Radiosity

General

Ray Tracing vs. Radiosity

• Ray Tracing and radiosity complement each other.
• Ray Tracing is good at:
  • Perfect specular reflections.
  • Refractions and transmissions.
  • Hard shadows.
• Ray Tracing is expensive for:
  • Diffuse inter-reflections.
  • Color bleeding.
  • Soft indirect light.
  • Scattering over many diffuse surfaces.
• Radiosity solves the diffuse-light problem.
• It is especially useful for indoor scenes where diffuse surfaces dominate.
• It is not efficient for specular reflection or refraction.

Object Space vs. Image Space

• Ray Tracing is an image-space method.
  • It traces rays per pixel.
  • It is tied to the current camera view.
• Radiosity is an object-space method.
  • It computes light transfer between scene patches.
  • It is view independent.
  • The camera can move without recomputing radiosity.
  • Recompute only when scene geometry, lights, or materials change.

Radiosity Setup

Patches

• Split the scene into surface patches.
• Patches can be uniform or adaptive.
• Light sources are also approximated as patches.
• Compute light energy transfer from patch to patch.
• Cost is roughly $O(n^2)$ for $n$ patches.
• The low-resolution radiosity result is reconstructed onto the final scene.

Radiosity Equation

• Radiosity $B_i$ is the energy leaving patch $i$ per unit area and time.
• It contains emitted light and reflected incoming light.

• $E_i$ is emission of patch $i$.
• $\rho_i$ is reflectance of patch $i$.
• $F_{ij}$ is the form factor from patch $i$ to patch $j$.
• $B_j$ is radiosity of patch $j$.

Form Factors

Meaning

• A form factor describes how much light leaving one patch reaches another patch.
• It depends on:
  • Distance.
  • Orientation.
  • Visible area.
  • Occlusion.
• Form factors are reciprocal because light transport is symmetric when patch properties are swapped.

Visibility

• Patches can be blocked by other geometry.
• Use a visibility factor $V_{ij}$.
• $V=1$ means visible.
• $V=0$ means blocked.
• Occlusion can remove transfer between patches.

Exact Solution

• Exact surface-to-surface form factors need a double integral over both patch areas.
• This is too expensive for practical rendering.
• Practical methods approximate the integral.

Computing Form Factors

Nusselt Analog

• Place a hemisphere above a patch point.
• Project the other surface onto the hemisphere.
• Project the hemisphere onto the base disk.
• Projected area divided by $\pi$ gives the form factor contribution.

Hemicube Method

• Approximate the hemisphere with a half cube.
• Center the hemicube above a patch point.
• Project scene polygons onto the cube faces.
• Use z-buffering for visibility.
• Sum the covered cell contributions.
• Advantage:
  • Uses standard graphics pipeline operations.
  • Can run on GPU.
• Disadvantage:
  • Discretization errors.
  • Aliasing, visibility, and proximity errors.
  • Higher resolution becomes slower.

Area Sampling

• Subdivide the destination patch into small samples.
• Cast rays between the source point and destination samples.
• Add a contribution only if the ray is visible.
• It is slower than hemicube rendering.
• Accuracy can be increased as needed.
• It is preferred in modern implementations.

Solving Radiosity

Linear System

• Every patch contributes one equation.
• All equations together form a large linear system.
• Direct solving is usually infeasible.
• Gaussian elimination is around $O(n^3)$.
• Iterative methods are preferred.

Gathering

• Compute one patch by gathering light from all other patches.
• Initialize emitting patches with $B=E$.
• Update patches repeatedly until convergence.
• Typical methods:
  • Jacobi.
  • Gauss-Seidel.

Shooting

• Choose one patch and shoot its unshot energy to all other patches.
• Initialize emitting patches with $B=E$.
• Often shoot the patch with the highest unshot energy first.
• Typical method:
  • Southwell.
  • Progressive radiosity.

Progressive Radiosity

• Iterative solutions produce usable images early.
• Every iteration improves the result.
• Stop when the image changes only slightly.
• Frame rate is inversely related to iteration cost.

Light Bounces

Neumann Series

• Radiosity can be interpreted as a sum of light bounces.
• $I$ term means emission.
• $A$ term means first bounce.
• $A^2$ term means second bounce.
• $A^k$ term means kth bounce.
• This is useful when only the first few bounces matter.

Patch Subdivision

Uniform Patches

• Uniform patches are simple.
• Low resolution creates artifacts:
  • Blocky shadows.
  • Discontinuities.
  • Rough indirect light.
• Increasing resolution everywhere is expensive.

Adaptive Patches

• Refine only where needed:
  • Shadow borders.
  • Strong light changes.
  • High-detail regions.
• Keep unimportant regions coarse.
• This gives a better quality-cost tradeoff.

Reconstruction

• Mesh surfaces into patches.
• Compute form factors.
• Solve the radiosity equation.
• Reconstruct or interpolate the low-resolution result.
• Remember:
  • Radiosity is best for diffuse, soft, indirect light.
  • Ray Tracing is best for specular effects, refraction, and hard shadows.