[GAMES202] Lecture Notes 6

IBL Shadowing

Different Perspetives

A many-light problem? (cost linearly to nLights)

A sampling problem? 1 to figure out an arbitrarily complex visibility $V(w_i)$; 2 $V$ needs to be extracted from $L_o=\int_\Omega ^+ fL_iV\cos\theta_i dw_i$ ? ($V$ may not satisfy the accurate condition of the approximation!)

Solutions

Industrial Solution (a compromise): Generate one (or more) shadows from the brightest light sources (e.g. the sun)

Related Research: Imperfect shadow maps (shadow for indirect lighting) Light Cuts (many-lights in offline rendering) RTRT, Real-time Ray Tracing (path-tracing + denoising) PRT, Precomputed Radiance Transfer

Precomputed Radiance Transfer

Related papers Precomputation-Based Rendering Precomputed Radiance Transfer

Preliminaries

Convolutions Filtering: the minimum frequency of $f$ and $g$ is reserved.

Basis Functions and Reconstruction

Orthogonal Basis Functions

Projection to Orthogonal Functions

We can do Monte Carlo integration to get $c_i$.

Spherical Harmonics

All details see the supplement material: SH Lighting: The Gritty Details

Q: Why SH instead of 2D Fourier Transform? A: (I don’t know  ヽ(。>д<)ｐ) Guess: is it because low-order Fourier Transform is hard to represent the high frequency changes nearing the edges of 2D images? But why don’t use cube map instead?

Rotating SH Functions

If we wanted to rotate the projected function $\tilde{f}(x)\to R(\tilde{f})(x)$, we can rotate all it’s SH basis functions then reconstruct it.

A rotated SH basis function $R(B_i)(x)$ can be representated as a linear combination of the SH basis functions at the same order. Thus we can get the $R(\tilde{f}(x))$ projected at once, without do the sampling (i.e. integration with basis functions) again.

More over, the SH functions are rotationally invariant, i.e. rotate before SH projecting a function is the same as directly rotate the projected function. Specifically, if we rotate the original funtion $f(x)\to g(x)=R(f)(x)$ then SH project it to $\tilde g(x)$, we will get $\tilde{g}(x)=R(\tilde{f})(x)$.

Example: Diffuse IBL using SH?

Diffuse BRDF is smooth (low-frequency), so it acts like a low pass filter (when do dot product on frequency domain or convolution on spatial domain).

Thus when integrating diffuse BRDF with radiance: the resulting function value is always smooth (in convolution only the lowest frequency of the two functions is reserved).

Thus we don’t need a high-order SH to describe environment lighting when being integrated with a diffuse BRDF. (3 order SH IBL has $\leq 3\%$ RMS Error)

SH IBL in RTR (without texture, SH order $\leq 3$): $E(n) = n^TMn$

SH IBL with Shadows

Solving the triple-product integration: (3 cubemap / equirectangular images)

The two-part rendering equation perspective: Lighting: $L(i)$ Light transport: $T(p)=V(i)f(i,o)(n\cdot i)$

Assumption: Static scene(the light transport part), dynamic lighting (each environment map needs to be pre-computed) (rotating envmap $L$ can be performed once)

Basic Steps:

Precompute: project each lighting and light transport to SH basis space, usually we compute a SH vector for each vertex. At runtime, we can interpolate the shading or the SH vector for every point within the triangle.

Runtime: Diffuse case: constant BRDF

Glossy case: leave to next class…