Radiosity

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Project #5 due today
Final Exam (W 12/11 8:00 - 11:00 SN011)

When we began our discussion of illumination models, we were mostly concerned with direct illumination from light sources to surfaces in our scene. We did not consider the effects of other scene elements on the illummination of a given surface. This is unfortunate, because a surprizing percentage of the light reaching a surface does so via an indirect paths (such as reflection off of other surfaces) rather than a direct path from the light source. Illumination models which consider indirect illumination are called global illumination models.

Unlike most of the other rendering methods that we have discussed in this class to this point, radiosity methods are based on the principle of conservation of energy. In a closed environment the sum of all of the radiant energy emitted by light sources should equal the sum sum of the energy reflected and absorbed by the materials in the scene. This condition is called any energy equilibrium. Computing this equilibrium can be very complex, particularly when the reflection characteristics of the surfaces in the scene are complex.

In classical radiosity several simplifying assumptions are made to simplify this computation. These assumptions include:

1. All surfaces are diffuse or Lambertian reflectors
2. The emission of energy from light sources is uniformly distributed about a hemisphere situated over any given point of the surface. (i.e. the lighting is diffuse)
3. The change is surface reflectance is a slow-varying function that is adequately approximated by interpolation about a relatively few discrete points in the scene.

In order to compute radiosity, all that is needed is the geometry of the scene, the energy of the light sources (measured in radiosity units, energy per unit area per unit time), and the diffuse reflectivity at a few select points on every surface in the scene.

Fundamentals

In an enclosed environment, the total energy being absorbed and emitted at any instant is a constant. The energy emitted or absorbed by a surface is measured in units of energy flux per unit area, that is, energy per unit area per unit time. This is the radiosity of the surface.

Given a scene composed entirely of opaque diffusely reflecting surfaces (some of which may be diffuse light emitters) the energy at a single point x in the scene is given by the Lambertian illumination model:

I_out(x) is the reflected energy from point x
is the diffuse reflectivity at x
is the incident angle between the surafce and the incoming light energy
I_in() is the light energy arriving from direction throgh an infintesimal solid angle .

The equation above may be expressed as "the intensity reflected at a point x is dependent on the diffuse reflectivity and the integral of all intensities (weighted by the projected area of the emitter) incident upon that point from objects in the scene". I_out is what is usually thought of as the distribution of intensity over the scene; I_inis the intensity incident upon x through a small solid angle . Given the value of I_out(x) for all x in the scene, I_in may be computed for any given point x, incoming direction , and solid angle (about ) . This equation may be iterated. Given an initial I_{out_0} (where the 0 represents iteration, not exponentiation), a new value of may be computed for each point x in the scene, thereby generating a new intensity distribution . Clearly this process may be repeated as many times as desired.

It is clear that, given for a scene which is in energy equilibrium, iterating the given equation at every point will yield no change in I_out. In a scene not in energy equilibrium, the result is a new and different function, simulating the physical movement of radiation about the scene. Just as light levels in a room reach an equilibrium when the lights are turned on, we expect iteration of the above equation to converge to an equilibrium distribution.

In order to simplify the computation, a finite-element approach may be taken. The scene is decomposed into a collection of elements (often called patches in the radiosity literature), the union of which is all surfaces in the scene, and the pairwise intersection of which is empty. It is tacitly assumed that the emittivity, diffuse reflectivity, and total radiosity do not change across a patch.

The radiosity of a given patch is the emittivity of the patch plus the absorbed contributions of all other visible patches in the scene. The percentage of the energy radiated by a patch j and incident upon a patch i is given by the form factor (form factors are discussed in more detail later).

Radiosity Equations

B_i is the radiant energy reflected by the surface patch
E_i is the energy emmited by a illuminating surface patch
A_i is the area of a surface patch

This equation states that the power (energy per unit time) radiated from patch i is the result of the emissivity of the patch (multiplied by the area to give units of power) plus the sum of the power absorbed from all other patches in the scene. The absorbed power from a remote patch is a function of three quantities: the power emitted by the remote patch (A_jB_j), the percentage of that power reaching patch i (), and the percentage of the power incident upon patch i which is absorbed by patch i (p_i). Given the E_i, p_i, and F_i for a scene, a set of simultaneous equations may be formed that describe the equilibrium solution of the radiosity of each patch in the scene.

Form Factors

Radiosity computation is difficult because, in a geometrically complex scene, the movement of energy is complex. Many patches are either partially or completely mutually invisible, restricting the paths light may take in the scene. The form factor is the only geometric term in the radiosity formulation, so all the complexity of the scene is expressed by the array of form factors in the radiosity equation.

A form factor is a measure of the projected solid angle of one surface as seen be another. It includes contributions from both the relative orientations of the surafce patch and the visibility of patches.