MIT EECS 6.837, Durand and Cutler Local IlluminationMIT EECS 6.837, Durand and Cutler Outline•Introduction•Radiometry •Reflectance•Reflectance ModelsMIT EECS 6.837, Durand and Cutler The Big PictureLight sources emit photons.We assume that photons travel along straight lines.When these photons hit the surfaces they are absorbed (changed to heat).We assume ray optics as opposed to e.g. wave optics. Our eye collects these photons.MIT EECS 6.837, Durand and Cutler Radiometry•Energy of a photon•Radiant Energy of n photons•Radiation flux (electromagnetic flux, radiant flux)Units: WattsdtdQ=Φhhce1063.6 ⋅≈=λλ∑==niihcQ1λsm csJ/103 834⋅≈⋅− r f q q r r n ( ( i i r fr fi fi , , , q qMIT EECS 6.837, Durand and Cutler Radiometry is the measurement of the electromagnetic radiation in the ultraviolet, the visible, and the infrared frequency spectrum. Radiation flux (or radiant flux, electromagnetic flux), usually denoted with φ, is equivalent to power. It measures the time-rate flow of light energy where Q denotes the energy of a collection of photons across all wavelengths and t denotestime. The unit of flux is the Watt [W].MIT EECS 6.837, Durand and Cutler RadiometryωθωddAdL cos)(2Φ=steradianmeterWattUnits :2•Radiance–radiant flux per unit solid angle per unit projected area–Number of photons arrivingper time at a small areafrom a particular direction MIT EECS 6.837, Durand and Cutler This is the most fundamental concept in radiometry. It is a physical quantity equivalent tothe psychological concept of brightness observed by humans. It is usually denoted with theletter L and defined for all directions ω. It measures electromagnetic flux Φtravellinginthe small range of directions through the solid angle element d ⎤ and crossing an elementof projected area dAMIT EECS 6.837, Durand and Cutler Radiometry•Irradiance–differential flux falling onto differential area•Irradiance can be seen as a density of the incident flux falling onto a surface. •It can be also obtained by integrating the radiance over the solid angle.dAdEΦ=2:meterWattUnitsMIT EECS 6.837, Durand and Cutler Light Emission•Light sources: sun, fire, light bulbs etc.•Consider a point light source that emits light uniformly in all directions θnsourcelight the todistance sourcelight theofpower 4 4cos22−−ΦΦ=Φ=ddLdEsssππθSurfaceMIT EECS 6.837, Durand and Cutler Outline•Introduction•Radiometry •Reflectance•Reflectance ModelsMIT EECS 6.837, Durand and Cutler Reflection & Reflectance•Reflection -the process by which electromagnetic flux incident on a surface leaves the surface without a change in frequency.•Reflectance –a fraction of the incident flux that is reflected•We do not consider:–absorption, transmission, fluorescence–diffractionMIT EECS 6.837, Durand and Cutler Reflectance•Bidirectional scattering-surface distribution Function (BSSRDF)Surface Images removed due to copyright considerations.MIT EECS 6.837, Durand and Cutler Reflectance•Bidirectional scattering-surface distribution Function (BSSRDF)),,,(),,,(),,,,,,,(iiiiirrrrrrriirriiyxdyxdLyxyxSφθφθφθφθΦ=steradianmeterUnits 1:2SurfaceMIT EECS 6.837, Durand and Cutler Reflectance•Bidirectional Reflectance Distribution Function (BRDF)),(),(),,,(iiirrrrriirdEdLfφθφθφθφθ=steradianUnits1: q q Lr Li n r fi fr iMIT EECS 6.837, Durand and Cutler Isotropic BRDFs•Rotation along surface normal does not change reflectance),(),(),,(),,(diidrrdririrrirdEdLffφθφθφθθφφθθ==− q q Lr Li n r fd iMIT EECS 6.837, Durand and Cutler Anisotropic BRDFs•Surfaces with strongly oriented microgeometryelements•Examples: –brushed metals,–hair, fur, cloth, velvet Images removed due to copyright considerations.MIT EECS 6.837, Durand and Cutler Properties of BRDFs•Non-negativity•Energy Conservation•Reciprocity0),,,(≥rriirfφθφθ),( allfor 1),(),,,(iirrrriirdfφθφθµφθφθ≤∫Ω),,,(),,,(iirrrrriirffφθφθφθφθ=MIT EECS 6.837, Durand and Cutler How to compute reflected radiance?•Continuous version•Discrete version –npoint light sourcesiiiiririirirrrdndLfdEfLωωωωωωωωω )cos()( ),()(),()(⋅===∫∫ΩΩ211 4 cos),(),()(jsjjnjrijrjnjrijrrrdfEfLπθωωωωωΦ===∑∑==),(φθω=MIT EECS 6.837, Durand and Cutler Outline•Introduction•Radiometry •Reflectance•Reflectance ModelsMIT EECS 6.837, Durand and Cutler How do we obtain BRDFs?•Measure BRDF valuesdirectly•Analytic Reflectance Models–Physically-based models•based on laws on physics–Empirical models•“ad hoc” formulas that work Rotating Annuli Transmittance Detector Sample Area Light Source Source Driver Hoop Reflectance Detector Ward, Gregory J. “Measuring and Modeling Anisotropic Reflection.” ACM SIGGRAPH Computer Graphics. 26, no.2 (July 1992): 265-272. Image adapted from:MIT EECS 6.837, Durand and Cutler Ideal Diffuse Reflectance•Assume surface reflects equally in all directions.•An ideal diffuse surface is, at the microscopic level, a very rough surface.–Example: chalk, clay, some paintsSurfaceMIT EECS 6.837, Durand and Cutler Ideal Diffuse Reflectance•BRDF value is constant SurfaceθidBdAidAdBθcos=iriiriirirrrEfdEfdEfL=====∫∫ΩΩ)()(),()(ωωωωωnMIT EECS 6.837, Durand and Cutler •Ideal diffuse reflectors reflect light according to Lambert's cosine law.Ideal Diffuse ReflectanceIdeal diffuse reflectors reflect light according to Lambert's cosine law.Lambert's law determines how much of theincoming light energy is reflected.The reflected intensity is independent of the viewing direction. LAMBERT'S COSINE LAW http://escience.anu.edu.au/Image adapted from:MIT EECS 6.837, Durand and Cutler Ideal Diffuse Reflectance•Single Point Light Source–kd: The diffuse reflection coefficient.–n: Surface normal.–l: Light direction.2 4 )()(dkLsdrπωΦ⋅=lnSurfaceθlnMIT EECS 6.837, Durand and Cutler Ideal Diffuse Reflectance –More Details•If nand lare facing away from each other, n • lbecomes negative. •Using max( (n• l),0 ) makes sure that the result is zero.–From now on, we mean max() when we write •.•Do not forget to normalize your vectorsfor the dot product!MIT EECS 6.837, Durand and Cutler Ideal SpecularReflectance•Reflection is only at mirror angle.–View dependent–Microscopic surface elements are usually oriented in the same direction as the surface itself.–Examples: mirrors, highly polished metals.SurfaceθlnrθMIT EECS 6.837, Durand and Cutler A second surface type is called a specularreflector. When we look at a shiny surface, such as polished metal or a glossy car finish, wesee a highlight, or bright spot. Where this bright spot appears on thesurface is a function of where the surface is seen from. This type of reflectance is view dependent. At the microscopic level a specularreflecting surface is very smooth, and usually these microscopic surface elements are oriented in the same direction as the surface itself. Specularreflection is merely the mirror reflectionof the light source in a surface. Thus it should come as no surprise that it is viewer dependent, since if you stood in front of a mirror and placed your finger over the reflection of a light, you would expect that you could reposition your head to look around your finger and see the light again. An ideal mirror is a purely specularreflector. In order to model specularreflection we need to understand the physics of reflection.MIT EECS 6.837, Durand and Cutler Ideal SpecularReflectance•Special case of Snell’s Law–The incoming ray, the surface normal, and the reflected ray all lie in a common plane.Surfaceθllnrθrrlrlrrllnnnnθθθθ===sinsinMIT EECS 6.837, Durand and Cutler The angle that the reflected ray forms with the surface normal is determined by the angle that the incoming ray forms with the surface normal, and the relative speeds of light of the mediums in which the incident and reflected rays propagate according to the following expression. (Note: nland nrare the indices of refraction) Reflection is a very special case of Snell's Law where the incident light's medium and the reflected rays medium is the same. Thus we can simplify the expression to:MIT EECS 6.837, Durand and Cutler Non-ideal Reflectors•Snell’s law applies only to ideal mirror reflectors.•Real materials tend to deviate significantly from ideal mirror reflectors.•They are not ideal diffuse surfaces either …At this point we will introduce an empirical model that is consistent with our experience, at least to a crude approximation.In general, we expect most of the reflected light to travel in the direction of the ideal ray.However, because of microscopic surface variations we might expect some of the light tobe reflected just slightly offset from the ideal reflected ray. As we move farther and farther,in the angular sense, from the reflected ray we expect to see less light reflected.MIT EECS 6.837, Durand and Cutler Non-ideal Reflectors•Simple Empirical Model:–We expect most of the reflected light to travel in the direction of the ideal ray.–However, because of microscopic surface variations we might expect some of the light to be reflected just slightly offset from the ideal reflected ray. –As we move farther and farther, in the angular sense, from the reflected ray we expect to see less light reflected. r n l q1 http://escience.anu.edu.au/Image adapted from:MIT EECS 6.837, Durand and Cutler The PhongModel•How much light is reflected?–Depends on the angle between the ideal reflection direction and the viewer direction α.SurfaceθlnrθCameravαMIT EECS 6.837, Durand and Cutler The PhongModel•Parameters–ks: specularreflection coefficient–q: specularreflection exponentSurfaceθlnrθCameravα22 4 )( 4 )(cos)(dkdkLsqssqsrππαωΦ⋅=Φ=rvOne function that approximates this fall off is called the PhongIllumination model.This model has no physical basis, yet it is one of the most commonly used illumination models in computer graphics.The cosine term is maximum when the surface is viewed from the mirror direction and falls off to 0when viewed at 90 degrees away from it. The scalar nshinycontrols the rate of this fall off.MIT EECS 6.837, Durand and Cutler The PhongModel•Effect of the q coefficientThe diagram below shows the how the reflectance drops off in a Phongilluminationmodel.For a large value of the nshinycoefficient, the reflectance decreases rapidly with increasing viewing angle.MIT EECS 6.837, Durand and Cutler The PhongModellnlnrnlr−⋅==+)(2 cos 2θSurfaceθ22 4 ))( 4 )()(dkdkLsqssqsrππωΦ−⋅⋅==Φ⋅=lnlnvrv)(2(nθrlrMIT EECS 6.837, Durand and Cutler Blinn-Torrance Variation•Uses the halfway vector hbetween land v.||vl||vlh++=22 4 )( 4 )(cos)(dkdkLsqssqsrππβωΦ⋅=Φ=hnSurfacelnCameravhβMIT EECS 6.837, Durand and Cutler Jim Blinnintroduced another approach for computing Phong-like illumination based on the work of Ken Torrance.His illumination function uses the following equation:H is the normal to the (imaginary) surface that maximally reflects light in the V directionIn this equation the angle of speculardispersion is computed by how far the surface's normal is from a vector bisecting the incoming light direction and the viewing direction. On your own you should consider how this approach and the previous one differ.MIT EECS 6.837, Durand and Cutler PhongExamples•The following spheres illustrate specularreflections as the direction of the light source and the coefficient of shininess is varied.Blinn-TorrancePhongMIT EECS 6.837, Durand and Cutler The PhongModel•Sum of three components:diffuse reflection +specularreflection +“ambient”.SurfaceMIT EECS 6.837, Durand and Cutler Ambient Illumination•Represents the reflection of all indirect illumination.•This is a total hack!•Avoids the complexity of global illumination.arkL=)(ωMIT EECS 6.837, Durand and Cutler Putting it all together•PhongIllumination Model2 4 )()()()(dkkkLsqsdarπωΦ⋅+⋅+=rvlnMIT EECS 6.837, Durand and Cutler For Assignment 3•Variation on PhongIllumination ModeliqsdadrLkkLkL)()()()(rvln⋅+⋅+=ωMIT EECS 6.837, Durand and Cutler Adding color • Diffuse coefficients: – kd-red, kd-green, kd-blue • Specular coefficients: – ks-red, ks-green, ks-blue • Specular exponent: qMIT EECS 6.837, Durand and Cutler PhongDemoMIT EECS 6.837, Durand and Cutler FresnelReflection•Increasing specularitynear grazing angles.Source: Lafortuneet al. 97Can Phongmodel handle this case? (Images removed due to copyright considerations.)MIT EECS 6.837, Durand and Cutler Off-specular& Retro-reflection•Off-specularreflection–Peak is not centered at the reflection direction•Retro-reflection:–Reflection in the direction of incident illumination–Examples: Moon, road markings MIT EECS 6.837, Durand and Cutler The PhongModel•Is it non-negative?•Is it energy-conserving?•Is it reciprocal?•Is it isotropic?MIT EECS 6.837, Durand and Cutler Shaders(Materialclass)•Functions executed when light interacts with a surface•Constructor:–set shaderparameters •Inputs:–Incident radiance–Incident & reflected light directions–surface tangent (anisotropic shadersonly)•Output:–Reflected radianceMIT EECS 6.837, Durand and Cutler Questions?