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Radiosity Methods `LD*E`

Diffuse-diffuse interactions are modelled, with principles of radiative heat transfer.

Scene discretized into perfectly diffuse patches.

View dependent, radiosity map contains constant radiosity per patch in scene.

Derived from Area formulation of rendering equation.

\[B(x) = E(x) + R(x) \int_S B(x') F(x', x) dA \\ B(x)dA_i = E(x)dA_i + R_i \int_j B_j F_{ji} dA_j \\ B(x)dA = E(x)dA + R_i \int_j B_j F_{ij} dA_i \\ \text{radiosity $\times$ area = emitted power $+$ reflected power}\]

$R \in [0, 1]$ is reflectivity/albedo.
$F_ji$ is form factor of energy leaving $j$ and entering $i$.

Form factors and Reciprocity

$F_{ij}$: Energy leaving $i$ and striking $j$ $\div$ Total energy leaving $i$ in hemisphere.

\[F_{ij} = \frac{1}{A_i} \int_{A_i}\int_{A_j} \frac{\cos \phi_i \phi_j}{\pi r^2} d_{A_i}d_{A_j}\]

Approximation: The form factor from $i$ to $j$ can be approximated with the form factor from $dA_i$ to patch $j$.

\[F_{dA_iA_j} = \int_{A_j} \frac{\cos \phi_i \phi_j}{\pi r^2} d_{A_j}\] \[F_{ji}dA_j = F_{ij}dA_i $ \le 1 \quad \text{ for all } i,j\]

Hemicube

Delta form factor: The form factor on each pixel on the hemicube

Form factor: $F_{ij} = \sum_k F_k$ for each pixel $k$ covered by projection patch to $j$ on the hemicube.

Computing hemicube delta form factors:

Take the vector $v$ from bottom-center of hemicube to center of the pixel.
$r = \sqrt{\text{length}(v)}$
$\cos \phi_i = \text{norm}(v) \cdot (0, 0, 1)$
$\cos \phi_j = \text{norm}(v) \cdot \text{opposite of face direction }\pm x, \pm y, \pm z$
Compute area $\Delta A = \frac{2.0}{h} * \frac{2.0}{w}$

System of linear equations

From each patch’s radiosity equation $B_i = E_i + R_i \int_j B_j F_{ij}$, we can get a system of $n$ linear equations that we can represent in a matrix.

Gaussian elimination $O(n^3)$ is inefficient
Iterative methods (Jacobi iteration/Gauss-Seidel) $O(kn^2)$ for $k$ iterations

Iterative methods to solve $Ax = E$

Iteratively compute the equation below for each iteration $k$.

Jacobi:

\[x_{i, k+1} = (E_i - a_{i1}x_{i,k} - \dots - a_{i,i-1}x_{i-1, k} - a_{i,i+1}x_{i+1, k} - \dots - a_{i,n}x_{n, k})/a_{ii}\]

Gauss-Seidel:

\[x_{i, k+1} = (E_i - a_{i1}x_{i,\textcolor{red}{k+1}} - \dots - a_{i,i-1}x_{i-1, \textcolor{red}{k+1}} - a_{i,i+1}x_{i+1, k} - \dots - a_{i,n}x_{n, k})/a_{ii}\]

Patch sizes

Shooter patches:

many, too small: Too much computation
few, too large and far apart: Banding artifacts due to “multiple point light sources”

Gatherer patches:

too large: blocky and jagged edges that are not sharp. Solvable by adapative subdivision/substructuring

Adaptive subdivision

Use initial patches to compute radiosity
If local radiosity variation too large, subdivide that patch
Completely recompute radiosity
Repeat steps 2 and 3 until local radiosity is small enough.

Substructuring

Use initial patches to compute radiosity
If local radiosity variation too large, subdivide that patch
Compute only the form factors from each new subpatch to initial patches
Compute radiosity by $B_{iq} = E_i + R_i \sum_{j=1}^n B_jF_{(iq), j}$
Repeat steps 2 and 3 until local radiosity is small enough.

Radiosity Methods `LD*E`