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Light Transport II Computer Graphics - Light Transport - Hendrik Lensch Computer Graphics WS07/08 – Light Transport Overview • So far – simple shading • Today – Physics behind ray tracing – Physical light quantities – Perception of light – Light sources – Light transport simulation • Next lecture – Light-matter interaction – Reflectance function – Reflection models Computer Graphics WS07/08 – Light Transport What is Light ? • Ray – Linear propagation ⇒ Geometrical optics • Vector – Polarization ⇒ Jones Calculus: matrix representation • Wave – Diffraction, Interference ⇒ Maxwell equations: propagation of light • Particle – Light comes in discrete energy quanta: photons ⇒ Quantum theory: interaction of light with matter • Field – Electromagnetic force: exchange of virtual photons ⇒ Quantum Electrodynamics (QED): interaction between particles Computer Graphics WS07/08 – Light Transport What is Light ? • Ray – Linear propagation ⇒ Geometrical optics • Vector – Polarization ⇒ Jones Calculus: matrix representation • Wave – Diffraction, Interference ⇒ Maxwell equations: propagation of light • Particle – Light comes in discrete energy quanta: photons ⇒ Quantum theory: interaction of light with matter • Field – Electromagnetic force: exchange of virtual photons ⇒ Quantum Electrodynamics (QED): interaction between particles Computer Graphics WS07/08 – Light Transport Light in Computer Graphics • Based on human visual perception – Macroscopic geometry – Tristimulus color model – Psycho-physics: tone mapping, compression, … • Ray optics – Light: scalar, real-valued quantity – Linear propagation – Macroscopic objects – Incoherent light – Superposition principle: light contributions add up linearly – No attenuation in free space • Limitations – Microscopic structures (≈λ) – Diffraction, Interference – Polarization – Dispersion Computer Graphics WS07/08 – Light Transport Angle and Solid Angle θ the angle subtended by a curve in the plane, is the length of the corresponding arc on the unit circle. Ω,d ω the solid angle subtended by an object, is the surface area of its projection onto the unit sphere, Units for measuring solid angle: steradians [sr] Computer Graphics WS07/08 – Light Transport Solid Angle in Spherical Coordinates Infinitesimally small solid angle du = r dθ dv dv = r sinθ dφ dA = du dv = r 2 sinθ dθ dφ dA ⇒ dω,dΩ = = sinθ dθ dφ r 2 Finite solid angle φ1 θ1 ()φ Ω = ∫∫dφ sinθ dθ du φ0 θ0 ()φ Computer Graphics WS07/08 – Light Transport Projected Solid Geometry The solid angle subtended by a small surface patch S with area ΔA is obtained (i) by projecting it orthogonal to the vector r to the origin ΔAcosθ and (ii) dividing by the square of the distance to the origin: ΔAcosθ ΔΩ ≈ r 2 Computer Graphics WS07/08 – Light Transport Radiometry • Definition: – Radiometry is the science of measuring radiant energy transfers. Radiometric quantities have physical meaning and can be directly measured using proper equipment such as spectral photometers. • Radiometric Quantities – energy [watt second] n · hν (Photon Energy) – radiant power (total flux) [watt] Φ – radiance [watt/(m2 sr)] L – irradiance [watt/m2]E – radiosity [watt/m2] B – intensity [watt/sr] I Computer Graphics WS07/08 – Light Transport Radiometric Quantities: Radiance • Radiance is used to describe radiant energy transfer. • Radiance L is defined as – the power (flux) traveling at some point x – in a specified direction ω = (θ,φ), – per unit area perpendicular to the direction of travel, – per unit solid angle. • Thus, the differential power d2Φ radiated through the differential solid angled 2Φ = L(x,ω) dA cosθ dω dω, from the projected differential area dA cosθ is: ω d 2Φ = L(x,ω) dAcosθ dω dA Computer Graphics WS07/08 – Light Transport Spectral Properties • Wavelength – Since light is composed of electromagnetic waves of different frequencies and wavelengths, most of the energy transfer quantities are continuous functions of wavelength. – In graphics each measurement L(x,ω) is for a discrete band of wavelength only (often some abstract R, B, G) Computer Graphics WS07/08 – Light Transport Radiometric Quantities: Irradiance Irradiance E is defined as the total power per unit area (flux density) incident onto a surface. To obtain the total flux incident to dA, the incoming radiance Li is integrated over the upper hemisphere Ω+ above the surface: dΦ E ≡ dA ⎡ ⎤ dΦ = L (x,ω)cosθ dω dA ⎢ ∫ i ⎥ ⎣⎢Ω+ ⎦⎥ ππ/ 2 2 E = L (x,ω)cosθ dω = L (x,ω)cosθ sinθ dθ dφ ∫ i ∫∫ i Ω+ 0 0 Computer Graphics WS07/08 – Light Transport Radiometric Quantities: Radiosity Radiosity B is defined as the total power per unit area (flux density) leaving a surface. To obtain the total flux radiated from dA, the outgoing radiance Lo is integrated over the upper hemisphere Ω+ above the surface. dΦ B ≡ dA ⎡ ⎤ dΦ = L (x,ω)cosθ dω dA ⎢∫ o ⎥ ⎣Ω ⎦ ππ/ 2 2 B = L (x,ω)cosθ dω = L (x,ω)cosθ sinθ dθ dφ ∫ o ∫∫ o Ω 0 0 Computer Graphics WS07/08 – Light Transport Photometry • Photometry: – The human eye is sensitive to a limited range of radiation wavelengths (roughly from 380nm to 770nm). – The response of our visual system is not the same for all wavelengths, and can be characterized by the luminuous efficiency function V(λ), which represents the average human spectral response. – A set of photometric quantities can be derived from radiometric quantities by integrating them against the luminuous efficiency function V(λ). – Separate curves exist for light and dark adaptation of the eye. Computer Graphics WS07/08 – Light Transport Radiometry vs. Photometry Physics-based quantities Perception-based quantities Computer Graphics WS07/08 – Light Transport Perception of Light dΩ dΩ' r dA f l photons / second = flux = energy / time = power Φ rod sensitive to flux angular extend of rod = resolution (≈ 1 arc minute2) dΩ projected rod size = area dA ≈ l 2 ⋅dΩ Angular extend of pupil aperture (r ≤ 4 mm) = solid angle dΩ'≈ π ⋅ r 2 / l 2 flux proportional to area and solid angle Φ ∝ dΩ'⋅dA Φ radiance = flux per unit area per unit solid angle L = dΩ'⋅dA The eye detects radiance Computer Graphics WS07/08 – Light Transport Brightness Perception dΩ dΩ' r dA dA' f l 2 2 r As l increases: Φ0 ∝ dA⋅dΩ'= l dΩ⋅π 2 = const l • dA’ > dA : photon flux per rod stays constant • dA’ < dA : photon flux per rod decreases Where does the Sun turn into a star ? − Depends on apparent Sun disc size on retina ⇒ Photon flux per rod stays the same on Mercury, Earth or Neptune ⇒ Photon flux per rod decreases when dΩ’ < 1 arc minute (beyond Neptune) Computer Graphics WS07/08 – Light Transport Brightness Perception II Extended light source dΩ r l 2 Φ0 = L0 ⋅π ⋅r ⋅dΩ ⇒ Flux does not depend on distance l ⇒ Nebulae always appear b/w Computer Graphics WS07/08 – Light Transport Radiance in Space dΩ2 dΩ1 L1 L2 l dA1 dA2 Flux leaving surface 1 must be equal to flux arriving on surface 2 L1 ⋅dΩ1 ⋅dA1 = L2 ⋅dΩ2 ⋅dA2 dA dA From geometry follows dΩ = 2 dΩ = 1 1 l 2 2 l 2 dA ⋅dA Ray throughput T = dΩ ⋅dA = dΩ ⋅dA = 1 2 1 1 2 2 l 2 L1 = L2 The radiance in the direction of a light ray remains constant as it propagates along the ray Computer Graphics WS07/08 – Light Transport Point Light Source • Point light with isotropic radiance – Power (total flux) of a point light source ∀Φg= Power of the light source [watt] – Intensity of a light source • I= Φg/(4π sr) [watt/sr] – Irradiance on a sphere with radius r around light source: 2 2 • Er= Φg/(4π r ) [watt/m ] – Irradiance on some other surface A dA θ dΦ dω r E(x) = g = I dA dA Φ dAcosθ = g ⋅ dω 4π r 2dA Φ cosθ = g ⋅ 4π r 2 Computer Graphics WS07/08 – Light Transport Inverse Square Law Irradiance E: d2 d1 2 E1 d 2 E 1 = 2 E2 d1 E2 • Irradiance E: power per m2 – Illuminating quantity • Distance-dependent – Double distance from emitter: sphere area four times bigger • Irradiance falls off with inverse of squared distance – For point light sources Computer Graphics WS07/08 – Light Transport Light Source Specifications • Power (total flux) – Emitted energy / time • Active emission size – Point, area, volume • Spectral distribution – Thermal, line spectrum • Directional distribution – Goniometric diagram Computer Graphics WS07/08 – Light Transport Sky Light • Sun – Point source (approx.) – White light (by def.) • Sky – Area source – Scattering: blue • Horizon – Brighter – Haze: whitish • Overcast sky – Multiple scattering in clouds – Uniform grey Courtesy Lynch & Livingston Computer Graphics WS07/08 – Light Transport Light Source Classification Radiation characteristics Emitting area • Directional light • Volume – Spot-lights – neon advertisements – sodium vapor lamps –Beamers – Distant sources • Area – CRT, LCD display • Diffuse emitters – (Overcast) sky –Torchieres • Line – Frosted glass lamps – Clear light bulb, filament • Ambient light • Point – “Photons everywhere” – Xenon lamp –Arc lamp – Laser diode Computer Graphics WS07/08 – Light Transport Surface Radiance L(x,ω ) = L (x,ω ) + f (ω , x,ω ) L (x,ω ) cosθ dω o e o ∫ r i o i i i i Ω • Visible surface radiance L(x,ω o ) ω o – Surface position x ω i – Outgoing direction ω o θi – Incoming illumination direction ω • Self-emission i Le (x,ω o ) • Reflected light x – Incoming radiance from all L (x,ω ) directions i i – Direction-dependent reflectance (BRDF: bidirectional reflectance fr (ω i , x,ω o ) distribution function) Computer Graphics WS07/08 – Light Transport Ray Tracing L(x,ω ) = L (x,ω ) + f (ω , x,ω ) L (x,ω ) cosθ dω o e o ∫ r i o i i i i Ω • Simple ray tracing – Illumination from light sources only - local illumination (integral sum) – Evaluates angle-dependent reflectance function - shading
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