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18. Radiosity

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Reading Recommended : Watt, Chapter 10. 18. Radiosity : Optional M. F. Cohen and J. R. Wallace. Radiosity and Realistic Image Synthesis. Academic Press. 1993. 2 1 Physically-based rendering Physically-based rendering, cont'd Basic optics Why physically-based? Physics of light and color Insights into problem of rendering Geometrical optics Illumination engineering Ray \metaphor" Quest for realism Re ection and transmission A \grand challenge" problem in graphics Radiative transfer The light holo deck...someday Measurement: radiometry and photometry Transp ort theory and integral equations 4 3 Radiometry Radiance So far we have considered b ouncing of light around ro om In graphics and illumination engineering, the measure of using ray tracing. energy along a ray is called radiance. What is the fundamental quantity that is b ouncing around? Rougly sp eaking, radiance is the power per unit area per unit direction passing through a p oint p in a particular direction u: This quanitity is a function of some variables. What are they? Lp; u For the moment, we are ignoring wavelength dep endence. One of the most imp ortant prop erties of radiance: radiance is constant along a ray travel ling through empty space. 6 5 The BRDF The grand scheme en an incoming direction and an outgoing direction, only Giv Photons/light Physics p ortion of the light is re ected dep ending on the surface a Transport theory material prop erties. Mathematics Lp; u Lp; u out Integral equations in Radiometry u out Computer Science u in Algorithms p Numerics dA Surface balance equation This re ection function is called the BRDF, or Bi-directional Re ectance Distribution Function. Surface rendering equation Lp; u out f p; u ! u r in out Lp; u in use the sign here, b ecause, strictly sp eaking we need to I Radiosity equation include a di erential and a cosine. Let's stick with our intuitive de nition. 7 8 Surface balance equation The surface rendering equation We can decomp ose the radiance coming from a surface in Our ob jective is to solve for the radiance function, Lp; u out terms of: over all surfaces in all directions. [outgoing] = [emitted] + [re ected] + [transmitted] Given the geometric relationship between two patches: Lp; u = L p; u + L p; u + L p; u p out e out r out t out j n j j dA j In our derivations, we will ignore transmission. Lp ; u i out n i Lp ; u i in i p i dA i The equation we are trying to solve is: Lp ; u = L p ; u i out e i out Z + f p ; u ! u Gp ; p Lp ; u dA r i in out i j i in j M Where M refers to all surfaces in the scene and Gp ; p i j describ es how eciently light is transp orted from surface j to surface i. This equation is known as the surface rendering equation. 10 9 Radiosity Radiosity and re ectance Solving the full blown rendering equation is very hard. What We could just go ahead and work on solving the new happ ens when we have only di use surfaces in a scene? rendering equation, but we must ob ey some conventions rst... Light is re ected equally in all directions, so: Radiosity, B p, is the power per unit area leaving a surface. f p; u ! u = f p r in out r;d For a di use surface, it is related to the radiance by: In addition, the radiance from each p oint is equal in all directions: B p = L p d Lp; u = L p out d Re ectance, p, is the ratio of the power per unit area leaving a surface to the power per unit area striking a surface. After dropping angular dep endencies, the rendering equation b ecomes: Re ectance varies from 0 to 1 and is related to the BRDF of a di use surface by: Z f p Gp ; p L p dA L p = L p + r;d i i j o j j d i e i M p = f p r;d Thus, we now need to solve for a function L p over 2D d surfaces only. 11 12 The radiosity equation Intuition for the radiosity equation After making some subsitutions, we arrive at: To solve for the discrete radiosity function, we will break a scene into patches and solve a light transp ort equation. Z Gp ;p i j Bp dA B p = E p + p j j i i i M Z B p = E p +p Fp ;p Bp dA i i i i j j j M The discrete form of this equation is: A room containinfg a light and one object Break up the room into ”patches” X B = E + F B i i i ij j j Each patch receives light from other patches... ...and reflects light back to these patches. 13 14 Intuition for the radiosity equation, cont'd The form factor Consider the transp ort of energy from patch j to i: Consider the light impinging on one patch: r ij j j j i i i Transport path Geometrical notation Now let's vary the size, orientation, visibility, and distance between the patches: j i j j j j i i i i (a) (b) (c) (d) We can compute the re ected radiosity, B , by re ecting and i summing the contributions from every other patch: j occluder j X i B = E + F B i i i ij j i j (e) (f) The F are called the form factors and tells us how How do es the transp ort vary in each of a-f ? ij radiosity from one patch is transp orted to another. But what form do es it take? 15 16 The form factor, cont'd Matrix form of the radiosity equation Here again is the radiosity equation: Let's put all these terms together: X B = E + F B i i i ij j V A cos cos j ij j i j F = ij 2 r ij We can re-write this as: X B F B = E i i ij j i j Now we can write a matrix equation and solve! 2 3 2 2 3 3 B 1 F F F E 6 7 1 6 1 11 1 12 1 1n 6 7 1 7 6 7 6 6 7 7 6 7 6 6 7 7 6 7 6 6 7 7 B F F E 2 2 21 2 2n 2 6 7 6 6 7 7 6 7 6 6 7 7 6 7 6 6 7 7 6 7 6 6 7 7 6 7 6 6 7 7 = 6 7 6 6 7 7 6 7 6 6 7 7 6 7 6 6 7 7 6 7 6 6 7 7 6 7 6 6 7 7 6 7 6 6 7 7 F F n1 n1;1 n1 n1;n 6 7 6 6 7 7 4 5 4 4 5 5 F F 1 F B E n n1 n n;n1 n nn n n 17 18 Matrix form of the radiosity equation, cont'd Solving the radiosity equation If we de ne: Iterative metho ds will generally get you close to the solution 2 3 in fewer steps. F F 1 11 1 1n 6 7 6 7 6 7 6 7 6 7 6 7 6 7 One p ossibility is to do a von Neumann expansion: K = 6 7 6 7 6 7 6 7 6 7 4 5 F F n n1 n nn 1 2 3 I K = I + K + K + K + ::: Then we can re-write the equation as: so that we could compute a truncated series: I K B = E 2 3 B = E + KE + K E + K E + ::: 1 One way to solve this is to compute B = I K E directly. This is really exp ensive. The matrix, I K is generally 2 fairly dense O n interactions, and inversion requires 3 O n op erations. 19 20 Solving the radiosity equation, cont'd Solving the radiosity equation, cont'd We can write this as iterative matrix multiplication: Whenever we multiply a row of K by B, we can think of this as gathering energy into a patch. A faster converging metho d is based on the notion of 0 B = E \sho oting" energy from patches. 1 0 0 B = B + KB 2 1 1 B = B + KB 3 2 2 B = B + KB Note that each multiplication by K corresp onds to b ouncing the radiosity through the scene one more time. 2 This still converges fairly slowly and runs in O n assuming you can stop after a number of iterations much less than n. 22 21 Solving the radiosity equation, cont'd Solving the radiosity equation, cont'd The idea is: Another idea is to compute the interactions hierarchical ly. 1. Sort the patches by the amount of radiosity they If two patches are close together, then link them in the usual currently have. way. If they are further apart, then link their parent patches. 2. Sho ot the energy from the brightest patch to all the other patches. 3. Mark this patch as having zero \unshot" radiosity. 4. Cho ose the next patch with the largest unshot radiosity and iterate. This approach is called \progressive radiosity". The convergence is substantially faster, closer to O n in practice.
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