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Radiosity Radiosity

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Radiosity Radiosity Radiosity Radiosity Motivation: what is missing in ray-traced images? Indirect illumination effects Color bleeding Soft shadows Radiosity is a physically-based illumination algorithm capable of simulating the above phenomena in a scene made of ideal diffuse surfaces. Books: Cohen and Wallace, Radiosity and Realistic Image Synthesis, Academic Press Professional 1993. Sillion and Puech, Radiosity and Global Illumination, Morgan- Kaufmann, 1994. Indirect illumination effects Radiosity in a Nutshell Break surfaces into many small elements Light source Formulate and solve a linear system of equations that models the equilibrium of inter-reflected Eye light in a scene. Diffuse Reflection The solution gives us the amount of light leaving each point on each surface in the scene. Once solution is computed, the shaded elements can be quickly rendered from any viewpoint. Meshing (partition into elements) Radiosity Input geometry Change geometry Form-Factors Change light or colors Solution Render Change view 1 Radiometric quantities The Radiosity Equation Radiant energy [J] Assume that surfaces in the scene have been discretized into n small elements. Radiant power (flux): radiant energy per second [W] Assume that each element emits/reflects light Irradiance (flux density): incident radiant power per uniformly across its surface. unit area [W/m2] Define the radiosity B as the total hemispherical Radiosity (flux density): outgoing radiant power per flux density (W/m2) leaving a surface. unit area [W/m2] Let’’s write down an expression describing the total flux (light power) leaving element i in the Radiance (angular flux dedensity):nsity): radiant power per scene: unit projected area per unit solid angle [W/(m2 sr)] scene: total flux = emitted flux + reflected flux The Radiosity Equation The Form Factor Total flux leaving element i: Bi Ai The form factor Fji tells us how much of the flux Total flux emitted by element i: Ei Ai leaving element j actually reaches element i. Total reflected flux: 1 cosθ cosθ F = x y V (x, y)dA dA (reflectance of element i)*(the total incoming flux) ij ∫∫ 2 x y Ai xA∈∈A y π x − y total incoming flux = sum of contributions from all other ij elements in the scene ρi ∑ B j Aj Fji j The full radiosity equation is then: n Bi Ai = Ei Ai + ρi ∑ B j Aj Fji j=1 Form-Factor Computation Properties of Form Factors Reciprocity: Ai Fij = Aj Fji Additivity: Fi( j∪k ) = Fij + Fik Conservation of energy in a closed environment: n ∑ Fij = 1 j=1 2 The Radiosity Equation Finally... n The radiosity equation Bi Ai = Ei Ai + ρi ∑ B j Aj Fji A linear system of n equations in n unknowns: j=1 n ⎡1− ρ1F11 − ρ1F12 − ρ1F1n ⎤⎡B1 ⎤ ⎡E1 ⎤ Aj L Divide equation by A : Bi = Ei + ρi B j A Fji ⎢ ⎥⎢ ⎥ ⎢ ⎥ Divide equation by Ai: ∑ i − ρ F 1− ρ F B E j=1 ⎢ 2 21 2 22 M ⎥⎢ 2 ⎥ = ⎢ 2 ⎥ n ⎢ M O M ⎥⎢ M ⎥ ⎢ M ⎥ B = E + ρ B F Apply form-factor reciprocity: i i i ∑ j ij ⎢ − ρ F 1− ρ F ⎥⎢B ⎥ ⎢E ⎥ j=1 ⎣ n n1 L L n nn ⎦⎣ n ⎦ ⎣ n ⎦ We can write this using matrix notation: ⎡B1 ⎤ ⎡E1 ⎤ ⎡ ⎤⎡B1 ⎤ ⎢ ⎥ = ⎢ ⎥ + ⎢ ⎥⎢ ⎥ ⎢ M ⎥ ⎢ M ⎥ ⎢ ⎥⎢ M ⎥ ⎣⎢Bn ⎦⎥ ⎣⎢En ⎦⎥ ⎣⎢ ⎦⎥⎣⎢Bn ⎦⎥ The Radiosity Method Examples: Take as input a geometric model of the scene, with emission and reflection properties of each surface Step 1 - Meshing: Discretize input surfaces into a mesh of small elements Step 2 - Setup: Compute the form factors Fij Step 3 - Solution: Solve the resulting linear system of equations Step 4 - Display: Render shaded elements from any desired view point. These steps are often interleaved in practice. Examples: Examples: 3 Solving the Equation Progressive Radiosity The naive approach - Gaussian elimination While not converged: Requires O(n2) memory to store the matrix Select one element in the scene as the current light source Requires O(n3) time to solve the equation “Shoot” radiosity from the light source to the rest of the scene The solution process mimics the physical process of light A better approach - iterative solution propagation in the scene. Jacobi iteration Must take care not to shoot the same light more than once Gauss-Seidel iteration (keep track of “unshot radiosity”) Southwell relaxation (known as Progressive Radiosity) B[i] = Unshot[i] = E[i] while (not converged) { Due to special properties of the radiosity matrix, it is Choose i with largest Unshot[i]*A[i] possible to prove that these iterative methods are guaranteed to converge to the correct solution. Shoot(i) } Progressive Radiosity Progressive Radiosity B[i] = Unshot[i] = E[i] In each iteration the algorithm computes n form factors on while (not converged) { the fly, removing the O(n2) storage complexity. Choose i with largest Unshot[i]*A[i] Shoot(i) } Choosing the “brightest” shooter at each iteration makes the solution to converge rapidly during the first iterations. Shoot(i): for j = 1..n { It is possible to display the solution after each iteration, Compute the form factor FF[i,j] resulting in a progressive sequence of images. Delta[j] = ρ[j] FF[i,j] Unshot[i] A[i]/A[j] B[j] += Delta[j] Typically, there is no need to run until complete convergence. Unshot[j] += Delta[j] The process can be stopped after relatively few iterations. } Unshot[i] = 0 Progressive Radiosity Example: Progressive Radiosity + Ambient 0, 8, 16, 25, 50, 100 iterations 0, 8, 16, 25, 50, 100 iterations 4 Progressive Radiosity with Adaptive Meshing (0,8,16,50,100) Meshing (0,8,16,50,100) Form-Factor Computation Analytic as well as approximate formulas exist for various configurations. A closed form expression for a form factor between two polygons (Schroder 93): extremely complicated formula does not take into account occlusion A common approximation is to assume that the inner integral is constant for all locations x in element i: 1 cosθ cosθ F = x y V (x, y)dA dA ij A ∫∫ π r2 x y i xA∈∈Aijy cosθ cosθ ≈ x y V (x, y)dA ∫ π r2 y y∈Aj Form-Factor Computation Form-Factor Computation The remaining integral is approximated as a finite sum: cosθ cosθ F ≈ x y V (x, y)dA ij ∫ π r2 y y∈Aj cosθ cosθ ≈ x y V (x, y)ΔA ∑ π r2 y This approximation works well so long as the elements i and j are well-separated from each other (the distance between them is large relative to their sizes) Nusselt’’s Method Hemicube 5 6 7.
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