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58 2 Image formation

^ ^
v i•n = 1
^ ^
0 < v i•n < 1 ^ s i
^
n
^ v i v Œ
^ ^
v i•n = 0
v Œ
-v ŏ
180°
v ŏ
^ ^
v i•n < 0
(a) (b)

Figure 2.17 (a) The diminution of returned light caused by foreshortening depends on ˆv i · ˆn, the cosine of the
angle between the incident light direction ˆv i and the surface normal ˆn. (b) Mirror (specular) reflection: The
incident light ray direction ˆv i is reflected onto the specular direction ˆs i around the surface normal ˆn.

where
+
[ˆv i · ˆn] = max(0, ˆv i · ˆn). (2.88)

Specular reflection
The second major component of a typical BRDF is specular (gloss or highlight) reflection,
which depends strongly on the direction of the outgoing light. Consider light reflecting off a
mirrored surface (Figure 2.17b). Incident light rays are reflected in a direction that is rotated
by 180 around the surface normal ˆn. Using the same notation as in Equations (2.29–2.30),
◦
we can compute the specular reflection direction ˆs i as
T
ˆ s i = v − v ⊥ =(2ˆnˆn − I)v i . (2.89)

The amount of light reflected in a given direction ˆv r thus depends on the angle θ s =
cos −1 (ˆv r · ˆs i ) between the view direction ˆv r and the specular direction ˆs i . For example, the
Phong (1975) model uses a power of the cosine of the angle,

f s (θ s ; λ)= k s (λ) cos k e θ s , (2.90)

while the Torrance and Sparrow (1967) micro-facet model uses a Gaussian,
2 2
f s (θ s ; λ)= k s (λ) exp(−c θ ). (2.91)
s s
Larger exponents k e (or inverse Gaussian widths c s ) correspond to more specular surfaces
with distinct highlights, while smaller exponents better model materials with softer gloss.

Phong shading

Phong (1975) combined the diffuse and specular components of reflection with another term,
which he called the ambient illumination. This term accounts for the fact that objects are
generally illuminated not only by point light sources but also by a general diffuse illumination
corresponding to inter-reflection (e.g., the walls in a room) or distant sources, such as the