Fundamentals of Rendering - Radiometry / Photometry

Transcription

1 Fundamentals of Rendering - Radiometry / Photometry CMPT 461/761 Image Synthesis Torsten Möller

2 Today The physics of light Radiometric quantities Photometry vs/ Radiometry 2

3 Reading Chapter 5 of Physically Based Rendering by Pharr&Humphreys Chapter 2 (by Hanrahan) Radiosity and Realistic Image Synthesis, in Cohen and Wallace. pp and Chapter 13 in Principles of Digital Image Synthesis, by Glassner. Radiometry FAQ rpfaq.htm

4 The physics of light 4

5 Realistic Rendering Determination of Intensity Mechanisms Emittance (+) Absorption (-) Scattering (+) (single vs. multiple) Cameras or retinas record quantity of light

6 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?

7 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

8 Electromagnetic spectrum

9 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 light with atoms and molecules

10 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

11 Today The physics of light Radiometric quantities Photometry vs/ Radiometry 11

12 Radiometric quantities 12

13 Radiometry Science of measuring light Analogous science called photometry is based on human perception.

14 Radiometric Quantities Function of wavelength, time, position, direction, polarization. g(, t, r,, ) Assume wavelength independence No phosphorescence, fluorescence g(t, r,, ) Incident wavelength λ 1 exit λ 1 Steady State Light travels fast No luminescence phenomena - Ignore it g(r,, )

15 Result five dimensions Adieu Polarization Would likely need wave optics to simulate Two quantities Position (3 components) Direction (2 components) g(r, )

16 Radiometry - Quantities Energy Q Power Φ - J/s or W Energy per time (radiant power or flux) Irradiance E and Radiosity B - W/m 2 Power per area Intensity I - W/sr Power per solid angle Radiance L - W/sr/m 2 Power per projected area and solid angle

17 Radiant Energy - Q Think of 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

18 Radiation Blackbody Tungsten

19 Consequence

20 Fluorescent Lamps

21 Power / Flux - Φ Flow of energy (important for transport) Also - radiant power or flux. Energy per unit time (joules/s = ev/s) Unit: W - watts Φ = dq/dt

22 Intensity I Flux density per unit solid angle I = dφ dω Units watts per steradian intensity is heavily overloaded. Why? Power of light source Perceived brightness

23 Solid Angle Size of a patch, da, is Solid angle is dω = da r 2 da = (rsinθ dφ)(r dθ) = sinθdθdφ Compare with 2D - angle θ r l θ = l r φ - azimuth angle θ - polar angle

24 Solid Angle (contd.) Solid angle generalizes angle! Steradian Sphere has 4π steradians! Why? Dodecahedron 12 faces, each pentagon. One steradian approx equal to solid angle subtended by a single face of dodecahedron Wikipedia

25 Isotropic Point Source I = dφ dω = Φ 4π Even distribution over sphere How do you get this? Quiz: Warn s spot-light? Determine flux

26 Warns Spot Light θ S I(θ) = (S A) s = cos s θ A Quiz: Determine flux

27 Other Lights

28 Other kinds of lights

29 Radiant Flux Area Density Area density of flux (W/m 2 ) u = Energy arriving/leaving a surface per unit time interval per unit area da can be any 2D surface in space u = dφ da da

30 Radiant Flux Area Density E.g. sphere / point light source: u = dφ da = Φ 4πr 2 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

31 Irradiance E Power per unit area incident on a surface E = dφ da

32 Radiosity or Radiant Exitance B Power per unit area leaving surface Also known as Radiosity Same unit as irradiance, just direction changes B = dφ da

33 Irradiance on Differential Patch from a Point Light Source x Inverse square law Key observation: x s θ x x s 2dω = cosθda

34 Hemispherical Projection Use a hemisphere Η over surface to measure incoming/outgoing flux Replace objects and points with their hemispherical projection ω r

35 Radiance L Power per unit projected area per unit solid angle. L = d 2 Φ Units watts per (steradian m 2 ) da p dω We have now introduced projected area, a cosine term. L = d 2 Φ dacosθdω

36 Why the Cosine Term? projected into the solid angle! Foreshortening is by cosine of angle. Radiance gives energy by effective surface area. θ d cosθ d

37 Incident and Exitant Radiance Incident Radiance: L i (p, ω) Exitant Radiance: L o (p, ω) In general: L ( i p,ω) L ( o p,ω) At a point p - no surface, no participating media: p,ω L i ( ) = L ( o p, ω ) L i p p,ω ( ) L o p,ω ( ) p

38 Irradiance from Radiance Ω E p,n ( ) = L i p,ω ( )cosθ dω cosθ dω is projection of a differential area We take cosθ in order to integrate over the whole sphere L i ( p,ω) n p r

39 Radiance Fundamental quantity, everything else derived from it. Law 1: Radiance is invariant along a ray Law 2: Sensor response is proportional to radiance

40 Law1: Conservation Law! dω 1 dω 2 Total flux leaving one side = flux arriving other side, so L 1 dω 1 da 1 = L 2 dω 2 da 2

41 Conservation Law! dω 1 dω 2 therefore dω 1 = da 2 /r 2 and dω 2 = da 1 /r 2

42 Conservation Law! dω 1 dω 2 so Radiance doesn t change with distance!

43 Radiance at a sensor Sensor of a fixed area sees more of a surface that is farther away. However, the solid angle is inversely proportional to distance. dω = da /r 2 Response of a sensor is proportional to radiance. Ω A

44 Radiance at a sensor Response of a sensor is proportional to radiance. Φ = T = A A LdωdA T depends on geometry of sensor Ω Ω dωda = LT Ω A

45 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.

46 Radiometry vs. Photometry 46

47 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).

48 CIE curve Response is integral over all wavelengths Violet Green Red CIE, 1924, many more curves available, see

49 Radiometric Quantities Physical description Definition Symbol Units Radiometric quantity (also spectral * [*/m]) Energy Q e [J=Ws] Joule Radiant energy Power / flux dq/dt Φ e [W=J/s] Radiant power / flux Flux density dq/dadt E e [W/m 2 ] Irradiance Flux density dq/dadt M e = B e [W/m 2 ] Radiosity Angular flux density dq/dadωdt L v [W/m 2 sr] Radiance Intensity dq/dωdt I v [W/sr] Radiant intensity 49

50 Photometric Quantities Physical description Definition Symbol Units Photometric quantity (also spectral * [*/m]) Energy Q v [talbot] Luminous energy Power / flux dq/dt Φ v [lm (lumens) = talbot/s] Luminous power / flux Flux density dq/dadt E v [lux= lm/m 2 ] Illuminance Flux density dq/dadt [M v =] B v [lux] Luminosity Angular flux density dq/da Φ dωdt L v [lm/m 2 sr] Luminance Intensity dq/dωdt I v [cd (candela) = lm/sr] Quantities weighted by luminous efficiency function Radiant / luminous intensity 50

51 Typical Values of Illuminance Solar constant: 1,37 kw/m 2 Light source Illuminance[lux] Direct sun light 25, ,000 Day light 2,000 27,000 Sun set Moon light Stars TV studio 5,000 10,000 Lighting in shops/stores 1,000 5,500 Office illumination Home illumination Street lighting