9/13/2002 CS 461, Copyright G.D. HagerComputer Vision, Lecture 3Professor Hagerhttp://www.cs.jhu.edu/~hager9/13/2002 CS 461, Copyright G.D. HagerOutline for Today• Reflectance•Examples:– understanding surfaces– finding specularities– image-based rendering9/13/2002 CS 461, Copyright G.D. HagerREFLECTANCE MODELS• Description of how light energy incident on an object is transferred from the object to the camera sensorSurfaceSurface NormalHalfwayVectorIncidentLightLθαReflectedLightE9/13/2002 CS 461, Copyright G.D. HagerRadiometry• Core Questions:– how “bright” will surfaces be? – what is “brightness”?• measuring light• interactions between light and surfaces• Core idea - think about light arriving at a surface• around any point is a hemisphere of directions• Simplest problems can be dealt with by reasoning about this hemisphere9/13/2002 CS 461, Copyright G.D. HagerSolid Angle• Defined by analogy with angle (in radians)– solid angle subtended by a patch is the area covered by the patch on a unit sphere• The solid angle subtended by an infinitesimal patch area dA is given by• Another useful expression:dω=dA cosϑr2dω= sinϑdϑ()dφ()9/13/2002 CS 461, Copyright G.D. HagerRadiance• Measure the “amount of light” at a point, in a direction• Definition: Radiant power per unit foreshortened area per unit solid angle• Units: watts per square meter per steradian (wm-2sr-1)• Usually written as:• Crucial property: In a vacuum, radiance leaving p in the direction of q is the same as radiance arriving at q from p– hence the unitsLx,ϑ,ϕ()9/13/2002 CS 461, Copyright G.D. HagerIrradiance• 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 L(x,θ,φ) coming in from dωexperiences irradianceLx,ϑ,ϕ()cosϑdω9/13/2002 CS 461, Copyright G.D. HagerIrradiance• Crucial property: – Total power arriving at the surface is given by adding irradiance over all incoming angles • Total power isLx,ϑ,ϕ()cosϑsinϑdϑdϕΩ∫9/13/2002 CS 461, Copyright G.D. HagerSurfaces and the BRDF• Many effects when light strikes a surface -- could be:– absorbed– transmitted– reflected– scattered• Assume that– surfaces don’t fluoresce– surfaces don’t emit light (i.e. are cool)– all the light leaving a point is due to that arriving at that point9/13/2002 CS 461, Copyright G.D. HagerThe BRDF• Given assumptions, we can model effects at a surface with a record of outgoing vs incoming illumination– the Bidirectional Reflectance Distribution Function (BRDF)• Definition: – the ratio of the radiance in the outgoing direction to the incident irradiance• Units: Inverse steradians()()()()()()ωϑϕϑϕϑϕϑρϕϑωϑϕϑϕϑϕϑϕϑρdxLxxLdxLxLxiiiiiioobdoooiiiioooiioobdcos,,,,,,,,,cos,,,,,,,,,==9/13/2002 CS 461, Copyright G.D. HagerLambertian surfaces and albedo• for a Lambertian surface, BRDF is independent of angle.•Useful fact:ρbrdf=ρdπ() ()ωϑϕϑρdxLxxLiiiibrdfocos,,)(∫=Suppose now that we have an ideal distant pinhole sourcein a given direction. Then we if look at the entire power() ()sourcesourcebrdfiiiibrdfoLxdxLxxLϑρωϑϕϑρcos)(cos,,)( ==∫Ωalbedo9/13/2002 CS 461, Copyright G.D. HagerLAMBERTIAN REFLECTANCE MAPLAMBERTIAN MODELE = L ρ ρ ρ ρ COS θθθθYXZθθθθ(ps,qs,-1)(p,q,-1)2222111ssssqpqpqqppCOS++++++=θ9/13/2002 CS 461, Copyright G.D. HagerLAMBERTIAN REFLECTANCE MAPGrouping L and ρ as a constant , local surface orientations that produce equivalent intensities under the Lambertian reflectance map are quadratic conic section contours in gradient space.2222111ssssqpqpqqppLE++++++=ρ2222111ssssqpqpqqppI++++++=9/13/2002 CS 461, Copyright G.D. HagerLAMBERTIAN REFLECTANCE MAPps=0 qs=09/13/2002 CS 461, Copyright G.D. HagerLAMBERTIAN REFLECTANCE MAPps=0.7 qs=0.39/13/2002 CS 461, Copyright G.D. HagerLAMBERTIAN REFLECTANCE MAPps= -2 qs= -19/13/2002 CS 461, Copyright G.D. HagerAnother View• Recall that cos(θ) = n · s – n unit normal to surface– s unit vector directed toward source• So, we can also write L0= ρbrdfn· s• An interesting exercise (Example Use 1):– show that three pinhole light sources suffice to fully determine the photometric structure of a Lambertian surface– hint: combine ρ and n into a single (now not unit vector) quanitity and consider varying the direction s• A second interesting execise: show that given this you can synthesize a view of the scene from a new directionns9/13/2002 CS 461, Copyright G.D. HagerMonge Patches• Let’s assume (quite reasonably) that the surface is of the form (x,y,f(x,y))• It follows that the surface normal direction is (-df/dx, -df/dy,1) and the normal is the unitized version of this• Given a normal vector (a,b,c) at location x,y, it follows that– df/dx = a/c and df/dy = b/c– note also that it should be the case that d2f / dx dy = d2f /dy dx; we can use this to either check the data, or regularize our reconstruction• To find the height at a location u, v just do– f(u,v) = ∫0vdf/dy(0,y) dy + ∫0udf/dx(x,v) dx9/13/2002 CS 461, Copyright G.D. HagerSpecular surfaces• Another important class of surfaces is specular, or mirror-like.– radiation arriving along a direction leaves along the specular direction– reflect about normal– some fraction is absorbed, some reflected– on real surfaces, energy usually goes into a lobe of directions– can write a BRDF, but requires the use of funny functions9/13/2002 CS 461, Copyright G.D. HagerPhong’s model• There are very few cases where the exact shape of the specular lobe matters.•Typically:– very, very small --- mirror– small -- blurry mirror– bigger -- see only light sources as “specularities”– very big -- faint specularities• Phong’s model– reflected energy falls off withcosnδϑ()9/13/2002 CS 461, Copyright G.D. HagerLambertian + specular model• Widespread model– all surfaces are Lambertian plus specular component• Advantages– easy to manipulate– very often quite close true• Disadvantages– some surfaces are not• e.g. underside of CD’s, feathers of many birds, blue spots on many marine crustaceans and fish, most rough surfaces, oil films (skin!), wet surfaces– Generally, very little advantage in modeling behavior of light at a surface in