I am confused by measuring "number of rays". Say, we measure there are 100 rays hitting a shape. Can we insert 100 rays between the rays we found to make it 200? I am trying to understand why the answer is not infinity.
Also, in this case there are area units associated with the calculated quantity. What does it mean to have units for the number of rays?
The idea is more like "measuring the area covered by a set of rays" (which is a somewhat unintuitive idea since it's the area of a 4 dimensional thing.) So just as you might measure a finite 2D area, which is itself comprised of an infinite number of points, you can think of measuring a finite area for an infinite collection of rays...
One way to think about it is in terms of measuring rays in 2D by considering two parallel 1D lines (you might want to try sketching this). Mark off a segment on each line. Now these two segments define an infinite collection of rays-all rays that pass through both segments. We might define the measure of those rays as the product of the lengths of those two segments.
(That connects to the "two plane" parameterization in Pat's original light field paper, though it's 4D there: http://graphics.stanford.edu/papers/light/light-lores.pdf.)
Here, the area of the rays is measured by the product of a 2D area on a plane that they pass through times a 2D area of solid angle that their directions subtend.
Thanks a lot for the detailed explanation!