Radiometry and radiosity © 1996-2018 Josef Pelikán CGG MFF UK Praha [email protected] http://cgg.mff.cuni.cz/~pepca/ Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 1 / 34 Global illumination, radiosity based on physics – energy transport (light transport) in simulated environment – first usage of radiosity in image synthesis: Cindy Goral (SIGGRAPH 1984) ➨ radiosity is able to compute diffuse light, secondary lighting, .. ➨ basic radiosity cannot do sharp reflections, mirrors, .. time consuming computation – Radiosity: light propagation only, RT: rendering Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 2 / 34 Radiosity - examples © David Bařina (WiKi) Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 3 / 34 Basic radiometry I Radiant flux, Radiant power d Q Φ = [ W ] d t Number of photons (converted to energy) per time unit (100W bulb: ~1019 photons/s, eye pupil from a monitor: 1012 p/s) dt Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 4 / 34 Basic radiometry II Irradiance, Radiant exitance, Radiosity d Φ( x) E (x) = [ W/m2 ] d A( x) Photon areal density (converted to energy) incident or radiated per time unit dA dt dA Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 5 / 34 Basic radiometry III Radiance d 2 Φ(x ,ω) 2 L( x ,ω) = ⊥ [ W/m /sr ] d Aω ( x) d σ(ω) Number of photons (converted to energy) per time unit passing through a small area perpendicular to the direction w. Radiation is directed to a small cone around the direction w. Radiance is a quantity defined as a density with respect to dA and with respect to solid angle ds(w). Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 6 / 34 Radiance I ➨ received/emitted radiance in direction : 2 – Lin() (Le(), Lout()) [ W/(m · sr) ] 2 N d L x, out dA d cos d dB out d cos x dI dA dA cos Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 7 / 34 Solid angles dA r2 sin d d d dA d sin d d r r2 d [] .. steradian (sr) the whole sphere .. 4 sr Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 8 / 34 Radiance II Φ(x ,ω) ∝ d σ(ω) sensor lens d σ(ω) x ω pixel Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 9 / 34 Radiance III ⊥ Φ(x ,ω) ∝ d Aω (x) ⊥ sensor d Aω x lens ω n dA θ pixel ω ⊥ d Aω = d A⋅cosθ Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 10 / 34 Energy preservation law (ray / fiber) L1() L1 d1 dA1 L2 d 2 dA2 r emitted received d1 dA1 power power L2() d 2 dA2 Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 11 / 34 Energy preservation law (ray / fiber) L1() L1 d1 dA1 L2 d 2 dA2 r T d dA d dA d 1 1 2 2 dA 1 1 dA dA 1 2 r2 L2() ray capacity L1 L2 d 2 dA2 ray ... radiance L Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 12 / 34 Light measurement ➨ measured quantity is proportional to radiance from visible scene sensor: area A2 aperture: area A1 R Lin A, cos d dA Lin T A2 Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 13 / 34 BSDF (Local transfer function) („Bidirectional Scattering Distribution Function“, older term: BRDF) L ( ) n i i Lo(o) dswi o i d L (ω ) d L (ω ) f (ω →ω ) = o o = o o s i o d E (ω ) ⊥ i Li (ωi) cosθi d σ (ωi) Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 14 / 34 Helmholtz law (reciprocity) ➨ for real surfaces (physically plausible): f in out f out in ➨ general BSDF needs not be isotropic (invariant to rotation around surface normal) – metal surfaces polished in one direction, .. f in,in,out,out f in,in ,out,out Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 15 / 34 Local rendering equation L (x, ) n i i Lo(x,o) ray-cast dswi Ls(x,o) xi vacuum: Li(x,) = Lo(xM(x,w),-) Le(x,o) = L (y,-) x o own emission at x Lo( x ,ωo) = Le ( x ,ωo) + ⊥ +∫ Lo( y ,−ωi)⋅f s( x ,ωi →ωo)⋅d σ x (ωi) Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 16 / 34 Radiance received from a surface N y dA y yo Nx xi cosyo dA di x y 2 x cos cos Geometric term: G y,x yo xi x y 2 Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 17 / 34 Radiance received from a surface Lo x, o integral over all incoming directions Le x, o f x,i o Li x,i cosxi di Le x, o f x,i o Lo y,i G y,x dA S integral over an emitting surface (assumption: the whole surface S is visible from x) Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 18 / 34 Reflected light N z z N N y dA x zi y yo xo xi x if y sees x Terminology: L y,x Lo y,x y Li x, y x f y,x, z f x, y x z x Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 19 / 34 Indirect radiance equation 1 if y sees x V y,x 0 else L x, z Le x, z f y,x, z L y,x G y,x V y,x dA S BRDF geometric own (emitted) terms radiant exitance Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 20 / 34 Radiosity equation ➨ assumption – ideal diffuse (Lambertian) surface: – BRDF is not dependent on incoming/outgoing angles – outgoing radiance L(y,) independent on direction L x, z Le x, z f x L y,x G y,x V y,x dA S L x, z B x , Le x, z E x , f x x G y,x V y,x B x E x x B y dA S Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 21 / 34 Discrete solution B x E x x B y g y,x dA S G y,x V y,x where g y,x solution B is infinit-dimensional ➨ discretization of the task: – Monte-Carlo ray-tracing (dependent on camera) – classical radosity (finite/boundary elements FEM) Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 22 / 34 General radiosity method object surfaces divided into set of elements definition of knot points on elements – radiosity will be computed there choice of an approximation method and error metric – basis functions for convex blend from knot points coefficients of linear equation system – “form-factors” Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 23 / 34 General radiosity method solution of linear equation system – result: radiosity in knot points reconstruction of values on whole surfaces – linear blends using basis functions and knot point radiosities rendering of results (arbitrary view) – light is proportional to radiosity Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 24 / 34 Remarks step is performed in algorithm design phase – does not appear in an implementation some advanced methods do not strictly follow the sequence to – sometimes a computation flow goes back to some previous phase, some phases can be iterated,.. Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 25 / 34 Radiosity approximation constant bilinear quadratic (knots in (knots in (more knots centers) vertices) in centers..) Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 26 / 34 Constant elements ➨ on every element Ai constant reflectivity is assumed , radiosity B – average of B(x): – terminology: i, Bi for i = 1 .. N average over B x E x x B y g y,x dA area Ai S 1 N B E B g y,x dA dA i i i A j j i i j1 Ai Aj radiosity received in point x (lying on Ai) Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 27 / 34 Basic radiosity equation switching sum and integral: N 1 Bi Ei i Bj g y,x dAj dAi Ai j1 Ai Aj geometric term – form factor Fij (part of energy irradiated from Ai received directly by Aj) N B E B F W i i i j ij m2 j1 Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 28 / 34 Intuitive derivation N BiAi EiAi i BjAj Fji W j1 emitted power = own power + reflected power reciprocal rule: Aj Fji Ai Fij N 1 BiAi EiAi i Bj Fij Ai Ai j1 N B E B F W i i i j ij m2 j1 Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 29 / 34 System of linear equations N Bi i Bj Fij Ei i 1.. N j1 1 1F1,1 1F1,2 .. 1F1,N B1 E1 2F2,1 1 2F2,2 .. 2F2,N B2 E2 .. .. .. .. .. .. NFN,1 NFN,2 .. 1 NFN,N BN EN vector of unknown vars [Bi] Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 30 / 34 System of linear equations ➨ for planar (convex) surfaces: Fii = 0 – the diagonal contain only unit values ➨ nondiagonal items are usually very small (abs value) – matrix is “diagonally dominant” system is stable and can be solved by iterative methods (Jacobi, Gauss-Seidel) ➨ for light change (light sources) [Ei] system needs not to be fully re-computed, only reverse phase could be done Radiometry 2018 © Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 31 / 34 Radiosity to vertices Even in constant element approach usage of some color interpolation method is recommended (Gouraud) (B +B )/2 1 2 B2 B1 B B 2 1 ..