Computer graphics III – Rendering equation and its solution Jaroslav Křivánek, MFF UK
[email protected] Global illumination – GI Direct illumination 2 Review: Reflection equation “Sum” (integral) of contributions over the hemisphere: Lr (x,ωo ) = Li (x,ωi )⋅ fr (x,ωi → ωo )⋅cosθi dωi ∫ H (x) n ω Li(x, i) Emitted radiance Lo(x, ωo) dωi θ L (x,ω ) = L (x,ω ) θo i o o e o + Lr (x,ωo ) Lr(x, ωo) Total outgoing rad. Reflected rad. From local reflection to global light transport Reflection equation (local reflection) Lo (x,ωo ) = Le (x,ωo ) + Li (x,ωi )⋅ fr (x,ωi → ωo )⋅cosθi dωi ∫ H (x) Where does the incoming radiance Li(x, ωi) come from? From other places in the scene ! Lo( r(x, ωi), -ωi) r(x, ωi) ω = ω −ω = Li (x, i ) Lo (r(x, i ), i ) L (x,ω ) i i x Ray casting function CG III (NPGR010) - J. Křivánek 2015 From local reflection to global light transport Plug for Li into the reflection equation Lo (x, ωo ) = Le (x,ωo ) + ω −ω ⋅ ω → ω ⋅ θ ω ∫ Lo (r(x, i ), i ) fr (x, i o ) cos i d i H (x) Incoming radiance Li drops out Outgoing radiance Lo at x described in terms of Lo at other points in the scene CG III (NPGR010) - J. Křivánek 2015 Rendering equation Remove the subscript “o” from the outgoing radiance: ω = ω L (x, o ) Le (x, o ) + ω −ω ω → ω θ ω ∫ L(r(x, i ), i ) fr (x, i o )cos i d i H (x) Description of the steady state = energy balance in the scene Rendering = calculate L(x, ωo) for all points visible through pixels, such that it fulfils the rendering equation CG III (NPGR010) - J.