Radiometry, lights and surfaces Marc Pollefeys COMP 256 Slides from David Forsyth,,…

Last class Camera Models Camera with Lenses Sensing The Human Eye Pinhole Perspective Projection Affine Projection Camera with Lenses Sensing The Human Eye

Radiometry 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 hemisphere Warn students: most people find radiometry tricky. Key trick to understanding is to remember to think about what a point on the surface “sees” --- lie on your back and look up. We do this using a small hemisphere of directions put on top of a surface at a particular point.

Lambert’s wall In Lambert’s wall, we assume that the sky is overcast, that the horizontal plane is uniform, and that the vertical wall is black --- what is the distribution of brightness (whatever that is) on the ground? Ans: every point sees the same input hemisphere, and as a result each point must be the same.

More complex wall This is a revised Lambert’s wall. Now the wall is half-infinite in length. This means that points on the ground lying on a ray through p all see the same input hemisphere. A little simple geometry shows that those on the ray along the base of the wall see only half the input hemisphere than those on the ray parallel to the wall but pointing away from it. This means that the rays are isophotes -- worth using the word and explaining --- and we know which is brighter. The point is that quite simple reasoning allows us to determine what’s going on with very little formalism.

Foreshortening Principle: two sources that look the same to a receiver must have the same effect on the receiver. Principle: two receivers that look the same to a source must receive the same amount of energy. “look the same” means produce the same input hemisphere (or output hemisphere) Reason: what else can a receiver know about a source but what appears on its input hemisphere? (ditto, swapping receiver and source) Crucial consequence: a big source (resp. receiver), viewed at a glancing angle, must produce (resp. experience) the same effect as a small source (resp. receiver) viewed frontally. In DAF’s experience, this is the only way to think about foreshortening. It also explains why there are two cosines. Firstly, the receiver cares about what the source looks like, but also the source must care about what the receiver looks like. It’s a good idea to do a drawing on the whiteboard emphasizing how two different sources can appear exactly the same at a given receiver.

Measuring Angle To define radiance, we require the concept of solid angle The solid angle sub- tended 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 2D If I’m at P, and I look out, solid angle tells me how much of my view is filled with an object

Solid Angle of a Small Patch Later, it will be important to talk about the solid angle of a small piece of surface A

Measuring Light in Free Space Desirable property: in a vacuum, the relevant unit does not go down along a straight line. How do we get a unit with this property? Think about the power transferred from an infinitesimal source to an infinitesimal receiver. We have total power leaving s to r = total power arriving at r from s Also: Power arriving at r is proportional to: solid angle subtended by s at r (because if s looked bigger from r, there’d be more) foreshortened area of r (because a bigger r will collect more power)

Radiance light surface All this suggests that the light transferred from source to receiver should be measured as: Radiant power per unit foreshortened area per unit solid angle This is radiance 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 which was how we got to the unit Everybody finds radiance difficult at first glance. In DAF’s experience, the right thing to do is to

Radiance is constant along straight lines Power 1->2, leaving 1: Power 1->2, arriving at 2: But these must be the same, so that the two radiances are equal I always warn a class that this proof looks silly, but is of the first importance, and needs to be understood.

Spectral Quantities To handle color properly, it is important to talk about spectral radiance Defined at a particular wavelength, per unit wavelength: L(x,,) To get total radiance, integrate over spectrum: More about color later…

Irradiance, E light light surface surface 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,q,f) coming in from dw experiences irradiance Crucial property: Total power arriving at the surface is given by adding irradiance over all incoming angles --- this is why it’s a natural unit Total power is At this point, many students find it reassuring if one regularly emphasizes the similarity between integration and adding up. light light surface surface

Example: Radiometry of thin lenses

Reflectance We have all the things we need dealing with the transport of light Reflectance is all about the way light interacts with surfaces It is an entire field of study on its own The most important quantity is the BRDF…

Light at surfaces Many effects when light strikes a surface -- could be: absorbed transmitted skin reflected mirror scattered milk travel along the surface and leave at some other point sweaty skin Assume that surfaces don’t fluoresce e.g. scorpions, washing powder surfaces don’t emit light (i.e. are cool) all the light leaving a point is due to that arriving at that point I use skin’s tendency to look bright along the viewing direction to illustrate transmission in a surface layer. Skin also illustrates transmission and later reflection (e.g. from a vein under the surface). Fluorescence is a nuisance, because you have to keep track of illumination that arrives at one wavelength and leaves at another. It’s useful to point out that the simplest thing one can do is to keep track of what comes out vs what goes in *at one point*. Any more complex model (changing wavelength, travelling along a surface) requires a lot more parameters and is hard to keep track of.

The BRDF Assuming 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 point Can model this situation with the Bidirectional Reflectance Distribution Function (BRDF) the ratio of the radiance in the outgoing direction to the incident irradiance for an incoming direction

BRDF

BRDF Units: inverse steradians (sr-1) Symmetric in incoming and outgoing directions - this is the Helmholtz reciprocity principle Radiance leaving a surface in a particular direction: add contributions from every incoming direction You should explicitly parse the equation as “add up, over all directions, the irradiance coming in along that direction (hence the cosine) times the BRDF --- which is the ratio of radiance out to incoming irradiance”

Intermezzo - Helmholtz stereo Classic stereo assumption: same appearance from all viewpoints (=Lambertian) Doesn’t hold for general BRDF Idea (Zickler et al. ECCV’02), exploit reciprocity!

Suppressing Angles - Radiosity In many situations, we do not really need angle coordinates e.g. cotton cloth, where the reflected light is not dependent on angle Appropriate radiometric unit is radiosity total power leaving a point on the surface, per unit area on the surface (Wm-2) note that this is independent of the direction Radiosity from radiance? sum radiance leaving surface over all exit directions, multiplying by a cosine because this is per unit area not per unit foreshortened area

Radiosity Important relationship: radiosity of a surface whose radiance is independent of angle (e.g. that cotton cloth) It’s worth pointing out that this pi is because it’s *not* just the area of the hemisphere --- that cosine again. It’s also worth pointing out that pi turns up in an awful lot of places in radiometry, for this reason. We’ll see this sort of thing again in a few more slides.

Suppressing the angles in the BRDF BRDF is a very general notion some surfaces need it (underside of a CD; tiger eye; etc) very hard to measure ,illuminate from one direction, view from another, repeat very unstable minor surface damage can change the BRDF e.g. ridges of oil left by contact with the skin can act as lenses for many surfaces, light leaving the surface is largely independent of exit angle surface roughness is one source of this property

Directional hemispheric reflectance the fraction of the incident irradiance in a given direction that is reflected by the surface (whatever the direction of reflection) unitless, range is 0-1 Note that DHR varies with incoming direction e.g. a ridged surface, where left facing ridges are absorbent and right facing ridges reflect.

Lambertian surfaces and albedo For some surfaces, the DHR is independent of illumination direction too cotton cloth, carpets, matte paper, matte paints, etc. For such surfaces, radiance leaving the surface is independent of angle Called Lambertian surfaces (same Lambert) or ideal diffuse surfaces Use radiosity as a unit to describe light leaving the surface DHR is often called diffuse reflectance, or albedo for a Lambertian surface, BRDF is independent of angle, too. Useful fact: The ubiquitous pi again. Tell students that this derivation follows that a few slides back.

Specular 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 functions

Phong’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 with

Lambertian + specular Widespread model Advantages Disadvantages 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 modelling behaviour of light at a surface in more detail -- it is quite difficult to understand behaviour of L+S surfaces

Diffuse + Specular example cosn(q), q=2,10,100,1000 www.exaflop.org/docs/ lca/lca1.html

Next class: Sources Shadows and Shading F&P Chapter 5 upcoming assignment: photometric stereo