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Radiometry Outline What is Radiometry? Quantities Radiant energy, flux density Irradiance, Radiance Spherical coordinates, foreshortening Modeling surface properties: the BRDF Lambertian Specular Radiometry Geometry of perspective projection explains location of scene point in image, but what about its intensity and color? Radiometry is the measurement of electromagnetic radiation, primarily optical Frequencies from infrared to visible to ultraviolet Photometry quantifies camera/eye sensitivity Measuring Light For now, ignore the fact that light has multiple wavelengths; we’ll come back to this when we discuss color Fundamental quantities Radiant energy Q (measured in joules J) Proportional to number of photons (and photon frequency) Radiant flux/power © of a light source (watts W) Joules per second emitted Radiant flux density (Wm-2) Power per unit area through a surface (real or imaginary) from all directions Incoming and Outgoing Light at a Surface Irradiance E (Wm-2) Light arriving at a point on a surface from all visible directions An image samples the irradiance at the pinhole -2 Radiosity B (Wm ) Light leaving a surface in all directions Radiance Radiance L (Wm-2sr-1) Power at a point in space in a given direction, foreshortened Can be incoming or outgoing Does not attenuate with distance in vacuum What is stored in a pixel—the light energy arriving along a particular ray at a particular point. The more the surface is tilted away, the larger an area the light energy is distributed over. Foreshortening Lines, patches tilted with respect to the viewing direction present smaller apparent lengths, areas, respectively, to the viewer E.g., angle subtended by differential line segment in 2-D follows from formula for arc length a df r dl Spherical Coordinate Terminology Local coordinates for patch of surface Colatitude/polar angle q (q Longitude/azimuth f = 0 at normal n) Solid angle w (steradians sr): Area of surface patch on unit sphere (w = 4p for entire sphere) w 3-D Foreshortening For patches, just extend 2-D argument to areas: Foreshortening factor for light coming from (q, f) is just cosq Solid Angle in Spherical Coordinates Differential patch dqdf has smaller area closer to pole due to shrinking width of df Circumference at a given polar angle q is 2prsinq, so the correct patch area is dw = sinqdqdf r Computing Irradiance Integrate radiance over the hemisphere Thus, irradiance from a particular direction is Describing an Environment’s Radiance Radiance distribution 5-dimensional function (3 position + 2 direction variables) Environment map: Radiance distribution at a given point Plenoptic function: Time variable added Light field Radiance distribution for an object 4-dimensional function: All positions on image planes of all orthographic views of object BRDF Bidirectional Reflectance Distribution Function (BRDF): Ratio of outgoing radiance in one direction to incident irradiance BRDF Properties Energy leaving · energy arriving Symmetric in both directions (Helmholtz reciprocity) Generally, only difference between incident and emitted angle f is significant Dependence on absolute f Anisotropy Can view BRDF as probability that incoming photon will leave in a particular direction courtesy of S. Rusinkiewicz Things the BRDF neglects Surface non-uniformity Subsurface scattering Light moving under the surface and emerging elsewhere—e.g., marble, skin BRDF Transmission Light going through to other side BSSRDF courtesy of H. Jensen Reflectance Equation Radiance for a viewing direction given all incoming light This is proportional to the pixel brightness for that ray Lambertian Surfaces Diffuse/matte reflectance: Radiance leaving surface does not depend on angle Albedo r: Ratio of light reflected by an object to light received BRDF: f(.) =r/p Specular Surfaces Mirrorlike: Radiance only leaves along specular direction (reflection of incoming direction) BRDF: Specular lobe models shiny surface that is not perfect mirror