Fundamentals of Rendering - Radiometry / Photometry Image Synthesis Torsten Möller © Lastra/Machiraju/Möller/Weiskopf Today • The physics of light • Radiometric quantities • Photometry vs/ Radiometry © Lastra/Machiraju/Möller/Weiskopf 2 Reading • Chapter 5 of “Physically Based Rendering” by Pharr&Humphreys • Chapter 2 (by Hanrahan) “Radiosity and Realistic Image Synthesis,” in Cohen and Wallace. • pp. 648 – 659 and Chapter 13 in “Principles of Digital Image Synthesis”, by Glassner. • Radiometry FAQ https://employeepages.scad.edu/~kwitte/ documents/Photometry_FAQ.PDF © Lastra/Machiraju/Möller/Weiskopf 3 The physics of light © Lastra/Machiraju/Möller/Weiskopf 4 Realistic Rendering • Determination of Intensity • Mechanisms – Emittance (+) – Absorption (-) – Scattering (+) (single vs. multiple) • Cameras or retinas record quantity of light © Lastra/Machiraju/Möller/Weiskopf 5 Pertinent Questions • Nature of light and how it is: – Measured – Characterized / recorded • (local) reflection of light • (global) spatial distribution of light • Steady state light transport? © Lastra/Machiraju/Möller/Weiskopf 6 What Is Light ? • Light - particle model (Newton) – Light travels in straight lines – Light can travel through a vacuum (waves need a medium to travel) • Light – wave model (Huygens): electromagnetic radiation: sinusoidal wave formed coupled electric (E) and magnetic (H) fields – Dispersion / Polarization – Transmitted/reflected components – Pink Floyd © Lastra/Machiraju/Möller/Weiskopf 7 Electromagnetic spectrum © Lastra/Machiraju/Möller/Weiskopf 8 Optics • Geometrical or ray – Traditional graphics – Reflection, refraction – Optical system design • Physical or wave – Diffraction, interference – Interaction of objects of size comparable to wavelength • Quantum or photon optics – Interaction of© Lastra/Machiraju/Möller/Weiskopf light with atoms and molecules 9 Nature of Light • Wave-particle duality – Wave packets which emulate particle transport. Explain quantum phenomena. • Incoherent as opposed to laser. Waves are multiple frequencies, and random in phase. • Polarization – Ignore. Un-polarized light has many waves summed all with random orientation © Lastra/Machiraju/Möller/Weiskopf 10 Radiometric quantities © Lastra/Machiraju/Möller/Weiskopf 11 Radiometry • Science of measuring light • Analogous science called photometry is based on human perception. © Lastra/Machiraju/Möller/Weiskopf 12 Radiometric Quantities • Function of wavelength, time, position, direction, polarization. g(⇥, t, r, ⇤, γ) • Assume wavelength independence g(t, r, ⇥, γ) – No phosphorescence, fluorescence – Incident wavelength λ1 exit λ1 • Steady State g(r, ⇥, γ) – Light travels fast – No luminescence phenomena - Ignore it © Lastra/Machiraju/Möller/Weiskopf 13 Result – five dimensions • Adieu Polarization g(r, ω) – Would likely need wave optics to simulate • Two quantities – Position (3 components) – Direction (2 components) © Lastra/Machiraju/Möller/Weiskopf 14 Radiometry - Quantities • Energy Q • Power Φ - J/s or W – Energy per time (radiant power or flux) • Irradiance E and Radiosity B - W/m2 – Power per area • Intensity I - W/sr – Power per solid angle • Radiance L - W/sr/m2 – Power per projected area and solid angle © Lastra/Machiraju/Möller/Weiskopf 15 Radiant Energy - Q • Think of a photon as carrying quantum of energy hc/λ = hf (photoelectric effect) – c is speed of light – h is Planck’s constant – f is frequency of radiation • Wave packets • Total energy, Q, is then energy of the total number of photons • Units - joules or eV © Lastra/Machiraju/Möller/Weiskopf 16 Radiation Blackbody Tungsten © Lastra/Machiraju/Möller/Weiskopf 17 Consequence © Lastra/Machiraju/Möller/Weiskopf 18 Fluorescent Lamps © Lastra/Machiraju/Möller/Weiskopf 19 Power / Flux - Φ • Flow of energy (important for transport) • Also known as radiant power or flux • Energy per unit time (joules/s = eV/s) • Unit: W — watts • Φ = dQ/dt © Lastra/Machiraju/Möller/Weiskopf 20 Intensity I • Flux density per unit solid angle dΦ I = dω • Units – watts per steradian • “intensity” is heavily overloaded. Why? – Power€ of light source – Perceived brightness © Lastra/Machiraju/Möller/Weiskopf 21 Solid Angle • Size of a patch, dA, is dA = (rsinθ dφ)(r dθ) • Solid angle is dA dω = = sinθdθdφ r2 € • Compare with 2D angle l θ = € r l θ r φ - azimuth angle € © Lastra/Machiraju/Möller/Weiskopf θ - polar angle 22 € € € Solid Angle (contd.) • Solid angle generalizes angle! • Steradian • Sphere has 4π steradians! Why? © Wikipedia • Dodecahedron – 12 faces, each pentagon. • One steradian approx equal to solid angle subtended by a single face of dodecahedron © Lastra/Machiraju/Möller/Weiskopf 23 Isotropic Point Source dΦ Φ I = = dω 4π • Even distribution over sphere • How do you get this? • Quiz:€ Warn’s spot-light? Determine flux © Lastra/Machiraju/Möller/Weiskopf 24 Warns Spot Light S θ I(θ) = (S ⋅ A)s = coss θ A • Quiz:€ Determine flux © Lastra/Machiraju/Möller/Weiskopf 25 Other Lights © Lastra/Machiraju/Möller/Weiskopf 26 Other kinds of lights © Lastra/Machiraju/Möller/Weiskopf 27 Radiant Flux Area Density • Area density of flux (W/m2) • u = Energy arriving/leaving a surface per unit time interval per unit area • dA can be any 2D surface in space dΦ u = dA dA € © Lastra/Machiraju/Möller/Weiskopf 28 Radiant Flux Area Density • E.g. sphere / point light source: dΦ Φ u = = dA 4πr2 • Flux falls off with square of the distance • Bouguer’s classic experiment: – Compare a light source and a candle € – Intensity is proportional to ratio of distances squared © Lastra/Machiraju/Möller/Weiskopf 29 Irradiance E • Power per unit area incident on a surface dΦ E = dA € © Lastra/Machiraju/Möller/Weiskopf 30 Radiosity or Radiant Exitance B • Power per unit area leaving surface • Also known as Radiosity • Same unit as irradiance, just direction changes dΦ B = dA € © Lastra/Machiraju/Möller/Weiskopf 31 Irradiance on Differential Patch from a Point Light Source x θ • Inverse square law xs • Key observation: 2 x − x s dω = cosθdA © Lastra/Machiraju/Möller/Weiskopf 32 € Hemispherical Projection • Use a hemisphere Η over surface to measure incoming/outgoing flux • Replace objects and points with their hemispherical projection ω r © Lastra/Machiraju/Möller/Weiskopf 33 Radiance L • Power per unit projected area per unit solid angle. d 2Φ L = dAp dω • Units – watts per (steradian m2) • We have€ now introduced projected area, a cosine term. d 2Φ L = dAcosθdω © Lastra/Machiraju/Möller/Weiskopf 34 € Why the Cosine Term? • projected into the solid angle! • Foreshortening is by cosine of angle. • Radiance gives energy by effective surface area. d cosθ θ d © Lastra/Machiraju/Möller/Weiskopf 35 Incident and Exitant Radiance • Incident Radiance: Li(p, ω) • Exitant Radiance: Lo(p, ω) • In general: Li (p,ω) ≠ Lo(p,ω) • At a point p - no surface, no participating media: Li (p,ω) = Lo(p,−ω) € Lo p,ω Li (p,ω) ( ) € p p © Lastra/Machiraju/Möller/Weiskopf 36 € € Irradiance from Radiance E(p,n) = ∫ Li (p,ω)cosθ dω Ω • |cosθ|dω is projection of a differential area L p,ω € • We take |cosθ| in order i ( ) n to integrate over the whole sphere € r p © Lastra/Machiraju/Möller/Weiskopf 37 Radiance • Fundamental quantity, everything else derived from it. • Law 1: Radiance is invariant along a ray • Law 2: Sensor response is proportional to radiance © Lastra/Machiraju/Möller/Weiskopf 38 Law1: Conservation Law ! dω1 dω2 € Total flux leaving one side = flux arriving other side, so € L1dω1dA1 = L2dω2dA2 © Lastra/Machiraju/Möller/Weiskopf 39 € Conservation Law ! dω1 dω2 € d dA /r2 and d dA /r2 € ω1 = 2 ω2 = 1 therefore € € © Lastra/Machiraju/Möller/Weiskopf 40 Conservation Law ! dω1 dω2 € so € Radiance doesn’t change with distance! © Lastra/Machiraju/Möller/Weiskopf 41 Radiance at a sensor • Sensor of a fixed area sees more of a surface that is farther away. Ω • However, the solid angle is inversely proportional to distance. dω = dA /r2 • Response of a sensor is proportional to radiance. € A © Lastra/Machiraju/Möller/Weiskopf 42 Radiance at a sensor • Response of a sensor is proportional to radiance. Ω Φ = ∫ ∫ LdωdA = LT A Ω T = ∫ ∫ dωdA A Ω • T depends on geometry of sensor A € © Lastra/Machiraju/Möller/Weiskopf 43 Thus … • Radiance doesn’t change with distance – Therefore it’s the quantity we want to measure in a ray tracer. • Radiance proportional to what a sensor (camera, eye) measures. – Therefore it’s what we want to output. © Lastra/Machiraju/Möller/Weiskopf 44 Radiometry vs. Photometry © Lastra/Machiraju/Möller/Weiskopf 45 Photometry and Radiometry • Photometry (begun 1700s by Bouguer) deals with how humans perceive light. • Bouguer compared every light with respect to candles and formulated the inverse square law ! • All measurements relative to perception of illumination • Units different from radiometric units but conversion is scale factor -- weighted by spectral response of eye (over about 360 to 800 nm). © Lastra/Machiraju/Möller/Weiskopf 46 CIE curve • Response is integral over all wavelengths 1.200 1.000 0.800 0.600 0.400 0.200 0.000 350 450 550 650 750 Violet Green Red © CIE, 1924, many more curves available, see http://cvision.ucsd.edu/lumindex.htm © Lastra/Machiraju/Möller/Weiskopf 47 Radiometric Quantities Physical Definition Symbol Units Radiometric quantity description (also “spectral *” [*/m]) Energy Qe [J=Ws] Joule Radiant energy Power / flux dQ/dt Φe [W=J/s] Radiant power / flux 2 Flux density dQ/dAdt Ee [W/m ] Irradiance 2 Flux density dQ/dAdt Me= Be [W/m ] Radiosity 2 Angular flux dQ/dAdωdt Le [W/m sr] Radiance density Intensity dQ/dωdt Ie [W/sr]