CSCE 441 Computer Graphics: Radiosity Jinxiang Chai

Rendering: Illumination Computing Direct (local) illumination Light directly from light sources No shadows Indirect (global) illumination Transparent, reflective surfaces, and hard shadows (Ray tracing) Diffuse interreflections, color bleeding, and soft shadow (radiosity)

Rendering: Illumination Computing Direct (local) illumination Light directly from light sources No shadows Indirect (global) illumination Transparent, reflective surfaces, and hard shadows (Ray tracing) Diffuse interreflections, color bleeding, and soft shadow (radiosity)

Review: Ray Tracing Assumption The illumination of a point is determined by - illumination/shadow ray (direct lighting from light sources)

Review: Ray Tracing Assumption The illumination of a point is determined by - illumination/shadow ray (direct lighting from light sources)

Review: Ray Tracing Assumption The illumination of a point is determined by - illumination/shadow ray (direct lighting from light sources) - reflection ray (light reflected by an object)

Review: Ray Tracing Assumption The illumination of a point is determined by - illumination/shadow ray (direct lighting from light sources) - reflection ray (light reflected by an object) - transparent ray (light passing through an object)

Review: Ray Tracing Assumption The illumination of a point is determined by - illumination/shadow ray (direct lighting from light sources) - reflection ray (light reflected by an object) - transparent ray (light passing through an object)

Ray Tracing Assumption The illumination of a point is determined by - illumination/shadow ray (direct lighting from light sources) - reflection ray (light reflected by an object) - transparent ray (light passing through an object)

Pros and Cons of Ray Tracing Advantages of ray tracing All the advantages of the local illumination model Also handles shadows, reflection, and refraction Disadvantages of ray tracing Computational expense No diffuse inter-reflection between surfaces (i.e., color bleeding) Not physically accurate Radiosity handles these shortcomings for diffuse surfaces!

Radiosity vs. Local Illumination

Radiosity

Physical Image vs. Radiosity Rendering

Radiostiy Definition: The radiant (luminous) exitance is the radiant flux per unit area leaving a surface.

Radiosity The radiosity model computes radiant-energy interactions between all the surfaces in a scene

Radiosity: Key Idea #1

Diffuse Surface

Radiosity: Key Idea #2

Constant Surface Approximation

Radiosity Equation

Radiosity Algorithm

Energy Conservation Equation

The total rate of radiant energy leaving surface i per unit square

Energy Conservation Equation The rate of energy emitted from surface i per unit area - zero if surface i is not a light source

Energy Conservation Equation Reflectivity factor Percent of incident light that is reflected in all directions

Energy Conservation Equation Form factor Fractional amount of radiant energy from surface j that reaches surface i

Compute Form Factors The form factor specifies the fraction of the energy leaving one patch and arriving at the other. In other words, it is an expression of radiant exchange between two surface patches!

Compute Form Factors Radiant energy reaching A y from A x Radiant energy leaving A x in all directions The form factor specifies the fraction of the energy leaving one patch and arrives at the other. In other words, it is an expression of radiant exchange between two surface patches!

Form Factor: Reciprocity

Radiosity Equation Radiosity for each polygon Linear system: - : radiosity of patch I (unknown) - : emission of patch I (known) - : reflectivity of patch I (known) - : form-factor (known)

Linear System A X =B

Radiosity Algorithm

Form Factors for Infinitesimal Surfaces

Form Factors for Subdivided Patches

Form Factor: How to compute? Closed Form - anlytical Hemicube

Form Factor: Analytical

Form Factor: How to compute? Closed Form - anlytical Hemicube

Form Factor: Nusselt Analog Nusselt developed a geometric analog which allows the simple and accurate calculation of the form factor between a surface and a point on a second surface. 3D diagram

Form Factor: Nusselt Analog The form factor is, then, the area projected on the base of the hemisphere divided by the area of the base of the hemisphere, or (A/B) A B 2D diagram

Form Factor: Nusselt Analog

Form Factor: HemiCube

Project path on hemicube Add hemicube cells to compute form factors A B 2D diagram

Delta Form Factor: Top Face Top of hemicube

Delta Form Factors: Side Faces Side of hemicube

The Hemicube in Action

Form Factors: HemiCube

Form Factors

Radiosity Algorithm

How to Solve Linear System? Matrix conversion Iterative approaches - Jacobian (gathering) - Gauss-Seidel (gathering) - progressive refinement (shooting)

Matrix Conversion - Computational cost: O(N 3 ) - Very slow for a large set of polygons

Iterative Approaches

Jacobian Iterations For all patches i, i=1,…,N, While not converged: for all patches i=1,…,N

Jacobian Iterations For all patches i, i=1,…,N, While not converged: for all patches i=1,…,N Update of one patch requires evaluation of N Form Factors What’s the computational cost?

Successive Approximation

Rendering - The final Φ i 's can be used in place of intensities in a standard renderer (Gouraud) - Radiosities are constant over the extent of a patch - A standard renderer requires vertex intensities (or radiosities) - If the radiosities of surrounding patches are know, vertex radiosities can be estimated using bilinear interpolation

Vertex Intensity: Bilinear Interpolation

Consolation Room

Theatre

Steel Mills

Radiosity: Benefit Global illumination method: modeling diffuse inter- reflection Color bleeding: a red wall next to a white one casts a reddish glow on the white wall near the corner Soft shadows – an “area” light source casts a soft shadow from a polygon No ambient term hack, so when you want to look at your object in low light, you don’t have to adjust parameters of the objects – just the intensities of the lights! View independent: it assigns a brightness to every surface

Radiosity: Limitation Radiation is uniform in all directions Radiosity is piecewise constant – usual renderings make this assumption, but then interpolate cheaply to fake a nice-looking answer – this introduces quantifiable errors No surface is transparent or translucent Reflectivity is independent of directions to source and destination