Image Processing 11. Radiometry and Shading Models Aleix M. Martinez [email protected] Radiometry • By understanding how light travels from a source to a surfaces and how this creates a brightness pattern, we will be able to estimate additional data from an image. • Our goal is to understand the principals and how these can be used. • We will used them to build what’s call shading models. Illumination A 3D object illuminated with a light source. ^n ^s Intensity at a image point I(u,v) (u,v)T. 1 Hemisphere of Direction Hemisphere of Direction • Questions: – how “bright” will a surfaces be (luminance)? – what is “brightness”? • measuring light, • interactions between light and surfaces. • Core idea: think the light arriving at a surface around any point defines a hemisphere of directions. • Simplest problems can be dealt with by reasoning about this hemisphere. 2 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 a patch ì dAcosJ area dA is given by: ïdw = , í r 2 îïdw = sinJ(dJ)(df). Radiance • The power (amount of energy per unit time) traveling at some point in a specified direction, per unit area perpendicular to the direction of travel, per unit solid angle. L(P,q,f) • Units: watts per square meter per steradian (wm-2sr-1). Irradiance • Incident power per unit area not foreshortened. • A surface experiencing radiance L(x,q,f) coming in from dw experiences irradiance: L(P,q,f)cosq dw. 3 BRDF • The BRDF (Bidirectional Reflectance Distribution Function) is the ratio between the incoming radiance and the outgoing irradiance at a point P: Lo (P,qo ,fo ) rbd (qo ,fo ,qi ,fi ) = . Li (P,qi ,fi ) cosqi dw • This is given by the properties of the object material. BRDF Original image Change the BRDF to make the skin look like tanned, with added facial hair, or darker. 4 Radiosity • The total power leaving a point on a surface per unit area on the surface (Wm-2). B P = L P,q,f cosq dw. ( ) ò o ( ) W • Note that this is independent of the direction. Radiosity and Constant Radiance • Radiosity of a surface whose radiance is independent of angle (e.g. that cotton cloth): B x = L x,J,j cosJdw ( ) ò o ( ) W = L x cosJdw o ( )ò W p 2 2p = L x cosJ sinJdjdJ o ( ) ò ò 0 0 = pLo (x). Albedo • A common, reasonable assumption is that the light leaving a surface is independent of the exit angle. • Directional Hemisphere Reflectance: the fraction of incident irradiance in a given direction that is reflected back, whatever the direction of reflection. Lo (P,qo ,fo )cosqo dwo òW r ph (qi ,fi ) = Li (P,qi ,fi ) cosqi dw = rdb (q0 ,f0 ,qi ,fi )cosqo dwo. òW 5 • The second most common assumption is that this directional hemisphere reflectance function does not depend on the direction of the illumination (i.e., most directions produce the same illumination effect). • This is reasonable if the object is convex. rd = rbd (q0 ,f0 ,qi ,fi )cosqo dwo òW = r cosqo dwo = rp. òW ALBEDO Sources and shading • How bright (or what color) • General idea: are objects? • One more definition: ìradiosity due to ü B(x) = E(x) + ò í ý dw Exitance of a source is W îincoming radianceþ – the internally generated power radiated per unit area on the radiating surface. • But what aspects of the • similar to radiosity: a incoming radiance will we source can have both model? – radiosity, because it reflects, – exitance, because it emits. Radiosity due to a point source 2 æ e ö p è dø 6 Radiosity due to a point source • Radiosity is B(x)= pLo (x) = rd (x) ò Li (x,w)cosqidw W = rd (x) ò Li (x,w)cosqidw D » rd (x)(solid angle)(Exitance term)cosqi r (x)cosq = d i (Exitance term and some constants) r(x)2 Standard nearby point source model • N is the surface normal • r is diffuse albedo • S is source vector - a vector from x to the source, whose length is the intensity term – works because a dot-product is basically a cosine æ N(x)× S(x)ö r x ç ÷ d ( )ç 2 ÷ è r(x) ø Standard distant point source model • 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: rd (x)(N(x)× Sd (x)) 7 Shadows cast by a point source 8.