,v)$

\[(2\pi u, 2\cos^{-1}(\sqrt{1-v}))\rightarrow (\theta,\phi)\]

function sphere_sample(N) {
    const positions = new Float32Array(N * 3);
    for (let i = 0; i < N; i++) {
        // uniform sample from [0, 1) * [0, 1)
        const u = Math.random();
        const v = Math.random();
        // transform to polar
        const theta = 2 * PI * u;
        const phi = 2 * Math.acos(Math.sqrt(1 - v));
        // transform to xyz
        positions[3 * i] = Math.cos(theta) * Math.sin(phi);
        positions[3 * i + 1] = Math.sin(theta) * Math.sin(phi);
        positions[3 * i + 2] = Math.cos(phi);
    }
    return positions;
}

SH Projection

The projection onto SH is hence

\[c_l^m = \int_S f(s) y_l^m(s)ds\]

which can be evaluated using Monte-Carlo integration (since we have our unit sphere uniform sampling).

and the reconstruction is

\[\tilde f(s) = \sum_{l=0}^{n-l}\sum_{m=-l}^l c_l^m y_l^m(s) = \sum_{i=0}^{n^2} c_i y_i(s)\]

where $i = l(l+1)+m$ is used to flatten the nested indexes into 1D.

Thus, we can connect everything together.

Properties

Rotation Invariance

SH functions are orthonormal, thus suppose that $g:= f\circ R$ where $R\in SO(3)$, then $\tilde g = \tilde f \circ R$ is the same approximation of $g$. Thus, if we do rotations of the light field (viewing from different angles, or in a dynamic lighting environment).

Integration to dot product

For the reflectance models, the light is an integral over a sampled area

\[\int_S L(s) t(s)ds\]

where $L$ is the incoming light from direction $s$, and $t$ is the transfer function (e.g. BRDF). However, if we use spherical harmonics to approximate $L$ and $t$, and take $L_i, t_i$ be the SH coefs. Then we have that

\[\int_S \tilde L(s) \tilde t(s)ds = \sum_{i=0}^{n^2} L_i t_i\]

which is the dot product of two coef vectors.