Intensity-Based and Polarisation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 142

Fig. 3.3 Irradiance dE illum
due to illumination by a light
source with radiance L illum
on a surface element covering
a solid angle element dΩ i

'foreshortening' and furthermore notes that for many surfaces the BRDF does not
depend separately on the angles φ i and φ e but only on the difference (φ e − φ i ) .
This is true for diffusely and specularly reflecting surfaces but e.g. not for surfaces
containing oriented microstructures. In the case of dependence of the BRDF on
the difference (φ e − φ i ) it is often convenient to express the BRDF in terms of

the incidence angle θ i , the emission angle θ e , and the 'phase angle' α between the
direction s of incident light and the direction v into which the reflected light is
emitted (Hapke, 1981 )(cf.Fig. 3.2 and Sect. 3.2.2.1 ).
Horn ( 1986 ) points out that the amounts of radiation energy exchanged between
two surfaces must always be identical in the state of thermal equilibrium. Otherwise,
the surface temperatures would change, leading to a deviation from thermal equi-
librium which would contradict the second law of thermodynamics. A physically
meaningful BRDF must therefore fulfil the Helmholtz reciprocity condition:
f(θ i ,φ i ,θ e ,φ e )

f(θ e ,φ e ,θ i ,φ i ). (3.7)
In this context, an important special case is the ideal Lambertian surface, which
appears equally bright from all directions while reflecting all incident light. For
such a surface, the BRDF must be constant, i.e. independent of the angles θ i , φ i , θ e ,
and φ e . To determine the value of that constant, we follow the derivation outlined
by Horn ( 1986 ). The integral of the radiance of the surface over all directions must
be equal to the total irradiance, leading to
π

=

π/ 2

f(θ i ,φ i ,θ e ,φ e )E(θ i ,φ i ) cos θ i sin θ e cos θ e dθ e dφ e = E cos θ i . (3.8)

− π 0

By taking into account that 2 sin θ e cos θ e = sin 2 θ e , it follows that πf = 1 for an

ideal Lambertian surface, corresponding to

1
π .

f Lambert (θ i ,φ i ,θ e ,φ e ) = (3.9)

Hence, the radiance L is obtained from the irradiance E 0 by L E 0 /π .

A BRDF model which is widely used in the domain of computer graphics to
represent smooth surfaces with a diffuse and a specular reflectance component is
introduced by Phong ( 1975 ). The Phong BRDF is given by the weighted sum of a
Lambertian BRDF component and a specular BRDF component according to

=

( cos θ r ) m
cos θ i

f spec

Phong (θ i ,θ r )

= σ (3.10)

with θ r as the angle between the viewing direction v and the direction into which an
ideal mirror would reflect the incident light (Dorsey et al., 2008 ). The parameter σ