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56 2 Image formation

^
n
^
n ^ v r
^ v i
^
ș i ș r d y
ij r
ij i
^
d x
(a) (b)

Figure 2.15 (a) Light scatters when it hits a surface. (b) The bidirectional reflectance distribution function
(BRDF) f(θ i ,φ i ,θ r ,φ r ) is parameterized by the angles that the incident, ˆv i , and reflected, ˆv r , light ray directions
ˆ ˆ
make with the local surface coordinate frame (d x , d y , ˆn).

direction ˆv i is emitted in a reflected direction ˆv r (Figure 2.15b). The function can be written
in terms of the angles of the incident and reflected directions relative to the surface frame as

f r (θ i ,φ i ,θ r ,φ r ; λ). (2.81)

The BRDF is reciprocal, i.e., because of the physics of light transport, you can interchange
the roles of ˆv i and ˆv r and still get the same answer (this is sometimes called Helmholtz
reciprocity).
Most surfaces are isotropic, i.e., there are no preferred directions on the surface as far
as light transport is concerned. (The exceptions are anisotropic surfaces such as brushed
(scratched) aluminum, where the reflectance depends on the light orientation relative to the
direction of the scratches.) For an isotropic material, we can simplify the BRDF to

f r (θ i ,θ r , |φ r − φ i |; λ) or f r (ˆv i , ˆv r , ˆn; λ), (2.82)

since the quantities θ i , θ r and φ r − φ i can be computed from the directions ˆv i , ˆv r , and ˆn.
To calculate the amount of light exiting a surface point p in a direction ˆv r under a given
lighting condition, we integrate the product of the incoming light L i (ˆv i ; λ) with the BRDF
(some authors call this step a convolution). Taking into account the foreshortening factor
+
cos θ i , we obtain

+
L r (ˆv r ; λ)= L i (ˆv i ; λ)f r (ˆv i , ˆv r , ˆn; λ) cos θ i dˆv i , (2.83)
where
+
cos θ i = max(0, cos θ i ). (2.84)
If the light sources are discrete (a finite number of point light sources), we can replace the
integral with a summation,

+
L r (ˆv r ; λ)= L i (λ)f r (ˆv i , ˆv r , ˆn; λ) cos θ i . (2.85)
i
BRDFs for a given surface can be obtained through physical modeling (Torrance and
Sparrow 1967; Cook and Torrance 1982; Glassner 1995), heuristic modeling (Phong 1975), or