# Radiometry. Computer Graphics CMU /15-662, Fall 2015

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1 Radiometry Computer Graphics CMU /15-662, Fall 2015

2 Last time we discussed light & color Image credit: Licensed under CC BY-SA 3.0 via Commons

3 How do we quantify measurements of light?

4 Radiometry System of units and measures for measuring EM radiation (light) Geometric optics model of light - Photons travel in straight lines - Represented by rays - Wavelength << size of objects - No diffraction, interference,

5 What do we want to measure (and why?) Many physical processes convert energy into photons - E.g., incandescent lightbulb turns heat into light (blackbody radiation) - Nuclear fusion in stars (sun!) generates photons - Etc. Each photon carries a small amount of energy Want some way of recording how much energy Energy of photons hitting an object ~ brightness - Film, eyes, CCD sensor, sunburn, solar panels, - Need this information to make accurate (and beautiful!) images Simplifying assumption: steady state process - How long does it take for lighting to reach steady state?

6 Names don t constitute knowledge! (Richard Feynman)

7 What does light propagation look like? Can t see it with the naked eye! Instead, repeat same experiment many times, take snapshot at slightly different offsets each time.

8 Imagine every photon is a little rubber ball hitting the scene: How can we record this process? What information should we store?

9 Radiant energy is total # of hits One idea: just store the total number of hits that occur anywhere in the scene, over the complete duration of the scene. Morally, this quantity captures the total energy of all the photons hitting the scene*. Radiant energy : 40 *Eventually we will care about constants & units. But these will not help our conceptual understanding

10 Radiant flux is hits per second * For illumination phenomena at the level of human perception, usually safe to assume equilibrium is reached immediately. So, rather than record total energy over some (arbitrary) duration, may make more sense to record total hits per second (Takes Keynote.05s to display each hit!) Estimate of radiant flux : 40 hits/2s = 20 hits/s *Very large for, say, Michael Jackson or Rihanna

11 Irradiance is #hits per second, per unit area Typically we want to get more specific than just the total To make images, also need to know where hits occurred So, compute hits per second in some really small area, divided by area: Estimate of radiant energy density : 2/ε 2

12 Image generation as irradiance estimation From this point of view, our goal in image generation is to estimate the irradiance at each point of an image (or perhaps the total radiant flux per pixel ):

13 Recap so far Radiant Energy (total number of hits) Radiant Energy Density (hits per unit area) Radiant Flux (total hits per second) Radiant Flux Density a.k.a. Irradiance (hits per second per unit area)

14 Ok, but how about some units

15 Measuring illumination: radiant energy How can we be more precise about the amount of energy? Sensor Said we were just going to count the number of hits, but do all hits contribute same amount of energy? Energy of a single photon: Planck s constant speed of light wavelength (color!) Q: What are units for a photon? (Joules times seconds) (meters per second)

16 Aside: Units are a powerful debugging tool!

17 Measuring illumination: radiant flux (power) Flux: energy per unit time (Watts) received by the sensor (or emitted by Sensor the light) Q = lim!0 t = dq dt apple J s Watts Can also go the other direction: time integral of flux is total radiant energy Z t1 Q = (t)dt t 0 (Units?)

18 Measuring illumination: irradiance Radiant flux: time density of energy Irradiance: area density of flux A Given a sensor of with area A, we can consider the average flux over the entire sensor area: A Irradiance (E) is given by taking the limit of area at a single point on the sensor: E(p) = lim!0 (p) A = d (p) da apple W m 2

19 Recap, with units Radiant Energy (total number of hits) Joules (J) Radiant Energy Density (hits per unit area) Joules per square meter (J/m 2 ) Radiant Flux (total hits per second) Joules per second (J/s) = Watts (W) Radiant Flux Density a.k.a. Irradiance (hits per second per unit area) Watts per square meter (W/m 2 )

20 What about color? How might we quantify, say, the amount of green?

21 Spectral power distribution Describes irradiance per unit wavelength (units?) Figure credit: Energy per unit time per unit area per unit wavelength

22 Given what we now know about radiant energy Why do some parts of a surface look lighter or darker?

23 Why do we have seasons? Sun Summer (Northern hemisphere) Winter (Northern hemisphere) Earth s axis of rotation: ~23.5 off axis [Image credit: Pearson Prentice Hall]

24 Beam power in terms of irradiance Consider beam with flux incident on surface with area A irradiance (energy per time, per area) radiant flux (energy per time) E = A A = EA

25 Projected area Consider beam with flux incident on angled surface with area A A A 0 A = A 0 cos A = projected area of surface relative to direction of beam

26 Lambert s Law Irradiance at surface is proportional to cosine of angle between light direction and surface normal. A A 0 A = A 0 cos E = A 0 = cos A

27 N-dot-L lighting Most basic way to shade a surface: take dot product of unit surface normal (N) and unit direction to light (L) double surfacecolor( Vec3 N, Vec3 L ) { return dot( N, L ); } N L (Q: What s wrong with this code?)

28 N-dot-L lighting Most basic way to shade a surface: take dot product of unit surface normal (N) and unit direction to light (L) double surfacecolor( Vec3 N, Vec3 L ) { return max( 0., dot( N, L )); } N L

29 Example: directional lighting Common abstraction: infinitely bright light source at infinity All light directions (L) are therefore identical L L L L L

30 Isotropic point source Slightly more realistic model total radiant flux intensity Z = S 2 I d! =4 I p I = 4

31 Irradiance falloff with distance Assume light is emitting flux in a uniform angular distribution r 2 Compare irradiance at surface r 1 of two spheres: E 1 = 4 r 2 1! =4 r 2 1E 1 E 2 = 4 r 2 2! =4 r 2 2E 2 E 2 E 1 = r2 1 r 2 2

32 What does quadratic falloff look like? Single point light, move in 1m increments: things get dark fast!

33 Angles and solid angles Angle: ratio of subtended arc length on circle to radius - = l r - Circle has 2 radians r l Solid angle: ratio of subtended area on sphere to radius squared - = A r 2 - Sphere has 4 steradians r A

34 Solid angles in practice Sun and moon both subtend ~60µ sr as seen from earth Surface area of earth: ~510M km 2 Projected area: 510Mkm 2 60µsr 4 sr = km 2

35 Differential solid angle da =(r d )(r sin d ) =r 2 sin d d d! = da r 2 =sin d d

36 Differential solid angle Z Z = d! S 2 Z Z Z 2 Z = Z 0 Z 0 sin d d =4

37 ! as a direction vector! Will use! to donate a direction vector (unit length)

38 Measuring illumination: radiance Radiance is the solid angle density of irradiance L(p,!) = lim!0 E! (p)! = de!(p) d! apple W m 2 sr where denotes that the differential surface area is E!! oriented to face in the direction da d! In other words, radiance is energy along a ray defined by origin point p and direction!

39 Radiance functions Equivalently, L(p,!) = de(p) d! cos = d2 (p) da d! cos Previous slide described measuring radiance at a surface oriented in ray direction - cos(theta) accounts for different surface orientation d! da

40 Field radiance: the light field Light field = radiance function on rays Radiance is constant along rays * Spherical gantry: captures 4D light field (all light leaving object) L(x, y,, ) (, ) * in a vacuum

41 Surface radiance Need to distinguish between incident radiance and exitant radiance functions at a point on a surface L i (p 1,! 1 ) L o (p 2,! 2 ) (p 1 p 2, In general: L i (p,!) 6= L o (p,!)

42 Properties of radiance Radiance is a fundamental field quantity that characterizes the distribution of light in an environment - Radiance is the quantity associated with a ray - Rendering is all about computing radiance Radiance is invariant along a ray in a vacuum A pinhole camera measures radiance

43 Irradiance from the environment Computing flux per unit area on surface, due to incoming light from all directions. de(p,!) =L i (p,!) cos d! Z Contribution to irradiance from light arriving from direction! E(p,!) = H 2 L i (p,!) cos d! d! Light meter da

44 Simple case: irradiance from uniform hemispherical source E(p) = Z = L H 2 L d! Z 2 0 Z 2 0 cos sin d d = L d! da

45 Example of hemispherical light source

46 Ambient occlusion Assume spherical (vs. hemispherical) light source, at infinity Irradiance is now rotation, translation invariant Can pre-compute, bake into texture to enhance shading without AO map with AO map ambient occlusion map

47 Screen-space ambient occlusion Credit: NVIDIA

48 Screen-space ambient occlusion Credit: NVIDIA

49 Screen-space ambient occlusion Credit: NVIDIA

50 Irradiance from a uniform area source (source emits radiance L) Z E(p) = L(p,!) cos d! HZ 2 O =L =L? cos d! Projected solid angle:? - Cosine-weighted solid angle - Area of object O projected onto unit d!? = cos d! sphere, then projected onto plane

51 Uniform disk source (oriented perpendicular to plane) Geometric Derivation Algebraic Derivation (using projected solid angle) Z 2 Z? = 0 0 =2 sin2 2 cos sin d d 0 = sin 2 sin? = sin 2

52 Examples of Area Light Sources Generally softer appearance than point lights: and better model of real-world lights!

53 Measuring illumination: radiant intensity Power per solid angle emanating from a point source apple W I(!) = d d! sr

54 More realistic light models via goniometry Goniometric diagram measures light intensity as function of angle.

55 Photometry: light + humans All radiometric quantities have equivalents in photometry Dark adapted eye (scoptic) Daytime adapted eye (photoptic) Photometry: accounts for response of human visual system V ( ) to electromagnetic radiation Luminance (Y) is photometric quantity that corresponds to radiance: integrate radiance over (nm) all wavelengths, weight by eye s luminous efficacy curve, e.g.: Z 1 Y (p,!) = 0 L(p,!, ) V ( )d

56 Radiometric and photometric terms Physics Radiometry Photometry Energy Radiant Energy Luminous Energy Flux (Power) Radiant Power Luminous Power Flux Density Irradiance (incoming) Radiosity (outgoing) Illuminance (incoming) Luminosity (outgoing) Angular Flux Density Radiance Luminance Intensity Radiant Intensity Luminous Intensity

57 Photometric Units Photometry MKS CGS British Luminous Energy Talbot Talbot Talbot Luminous Power Lumen Lumen Lumen Illuminance Luminosity Lux Phot Footcandle Luminance Nit, Apostlib, Blondel Stilb Lambert Footlambert Luminous Intensity Candela Candela Candela Thus one nit is one lux per steradian is one candela per square meter is one lumen per square meter per steradian. Got it? James Kajiya

58 Next time More toward our goal of realistic rendering Materials, scattering, etc.

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