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

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

3. 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.

4. 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.

5. 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.

6. 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.

7. 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

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

9. 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)

10. 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

11. 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.

12. 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…

13. 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

14. Example: Radiometry of thin lenses

15. 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…

16. 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.

17. 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

18. BRDF

19. 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”

20. 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!

21. 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

22. 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.

23. 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

24. 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.

25. 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.

26. 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

27. 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

28. 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

29. Diffuse + Specular example
cosn(q), q=2,10,100,1000
lca/lca1.html

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