Radiometry Radiometry Computer Vision I CSE5A ecture 5-Part II Read Chapter 4 of Ponce & Forsyth Solid Angle Irradiance Radiance BRDF ambertian/phong BRDF Measuring Angle Solid Angle By analogy with angle (in radians), the solid angle subtended by a region at a point is the area projected on a unit sphere centered at that point The solid angle subtended by an object from a point P is the area of the projection of the object onto the unit sphere centered at P. Measured in steradians, sr Definition is analogous to projected angle in D If I m at P, and I look out, solid angle tells me how much of my view is filled with an object The solid angle subtended by a patch area da is given by dω = dacosϑ r Radiance Power is energy per unit time Radiance: Power traveling at some point in a specified direction, per unit area perpendicular to the direction of travel, per unit solid angle Symbol: (θ,φ) Units: watts per square meter per steradian : w/(m sr 1 ) q x da dw P = ( dacosθ ) dω Radiance transfer What is the power received by a small area da at distance r from a small emitting area da 1? P = da1 cosθ1dω1 > = da 1dA cosθ1 cosθ r From definition of radiance P = ( dacosθ ) dω From definition of solid angle dacosθ dω = r 1
How much light is arriving at a surface? Sensible unit is Irradiance Incident power per unit area not foreshortened This is a function of incoming angle. A surface experiencing radiance (θ,φ) coming in from solid angle dω experiences irradiance: ( θ, φ ) cosφdω (θ,φ) φ x Irradiance Crucial property: Total power arriving at the surface is given by adding irradiance over all incoming angles Total power is Ω ( θ, φ) cosφ sinθdθd φ x Intermezzo: Camera s sensor Measured pixel intensity is a function of irradiance integrated over pixel s area over a range of wavelengths For some time I = t λ x y E( y, λ, t) s( y) q( λ) dydxdλ dt Ideally, it s linear to the radiance, but the camera response C(.) may not be linear I C = t λ x y E( y, λ, t) s( y) q( λ) dydxdλ dt Image sensor Two types : 1. CCD. CMOS CCD separate photo sensor at regular positions no scanning charge-coupled devices (CCDs) : interline transfer and frame transfer photosensitive storage CMOS Each photo sensor has its own amplifier More noise (reduced by subtracting black image) ower sensitivity (lower fill rate) Uses standard CMOS technology Allows to put other components on chip Smart pixels CCD vs. CMOS Mature technology Specific technology production cost power consumption er fill rate Blooming Sequential readout Recent technology Standard IC technology Cheap ow power ess sensitive Per pixel amplification Random pixel access Smart pixels On chip integration with other components
We consider concepts: Color Cameras 1. Prism (with sensors). Filter mosaic. Filter wheel Prism colour camera Separate light in beams using dichroic prism Requires sensors & precise alignment Good color separation and X Filter mosaic Coat filter directly on sensor Filter wheel Rotate multiple filters in front of lens Allows more than colour bands Demosaicing (obtain full colour & full resolution image) Only suitable for static scenes Prism vs. mosaic vs. wheel approach # sensors Separation Cost Framerate Artefacts Bands Prism ow Mosaic 1 Average ow Aliasing Wheel 1 Good Average ow Motion or more new color CMOS sensor Foveon s X smarter pixels better image quality -end cameras ow-end cameras Scientific applications
Example: Radiometry of thin lenses ight at surfaces δa'cosα δacos β δω = = ( z' / cosα ) ( z / cos β ) π d cosα π d Ω = = cosα 4 ( z / cosα ) 4 z δa cosα z = δa' cos β z' π d δp = ΩδA cos β = δacos α cos β 4 z δp π d δa E = = cos α cos β δa' 4 z δa' d E = π cos 4 α 4 z E: Image irradiance : emitted radiance d : ens diamter Z: depth α: Angle of patch from optical axis Many effects when light strikes a surface -- could be: transmitted Skin, glass reflected mirror scattered milk travel along the surface and leave at some other point absorbed sweaty skin Assume that surfaces don t fluoresce e.g. scorpions, detergents surfaces don t emit light (i.e. are cool) all the light leaving a point is due to that arriving at that point With assumptions in previous slide Bi-directional Reflectance Distribution Function ρ(θ in, φ in ; θ out, φ out ) BRDF Ratio of incident irradiance to emitted radiance Function of Incoming light direction: θ in, φ in Outgoing light direction: θ out, φ out ρ ( x θ, φ ; θ, φ ) ; in in out out = (θ in,φ in ) o( x; θout, φout ) ( x; θ, φ ) cosφ dω i in in in ^n (θ out,φ out ) Surface Reflectance Models Common Models ambertian Phong Physics-based Specular [Blinn 1977], [Cook-Torrance 198], [Ward 199] Diffuse [Hanrahan, Kreuger 199] Generalized ambertian [Oren, ayar 1995] Thoroughly Pitted Surfaces [Koenderink et al 1999] Phenomenological [Koenderink, Van Doorn 1996] Arbitrary Reflectance on-parametric model Anisotropic on-uniform over surface BRDF Measurement [Dana et al, 1999], [Marschner ] ambertian (Diffuse) Surface BRDF is a constant called the albedo Emitted radiance is OT a function of outgoing direction i.e. constant in all directions. For lighting coming in single direction, emitted radiance is proportional to cosine of the angle between normal and light direction r =. ω ι 4
ambertian reflection Specular Reflection: Smooth Surface ambertian Matte Diffuse ight is radiated equally in all directions Rough Specular Surface Phong obe Gonioreflectometers Gonioreflectometers Three degrees of freedom spread among light source, detector, and/or sample Three degrees of freedom spread among light source, detector, and/or sample 5
Gonioreflectometers Ward s BRDF Measurement Setup Collect reflected light with hemispherical (should be ellipsoidal) mirror [SIGGRAPH 9] Can add fourth degree of freedom to measure anisotropic BRDFs Marschner s ImageImage-Based BRDF Measurement Ward s BRDF Measurement Setup For uniform BRDF, capture -D slice corresponding to variations in normals Result: each image captures light at all exitant angles BRDF ot Always Appropriate ight sources and shading How bright (or what colour) are objects? One more definition: Exitance of a source is the internally generated power radiated per unit area on the radiating surface Similar to irradiance BRDF BSSRDF http://graphics.stanford.edu/papers/bssrdf/ (Jensen, Marschner, evoy, Hanrahan) 6
Radiosity due to a point sources Standard nearby point source model small, distant sphere radius ε and exitance E, which is far away subtends solid angle of about π ε d ρ d ( x) ( x) S( x) ( ) r x S is the surface normal ρ is diffuse (ambertian) albedo S is source vector - a vector from x to the source, whose length is the intensity term works because a dotproduct is basically a cosine Standard distant point source model ine sources Issue: nearby point source gets bigger if one gets closer the sun doesn t for any reasonable binding of closer Assume that all points in the model are close to each other with respect to the distance to the source. Then the source vector doesn t vary much, and the distance doesn t vary much either, and we can roll the constants together to get: S ( x) ( ( x) S ( x) ) ρ d d radiosity due to line source varies with inverse distance, if the source is long enough Area sources Examples: diffuser boxes, white walls. The radiosity at a point due to an area source is obtained by adding up the contribution over the section of view hemisphere subtended by the source change variables and add up over the source See Forsyth & Ponce or a graphics text for details. 7