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Fundametals of Rendering Radiometry / Photometry CIS 782 Advanced Computer Graphics Raghu Machiraju © Lastra/Machiraju/Möller 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 http://www.optics.arizona.edu/Palmer/rpfaq/rp faq.htm © Lastra/Machiraju/Möller Realistic Rendering • Determination of Intensity • Mechanisms – Emittance (+) – Absorption (-) – Scattering (+) (single vs. multiple) Observer Light • Cameras or retinas record quantity of light © Lastra/Machiraju/Möller 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 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: sinusiodal wave formed coupled electric (E) and magnetic (H) fields – Diffraction / Polarization – Transmitted/reflected compoments – Pink Floyd © Lastra/Machiraju/Möller QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Electromagnetic spectrum © Lastra/Machiraju/Möller Optics • Three kinds • Geometrical or ray – Traditional graphics – Reflection, refraction – Optical system design • Physical or wave – Dispersion, interference – Interaction of objects of size comparable to wavelength • Quantum or photon optics – Interaction of light with atoms and molecules © Lastra/Machiraju/Möller 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 Radiometry • Science of measuring light • Analogous science called photometry is based on human perception. © Lastra/Machiraju/Möller Radiometric Quantities • Function of wavelength, time, position, direction, polarization. g(,t,r,, ) • Assume wavelength independence – No phosphorescence – Incident wavelength 1 exit 2 • Steady State – Light travels fast – No luminescence - Ignore it © Lastra/Machiraju/Möller g(t,r,, ) g(r,, ) Result – five dimensions • Adieu Polarization – Would likely need wave optics to simulate • Two quantities – Position (3 components) – Direction (2 components) © Lastra/Machiraju/Möller g(r, ) Radiometry - Quantities • Energy Q • Power – Energy per time • Irradiance E and Radiosity B – Power per area • Intensity I – Power per solid angle • Radiance L – Power per projected area and solid angle © Lastra/Machiraju/Möller Radiant Energy - Q • Think of photon as carrying quantum of energy hc/ = 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 Radiation Blackbody © Lastra/Machiraju/Möller Tungsten Consequence © Lastra/Machiraju/Möller Fluorescent Lamps © Lastra/Machiraju/Möller Power - • • • • • • Flow of energy (important for transport) Also - radiant power or flux. Energy per unit time (joules/s = eV/s) Unit: W - watts = dQ/dt Falls off with square of distance © Lastra/Machiraju/Möller Radiant Flux Area Density • Area density of flux (W/m2) • u = Energy arriving/leaving a surface per unit time interval • dA can be any 2D surface in space • E.g. sphere: u 2 4r d u dA dA © Lastra/Machiraju/Möller Irradiance E • Power per unit area incident on a surface d E dA © Lastra/Machiraju/Möller Radiosity or Radiant Exitance B • Power per unit area leaving surface • Also known as Radiosity • Same unit as irradiance, just direction changes d B dA © Lastra/Machiraju/Möller Intensity I • Flux density per unit solid angle d I d • Units – watts per steradian • “intensity” is heavily overloaded. Why? – Power of light source – Perceived brightness © Lastra/Machiraju/Möller Solid Angle • Size of a patch, dA, is dA (r sin d )( r d ) • Solid angle is dA d 2 sin dd r • Angle l r © Lastra/Machiraju/Möller Solid Angle (contd.) • • • • • Solid angle generalizes angle! Steradian Sphere has 4 steradians! Why? Dodecahedron – 12 faces, each pentagon. One steradian approx equal to solid angle subtended by a single face of dodecahedron QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. © Wikipedia QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. © Lastra/Machiraju/Möller Hemispherical Projection • Use a hemisphere H over surface to measure incoming/outgoing flux • Replace objects and points with their hemispherical projection r © Lastra/Machiraju/Möller Isotropic Point Source d I d 4 • Even distribution over sphere • How do you get this? • Quiz: Warn’s spot-light ? Determine flux s cos I( ) © Lastra/Machiraju/Möller Other kinds of lights © Lastra/Machiraju/Möller Other Lights © Lastra/Machiraju/Möller Irradiance on Differential Patch d cos EI 2 dA 4 x x s • Inverse square Law • How do you determine that ? • Iso-lux contours © Lastra/Machiraju/Möller x xs Radiance L • Power per unit projected area per unit solid 2 angle. d L 2 • Units – watts per (steradian m ) dAp d • We have now introduced projected area, a cosine term. © Lastra/Machiraju/Möller d L dAcosd 2 Projected Area • Ap = A (N • V) = A cos V N V © Lastra/Machiraju/Möller Why the Cosine Term? • Foreshortening is by cosine of angle. • Radiance gives energy by effective surface area. d cos d © Lastra/Machiraju/Möller Incident and Exitant Radiance • • • • Incident Radiance: Li(p, ) Exitant Radiance: Lo(p, ) In general: Li p, Lo p, p - no surface, no participating media Li p, Lo p, Li p, Lo p, p p © Lastra/Machiraju/Möller Irradiance from Radiance E p,n L p,cos d i • |cos|d is projection of a differential area • We take |cos| in order to integrate over the whole sphere n Li p, r © Lastra/Machiraju/Möller p 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 Law1: Conservation Law ! d1 dA1 2 d dA2 Total flux leaving one side = flux arriving other side, so L1d1dA1 L2d2dA2 © Lastra/Machiraju/Möller Conservation Law ! d1 dA1 2 d d1 dA2 /r 2 and dA2 d 2 dA1 /r therefore L1 (dA2 / r ) dA1 L2 (dA1 / r ) dA2 © Lastra/Machiraju/Möller 2 2 2 Conservation Law ! d1 dA1 2 so d dA2 L1 (dA2 / r 2 ) dA1 L2 (dA1 / r 2 ) dA2 L1 L2 Radiance doesn’t change with distance! © Lastra/Machiraju/Möller 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 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 but conversion is scale factor -- weighted by spectral response of eye (over about 360 to 800 nm). © Lastra/Machiraju/Möller CIE curve • Response is integral over all wavelengths v V ( )d 350 400 450 Violet 500 550 600 Green 650 700 750 Red © CIE, 1924, many more curves available, see http://cvision.ucsd.edu/lumindex.htm © Lastra/Machiraju/Möller Radiometry vs. Photometry © Lastra/Machiraju/Möller Next SPD XYZ Display Tone Reproduction RGB Dither © Lastra/Machiraju/Möller