Page 221 - Contributed Paper Session (CPS) - Volume 4

CPS2201 Mikhail L.
3. Main definition
In the most basic case, the bidirectional reflectance distribution function
(BRDF), ( , )) is a four-dimensional function that defines how light is

reflected at an opaque surface. The function takes a negative incoming light
direction, , and outgoing direction, , both defined with respect to the

surface normal , and returns the ratio of reflected radiance exiting along

to the irradiance incident on the surface from direction . The BRDF was first

defined by Nicodemus in Nicodemus (1965). The defining equation is:

( ) ( )
(1) ( , ) = =

( ) ( ) cos

where is radiance, or power per unit solid-angle-in-the-direction-of-a-ray
per unit projected-areaperpendicular-to-the-ray, E is irradiance, or power per
unit surface area, and is the angle between and the surface normal, .

The index indicates incident light, whereas the index indicates reflected
light.
Suppose we have measurements of a BRDF available for the set of incoming
angles

() () ()
(2)Ω = { } = {( , )} .

=1 =1

Here ≥ 1 is the total number of incoming angles where the
()
measurements were taken. Say that for an incoming angle { } we have
measurements available for angles from the set of reflection angles

(3) Ω = ⋃ Ω (),
=1
where

() ()
Ω () = { () } = {( () , () )} ,
=1 =1

where are { ()} =1 (possibly different) numbers of measurements taken
for corresponding incoming angles. Our aim is to infer the BRDF manifold (1)
from the above observations.
In general, the connection between the true BRDF and its measurements is
described via a stochastic transformation , i.e., ( , ) =

( ( , )), where : ℳ × × ℱ → ℱ , with ℳ = (, , ) is an (unknown)

4
4
measurable space, = (Π, , ℙ) is an unknown probability space, ℱ is the
4
space of all Helmholtz-invariant energy preserving 4-dimensional BRDFs, and
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