350 likes | 422 Views
This text delves into image synthesis and the illusion of viewing real-world objects on a monitor. It covers topics such as tone mapping, pixel tones, color perception, and light measurements. The content explores the perception of non-monochromatic light, the propagation of light, and the interaction between light and surfaces. The rendering equations and the Bidirectional Reflectance Distribution Function (BRDF) are also discussed. The text provides an in-depth analysis of light properties and radiance, sensitivity of measuring devices, and the modeling of eye and camera systems in rendering scenes.
E N D
RenderingProblem László Szirmay-Kalos
Image synthesis: illusion of watching real world objects monitor Tone mapping pixel S fr (’, x, ) We(x,) Le(x,) Color perception
Measuring the light: Flux • Power going through a boundary [Watt] • Number of photons • Spectral dependence: F [, +d] F ()
g() r() b() 400 500 600 700 Color perception perception: r, g, b r, g, b
Perception of non-monochromatic light F () r=F () r () d F (i) r (i) i g=F () g () d b =F () b () d
Representative wavelengths Fe() F () Light propagation: Linear functional: F () = T(Fe()) r=F () r () d = F (i) r (i) i = r= T(Fe( i))r (i) i
Measuring the directions: 2D 2D case Direction: angle from a reference direction Directional set: angle [rad] arc of a unit circle Size: length of the arc Total size: 2
Measuring the directions: 3D Direction: angles , from two reference directions Directional set: solid angle [sr] area of a unit sphere Size: size of the area Total size: 4
Size of a solid angle d d d dw = sin d d sin d
Solid angle in which a surface element is visible dA r dw dA cos dw = r2
Radiance: L(x,w) • Emitted power of a unit visible area in a unit solid angle [Watt/ sr/ m2] w dw dF L(x,w) = dF / (dA cos dw) dA
Light propagation between two infinitesimal surfaces: Fundamental law of photometry ’ r L dw dA dA’ dF emitter receiver dA cos dA’ cos ’ dF = LdA cos dw = L r2
Symmetry relation of the source and receiver dA cos dA’ cos ’ dF = L =LdA’ cos ’dw’ r2 dw’ ’ r dA dw’ dA’ dF emitter receiver
Light-surface interaction w dw w’ x Probability density of the reflection w(w’,x,w) dw = Pr{photon goes to w dw | comes from w’}
Reflection of the total incoming light w dw Fin (dw’) Fref (dw) dw’ w’ x Fref (dw) = Fe (dw) + Fin (dw’) w(w’,x,w)dw
Rewriting for the radiance Fref (dw)= LdA cos dw Fe(dw)= LedA cos dw Fin(dw’)= LindA cos ’dw’ Visibility function h(x,-w) L(x,w) ’ Lin =L(h(x,-w’),w’) w ’ x
Substituting and dividing bydA cos dw w(’,x,) cos L(x,w)=Le(x,w)+L(h(x,-w’),w’) cos ’dw’ w ’ w(’,x,) cos = fr (’,x,) x Bidirectional Reflectance Distribution Function BRDF: fr (’,x,) [1/sr]
Rendering equation L(x,w)=Le(x,w)+L(h(x,-w’),w’) fr(’,x,) cos’dw’ L(h(x,-w),w) h(x,-w) L(x,w) ’ ’ w x fr (’,x,) L = Le + tL
Rendering equation • Fredholm integral equation of the second kind • Unknown is a function • Function space: Hilbert space, L2 space • scalar product: L = Le + tL <u(x,w),v(x,w)> = S u(x,w) v(x,w) cos dwdx
Function space • Linear space (vector space) • addition, zero, multiplication by scalars • Space with norms • ||u||2 = <u,u >, ||u||1 = <|u|,1>, • ||u|| = max|u|, • Hilbert space: scalar product: • L2 space: finite square integrals
Measuring the light: radiance • Sensitivity of a measuring device: We(y,w’) L(y,w’) ’ Light beam reaches the device: 0/1 „probability” We(y,w’): effect of a light beam of unit power emitted at y in direction w’ Scaling factor
Measured values • Single beam : F(dw’) We(y,w’) = L(y,w’)cos dAdw’ We(y,w’) • Total measured value:SWe(y,w’)dF=S L(y,w’)We(y,w’) cosdw’dy = < L, We > = ML
Simple eye model Pupil: e Pupil: e Wp Wp y ’ Lp pixel ’ y F r Computer screen F Real world Lp=F / (e cosqeWp) C=1 /(e cosqeWp)if y is visible in Wp and ’ points from y to e 0 otherwise We(y,’)=
Simple eye model: pinhole camera Pupil: e dw’= decose /r2 dy= r2dp/ cos Wp y ’ ’ y Pinhole camera: e, ’ 0 r Lp= ML =S L(y,w’)We(y,w’) cosdw’dy yL(y, w’) C· cos·’ · dy = p L(h(eye, wp ),-wp) C ·cos ·e cose /r2 · r2dp/cos = p L(h(eye, wp ),-wp) ·Ce cose dp Proportional to the radiance! Camera constant: 1 /Wp
Why radiance Lp= pL(h(eye, wp ),-wp) /Wpdp The color of a pixel is proportional to the radiance of the visible points and is independent of the distance and the orientation of the surface!! F=LDA cosdw /r2 pixel DA r2 / cos r
Integrating on the pixel Sp pixel p p f dp= dp cos p/|eye-p|2 = dp cos3p/f 2 dp/Wp dp / Sp
Integrating on the visible surface pixel y r dp= dy cos /|eye-y|2 = dy g(y)
Measuring function F= S L(y,w’)We(y,w’) cosdw’dy = = pL(h(eye, wp ),-wp) /Wpdp= = SL(y,w’) · cos /|eye-y|2 /Wpdy (w-wyeye)/|eye-y|2 /Wpif y is visible in the pixel 0 otherwise We(y,’)= g(y)
Potential: W(y,w’) • The direct and indirect effects in a measuring device caused by a unit beam from y at ’ • The product of scaling factor C and the probability that the photon emitted at y in ’ reaches the device w’ y
Duality of radiance and potential • Light propagation = emitter-receiver interaction • radiance: intensity of emission • potential: intensity of detection
Potential equation w’ y C · Pr{ detection} = C · Pr{ direct detection} + C · Pr{ indirect detection} Pr{ indirect detection} = Pr{ detection from the new point | reflection to w}· Pr{ reflection to w} dw
Potential equation W(y,w’)=We(y,w’)+W(h(y,w’),w) fr(’,h(y,w’),)cosdw y W(y,w’) w’ q fr (’,h(y,w’),) h(y,w’) W = We + ’W
Measuring the light: potential w’ y Fe(dw’) Measured values of a single beam = Fe(dw’)W(y,w’) = Le(y,w’)cos dAdw’ W(y,w’) Total measured value = M’W= S W(x,w)dFe=S Le(x,w) W(x,w) cosdwdx = < Le , W>
Operators of the rendering and potential equations • Measuring a single reflection of the light: • Adjoint operators: F1=< Le , W> = < Le , ’We > F1=< L, We> = < Le ,We > < Le , ’We > = < Le ,We >
Rendering problem: <S,Le,We ,fr> pixel S fr (’, x, ) Le(x,) We(x,) F= S We(x,w) dF=S L(x,w) We(x,w) cos dwdx F= ML