Realistic Rendering Fundametals of Rendering - Radiometry / Photometry • Determination of Intensity • Mechanisms – Emittance (+) – Absorption (-) “Physically Based Rendering” – Scattering (+) (single vs. multiple) by Pharr & Humphreys • Cameras or retinas record quantity of light

•Chapter 5: Color and Radiometry •Chapter 6: Camera Models - we won!t cover this in class

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Pertinent Questions Electromagnetic spectrum

• Nature of light and how it is: – Measured – Characterized / recorded • (local) reﬂection of light • (global) spatial distribution of light

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Spectral Power Distributions Tristimulus Theory of Color e.g., Fluorescent Lamps

Metamers: SPDs that appear the same visually

Color matching functions of standard human observer International Commision on Illumination, or CIE, of 1931

“These color matching functions are the amounts of three standard monochromatic primaries needed to match the monochromatic test primary at the wavelength shown on the horizontal scale.” from Wikipedia “CIE 1931 Color Space” 782 782 Optics What Is Light ?

Three views • Light - particle model (Newton) •Geometrical or ray – Light travels in straight lines Traditional graphics – Light can travel through a – vacuum – Reﬂection, refraction (waves need a medium to – Optical system design travel in) – Quantum amount of energy Physical or wave • • Light – wave model (Huygens): – Dispersion, interference electromagnetic radiation: sinusiodal wave formed coupled – Interaction of objects of size comparable to electric (E) and magnetic (H) wavelength ﬁelds •Quantum or photon optics – Interaction of light with atoms and molecules

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Nature of Light Nature of Light

• Wave-particle duality • Coherence - Refers to frequencies of waves – Light has some wave properties: frequency, phase, orientation • Laser light waves have uniform frequency – Light has some quantum particle properties: quantum packets • Natural light is incoherent- waves are multiple frequencies, and random (photons). in phase.

• Dimensions of light • Polarization - Refers to orientation of waves. – Amplitude or Intensity – Polarized light waves have uniform orientation – Frequency – Natural light is unpolarized - it has many waves summed all with – Phase random orientation – Polarization – Focused feﬂected light tends to be parallel to surface of reﬂection

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Radiometry Radiometry Questions

• Science of measuring energy (light in our case) • Measure energy leaving a light source, as a function of direction • Analogous science called photometry is based on human perception. • Measure energy hitting a surface, in a particular direction

• Measure energy leaving a surface, in a particular direction

The energy is light, photons in this case 782 782 Solid Angle Terminology

First, need to define 3D angular units Energy Radiant energy

2D Power Energy per unit of time full circle - radians

3D full sphere - steradians

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Power on a surface Power in a direction

Radiant intensity Flux Radiant power passing through a surface Power per unit solid angle

Radiance Power radiated per unit Flux density Radiant power per unit area on a solid angle per unit surface projected source area

Incident ﬂux density Flux density arriving at a surface in all directions for rendering: important when considering the direction toward the camera Exitant ﬂux density Flux density leaving from a (aka Radiosity) surface in all directions

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Radiometry - Quantities Terms & Units Energy per time • Energy Q Power • Power " for a surface per unit Flux – Energy per time solid angle • Irradiance E and Radiosity B per unit surface area – Power per area Radiant • Intensity I Intensity Flux density – Power per solid angle (incident, irradiance, exitant, radiosity) • Radiance L per unit per unit – Power per projected area and solid angle projected solid angle source area Given radiance, you can Radiance integrate to get other terms 782 782 Radiant Energy - Q Power - !

• Think of photon as carrying quantum of energy • Flow of energy (important for transport) • Wave packets • Also - radiant power or ﬂux. • Total energy, Q, is then energy of the total number of • Energy per unit time (joules/s = eV/s) photons • Unit: W - watts • Units - joules or eV • • Falls off with square of distance

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Radiant Flux Area Density Irradiance E or simply ﬂux density

• Area density of ﬂux (W/m2) • Power per unit area incident on a surface • u = Energy arriving/leaving a surface per unit time interval

• dA can be any 2D surface in space

• E.g. sphere:

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Radiosity or Radiant Exitance B Intensity I

• Power per unit area leaving surface • Flux density per unit solid angle • Also known as Radiosity • Same unit as irradiance, just direction changes • Units – watts per steradian • Radiant intensity • “intensity” is heavily overloaded. Why? – Power of light source – Perceived brightness

782 782 Solid Angle Solid Angle (contd.)

• Size of a patch, dA, in terms of its Solid angle generalizes angle! angular direction, is • • Steradian • Solid angle is • Sphere has 4# steradians! Why? • Dodecahedron – 12 faces, each pentagon. • One steradian approx equal to solid angle subtended by a single face of dodecahedron

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Hemispherical Projection Isotropic Point Source

• Use a hemisphere $ over surface to measure incoming/outgoing ﬂux

• Replace objects and points with their hemispherical projection • Even distribution over sphere

! ! r

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Radiance L Projected Area

• Power per unit projected area per unit solid angle. • Units – watts per (steradian m2) • We have now introduced projected area, a cosine term.

V N V "

782 782 Why the Cosine Term? Incident and Exitant Radiance

• Foreshortening is by cosine of angle. • Incident Radiance: Li(p, %) Note that direction is always away from point • Radiance gives energy by effective surface area. • Exitant Radiance: Lo(p, %) • In general: • p - no surface, no participating media

d cos" "

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Irradiance from Radiance Reflected Radiance & BRDFs

dLo(p, ωo) ∝ dE(p, ωi)

N • |cos&|d% is projection of a differential area ωo • We take |cos&| in order to integrate over the whole ωi θi sphere n

dLo(p, ωo) dLo(p, ωo) fr(p, ωo, ωi) = = dE(p, ωi) Li(p, ωi) cos θidωi r p 782 782

Bidirectional Reflection Bidirectional Scattering Distribution Functions Distribution Functions

Bidirectional Reﬂection Distribution Function (BRDF) Reciprocity: fr(p, ωi, ωo) = fr(p, ωo, ωi) fr(p, ωo, ωi)

Energy Conservation: Bidirectional Transmittance Distribution Function (BTDF)

ft(p, ωo, ωi) fr(p, ωo, ω!) cos θ!dω! ≤ 1 !H2(n) Bidirectional Scattering Distribution Function (BSDF)

f(p, ωo, ωi) 782 782 Bidirectional Scattering Distribution Functions

dLo(p, ωo) = f(p, ωo, ωi)Li(p, ωi)| cos θi|dωi

Lo(p, ωo) = f(p, ωo, ωi)Li(p, ωi)| cos θi|dωi !S2(n)

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