Global Illumination: Radiosity

Global Illumination: Local vs. Global

• Global illumination captures indirect light source effects, unlike local illumination.

• Global illumination provides a more realistic rendering by considering all light interactions.

Ray Tracing and Path Tracing Review

• Both ray tracing and path tracing involve tracing light from the screen (eye) back to the light source.

• Ray tracing considers a single light path, while path tracing considers multiple paths.

• Path tracing uses methods like random walk or multi-column methods to select the best paths.

Radiosity Theory

• Radiosity is a view-independent method, unlike ray tracing and path tracing, which are view-dependent.

• It is designed for diffuse interactions, focusing on interactions between objects rather than light paths.

• Radiosity provides a different approach to solving illumination problems without requiring recursion.

• It directly calculates radiosity values without iterative estimation of integrals.

Radiosity Concept

• The scene is divided into triangle patches.

• The method is based on the conservation of energy.

• Green arrows represent light interactions between patches.

• Red and pink arrows indicate light emission and reflection by patches.

Wave Block and RGB Values

• Radiosity values depend on wavelengths, necessitating the calculation of RGB values.

• Each patch requires separate RGB value calculations due to different RGB waveforms.

• The three radiosity volumes (RGB) determine the final scene colors.

• The radioactive method calculates energy for each RGB color channel independently.

Patch Processing

• Patch processing calculates radiosity weights and involves separating the scene into patches.

• Small polygons can use polyamide directly.

• Large polygons need to be subdivided into smaller ones.

• Light is often inconsistent across large polygons, necessitating subdivision for realistic rendering.

Form Factors

• Form factor determines the proportion of energy sent from one patch to another.

• It is a key concept for calculating energy transfer between objects, not light tracing.

Radiosity Definitions and Calculations

• Radiosity is defined as energy per unit area leaving a patch per unit time.

• The energy is calculated using equations involving form factors, considering both emitted and reflected energy.

• $B_i$: Radiosity value for patch i.

• $dA_i$: Area of ​​patch i.

• $E_i$: Energy emitted from patch i.

• $R_i$: Reflected coefficient.

• $F_{ji}$: Form factor from patch j to patch i.

• $dA_i$: Area of ​​patch i.

Radiosity Equation

• Calculate energy transfer for each patch to render the entire scene.

• $B<em>i dA</em>i = E<em>i dA</em>i + R<em>i \sum</em>{j=1}^{n} F<em>{ji} B</em>j dA_j$

• This calculation is computationally expensive.

Reciprocity

• Due to reciprocity between patches I and j:

• $F<em>{ji} dA</em>j = F<em>{ij} dA</em>i$

• The main equation is simplified

• $B<em>i = E</em>i + R<em>i \sum</em>{j=1}^{n} F<em>{ij} B</em>j$

Linear System of Equations

• The problem is represented as a linear system of equations, solvable through iterative solvers.

Iterative Solver

• The equations can be solved iteratively using methods like progressive refinement.

Matrix Representation

• The linear system can be represented in matrix form for solving.

Radiosity Combination

• Radiosity value for a specific patch is a combination of its own emitted energy and the energy reflected from other patches.

Closed Form Solutions

• Closed-form solutions have a complexity, where n is the number of patches.

• $O(n^3)$

• This is computationally expensive for large n.

• Iterative solvers, such as the Jacobi method, are often used in practice.

Form Factor Calculation

• The form factor (f) is calculated based on the energy leaving one area (Aj) and arriving directly at another area (Ai).

• The cosine term represents the angle between the patches.

• $F<em>{i j}=\frac{\int</em>{A<em>{i}} \int</em>{A<em>{j}} \frac{\cos \theta</em>{i} \cos \theta<em>{j}}{\pi r^{2}} d A</em>{j} d A<em>{i}}{\int</em>{A<em>{i}} d A</em>{i}}$

Angle Considerations

• If the patches are at a right angle, there is no coupling.

• When directly facing each other and parallel, the energy transfer is maximized.

Simplified Calculations

• If area Ai is small compared to the distance r, the equation can be simplified.

Efficient Models for Computation

• Models can simplify the calculation process, such as different projections.

• These predefined models may reduce accuracy but improve efficiency, with minimal visual impact.

• The primary objective is to reduce computational complexity.

Type Cube Method

• This involves projecting a patch onto another to determine interactions.

• Interactions through a bounding box help simplify calculations.

Pixel Areas

• Delta factors are calculated for each pixel, representing the relationship between pixels and patches.

• Predefined discrete pixels are used to convert complex integral equations.

Delta Form Factor

• The delta form factor is calculated based on angles between light and pixel areas.

• $\Delta F = \frac{\cos \theta \Delta a}{\pi r^2}$

• Where:

  • $\theta$ is the angle between the light and also the pixel areas.

  • $\Delta a$ is the area of the pixels.

  • r is the distance.

Simplifications

• If the light is directly above, the equation simplifies based on area and distance.

• Pre-calculated delta numbers are saved in a dictionary to speed up computations.

Projections and Visibility

• Determine which patches are visible from the pixels.

• Ray tracing or depth buffers can identify occluded areas, similar to shadow mapping.

Relationship Between Pixels and Patches

• Identify the nearest visible patch using ray tracing or buffer methods.

• Calculate delta form factors for each patch and save them in a table for quick lookup.

Computational Issues

• Discrete computation methods may introduce errors, but these are often insignificant to the overall visual effect.

Reciprocity Issues

• Reciprocity means energy exchange is equivalent between patches.

• Only half the patches need to be calculated.

Interpolations

• Interpolation methods improve the final result if aliasing occurs.

• Linear or cubic interpolation can smooth the radiosity values, with cubic interpolation providing better results.

Artifacts and Discontinuities

• Discontinuities can occur between shadow and non-shadow parts.

• Accurate mesh separation can minimize these artifacts.

Adaptive Solutions

• Adaptive methods compare patch values nearby to smooth boundary interactions.

• Increase density or move boundaries to maximize mesh reflection of boundaries.

Radiosity Computations: Summary

• Viewpoint independent, focusing on object interactions.

• Divided into separate patches for computational efficiency.

• Form factors computed by different methods, including delta form factors.

• Matrix equations solved for each patch, emphasizing computational efficiency.