Lecture 23: RadiosityAnd he shall be as the light of the morning € K 2 Samuel 23:41. RadiosityRadiosity models the transfer of light between surfaces. Recursive ray tracing also models light bouncing off surfaces, but ray tracing makes several simplifying assumptions that give scenes a harsh, unnatural look. Ray tracing assumes that all light sources are point sources and that the ambient light is constant throughout the scene. These assumptions lead to stark images with sharp shadows. In contrast, radiosity models all surfaces as both emitters and reflectors. This approach softens the shadows and provides a more realistic model for ambient light.Informally, radiosity is the rate at which light energy leaves a surface. There are two contributions to radiosity: emission and reflection. Henceradiosity = emitted energy + reflected energy.For the purpose of display, we shall identify intensity with radiosity.Radiosity computations typically take much longer than recursive ray tracing because the model of light is much more complex. To simplify these computations, we shall model only diffuse reflections; we shall not attempt to model specular reflections with radiosity. Since radiosity replaces ambient and diffuse intensity, radiosity is view independent. Thus we can reuse the same computation for every viewpoint once we compute the radiosity of all the surfaces. View dependent calculations are required only to compute hidden surfaces.2. The Radiosity EquationsWe will begin with a very general integral equation called the Rendering Equation based on energy conservation. We will then repeatedly simplify and discretize this equation till we get a large system of linear equations -- the Radiosity Equations -- which we can solve numerically for the radiosity of each surface. Radiosity is identified with intensity, so once we have the radiosity for each surface we can render the scene.2.1 The Rendering Equation. Energy conservation for light is equivalent toTotal Illumination = Emitted Energy + Reflected Energy.We can rewrite these innocent looking words as an integral equation called the Rendering Equation.Rendering Equation€ I(x,′ x ) = E(x,′ x ) +ρ( x,′ x ,′ ′ x )S∫I(′ x ,′ ′ x )d′ ′ x (2.1)where€ I(x,′ x ) is the total energy passing from € ′ x to € x.€ E(x,′ x ) is the energy emitted directly from € ′ x to € x.€ ρ(x,′ x ,′ ′ x ) is the reflection coefficient -- the percentage of the energy transferred from € ′ ′ x to € ′ x that is passed on to € x.Essentially all of the computations in Computer Graphics that involve light are summarized in the Rendering Equation. Notice, in particular, that the Rendering Equation is precisely the set up for recursive ray tracing!2.2 The Radiosity Equation -- Continuous Form. The continuous form of the Radiosity Equation is just the Rendering Equation restricted to diffuse reflections. Once again by conservation of energy:Radiosity = Emitted Energy + Reflected Energy.Now, however, since we are dealing only with diffuse reflections, we can be more specific about the form of the reflected energy. Restricting to diffuse reflections leads to the following integral equation for radiosity.Radiosity Equation -- Continuous Form€ B(x) = E (x) +ρd(x) B(y)S∫cosθcos ′ θ πr2(x, y)V (x, y) dy(2.2)where€ B(x) is the radiosity at the point x, which we identify with the intensity or energy reflecting off a surface in any direction -- that is, the total power leaving a surface/unit area/solid angle. This energy is uniform in all directions, since we are assuming that the scene has only diffuse reflectors.€ E(x) is the energy emitted directly from a point x in any direction. This energy is uniform in all directions, since we are assuming that the scene has only diffuse emitters.€ ρd(x) is the diffuse reflection coefficient -- the percentage of energy reflected in all directions from the surface at a point x. By definition, € 0 ≤ρd(x) ≤1.€ V(x, y) is the visibility term:€ V(x, y) = 0 if x is not visible from y.€ V(x, y) = 1 if x is visible from y.€ θ = angle between the surface normal (N) at x and the light ray (L) to y.€ ′ θ = angle between surface normal (€ ′ N ) at y and the light ray (L) to x.€ r(x, y) = distance from x to y. 2In the Radiosity Equation, the term€ B(y)S∫cosθcos ′ θ πr2(x, y)V (x, y) dy = energy reaching the point x from all other points y,so€ ρd(x) B(y)S∫cosθcos ′ θ πr2(x, y)V (x, y)dy = total energy reflected from x. The factor € 1 / r2(x, y) models an inverse square law, since the intensity of light varies inversely as the square of the distance. The factors € cosθ,cos′ θ come from Lambert’s Law (see Lecture 16, Section 4) and represent projections of the flux onto the emitting and reflecting surfaces (see Figure 1 and the accompanying discussion). The appearance of the factor π in the denominator will be explained shortly below.To understand the cosine terms better, consider two small surface patches. Recall that the intensity (or radiosity) on any facet from any other facet is given by€ Ireceptor=Light DepositedUnit Area=Beam Cross SectionReceptor Facet Area× Isourcewhere € Isource=Light EmittedUnit Area=Beam Cross SectionSource Facet Area× Iemitter.But from Figure 1,€ Beam Cross SectionFacet Area= cos(θ),cos(′ θ ).so€ Ireceptor= cos(θ)cos(′ θ )Iemitter. Facet (Receptor) N BeamCross Section θθ€ ′ θ € ′ θ € ′ N Facet(Source) Figure 1: Lambert’s Law. Intensity is given by the ratio of the beam cross section to the facet area, which, in turn, is equal to the cosine of the angle between the beam and the surface normal. Two cosines appear: one for the source and one for the receptor. Thus Lambert’s Law accounts for the two cosines that appear in the Radiosity Equation.The factor π in the denominator of the second term on the right hand side of Equation (2.2) arises for the following reason. Recall that in the Rendering Equation 3€ ρ(′ x , x,′ ′ x ) = the percentage of the energy transferred from € ′ ′ x to € x that is passed on to a single point € ′ x . (Note that here we have