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46 CS488/688 Introduction to Computer Graphics 16.4 Radiosity [Last Used: Fall 1996 Final] • Briefly described the aspect of light modeled by radiosity techniques. • Briefly describe how radiosity models this aspect of light. • Although computing a radiosity solution is as slow (or slower) than ray tracing a single image, what advantage does radiosity have over raytracing? 16.5 Form Factors and Equations [Last Used Spring 1997 Final] The simple energy balance equation is ∫ B(x) = E(x) + ρ d (x) y∈S B(y) cos θ xy cos θ yx π ‖ x − y ‖ 2 V (x, y)dA y where B(x) is the total radiosity emanating from any surface point x in a scene S, E(x) gives the portion of the radiosity self-emitted from x, and the integral gives the portion of the radiosity received from all other surface points y and then re-emitted. V (x, y) = 1 if the line between surface points x and y is unobstructed by any surface, and V (x, y) = 0 otherwise. θ yx is the angle between the surface normal at y and the xy line, and θ xy is the angle between the surface normal at x and that same line. This equation assumes that all surfaces in the scene are ideal diffuse (Lambertian) reflectors. ρ d (x) is the reflection distribution function, which describes the proportion of incoming energy to any point that is re-emitted. In general, the above equation can only be solved numerically, and the result will only provide an approximation to B. A common way to begin the computation is to express the approximation to the (unknown) function B in terms of basis functions N i : B(x) ≈ ˜B(x) ≡ ∑ i B i N i (x) and to approximate the (known) function E in the same terms: E(x) ≈ Ẽ(x) ≡ ∑ i E i N i (x) Substituting ˜B and Ẽ into the energy balance equation will result in a residual r(x) no matter what values for B i are computed. The idea, however, is to find values for B i that make r(x) small in some way. Two approaches are common: (a) Make r(x j ) = 0 for a selection of points x j ∈ S; (b) Make ∫ x∈S r(x)N j(x)dA x = 0 for all basis functions N j . Assume all surfaces in the scene are decomposed into disjoint polygons P j of area A j and that N j is 1 on P j and 0 otherwise. Assume that x j is a selected point on P j and that ρ d (x) ≡ ρ d (x j ) ≡ ρ j on P j . 1. Show that approach (a) leads to linear equations ∑ B i = E i + ρ i F ij B j and give an expression for F ij . 2. Show that (b) leads to the same linear equations with a different expression for F ij . j
CS488/688 Introduction to Computer Graphics 47 16.6 Radiosity vs. Ray Tracing [Last Used: Fall 2002 Final] The energy balance equation gives an expression for the amount of energy (radiance) leaving a point on a surface in the scene. Essentially, it is a formula for L(x, θ x , φ x , λ), where x is a point on the surface, θ x is the incoming direction, φ x is the outgoing direction (both directions are 3-space vectors), and λ is the wavelength of light for which we are computing the energy leaving the surface. This equation involves an emission term, and a triple integral over all incoming directions and wavelengths of the incoming energy from the scene, with terms to account for the relevant geometry. The equation also accounts for the reflective properties of the surface in question. In general, not only is the equation impossible to solve analytically, we are unable to measure many of the properties that are required (such the reflective properties of the surface). For those who are interested, one version of this equation is given below. However, the equation itself is not needed to answer this question. 1. Describe what simpflifying assumptions and approximations are made in ray tracing to allow us to solve a simplified version of this equation (you do not have to give this equation). As a result of these assumptions, what visual effects are we unable to model, and what other artifacts are introduced in the final results? 2. Describe what simplifying assumptions and approximations are made in radiosity to allow us to solve a simplified version of this equation (you do not have to give the equation). As a result of these assumptions, what visual effects are we unable to model, and what other artifacts are introduced in the final results? Energy Balance Equation: L o (x, θ o x, φ o x, λ o ) = L e (x, θ o x, φ o x, λ o ) + ∫ π 2 • L o (x, θ o x, φ o x, λ o ) is the radiance – at wavelength λ o – leaving point x – in direction θ o x, φ o x 0 ∫ 2π ∫ λmax 0 λ min ρ bd (x, θ i x, φ i x, λ i , θ o x, φ o x, λ o ) cos(θ i x)L i (x, θ i x, φ i x, λ i )dλ i sin(θ i x)dφ i xdθ i x • L e (x, θ o x, φ o x, λ o ) is the radiance emitted by the surface from the point • L i (x, θ i x, φ i x, λ i ) is the incident radiance impinging on the point • ρ bd (x, θ i x, φ i x, λ i , θ o x, φ o x, λ o ) is the BRDF at the point – describes the surface’s interaction with light at the point • the integration is over the hemisphere above the point
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