Intensity-Based and Polarisation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 144

Fig. 3.4 ( a ) Sketch of the three reflectance components according to Nayar et al. ( 1991 )(but
without off-specular lobe), modelled by ( 3.14 ). The reflectance components are plotted for constant
illumination direction s and varying viewing direction v .( b ) Measured reflectance of a raw forged
iron surface for α = 75 ◦

Wöhler and d'Angelo 2009 ) a phenomenological BRDF model for rough metallic
surfaces is used which is an extension of the model by Phong ( 1975 ) and has been
chosen such that the empirically determined reflectance properties of the material
are well represented. Hence, the reflectance of a typical rough metallic surface is
assumed to consist of three components according to Nayar et al. ( 1991 ): a dif-
fuse (Lambertian) component, the specular lobe, and the specular spike. The diffuse
component is generated by multiple scattering processes in the surface material.
The specular lobe, which is caused by single reflection at the surface, has a max-
imum close to the mirror direction and may be rather broad. The specular spike
describes reflections in the mirror direction, where the strength decreases strongly
already for small deviations from the mirror direction. Figure 3.4 a illustrates the
three components of the reflectance function. Notably, in the model by Nayar et
al. ( 1991 ) the maximum of the specular lobe not necessarily coincides with the
mirror direction, while such 'off-specular' components are not regarded here. For
illustration, Fig. 3.4 b shows a reflectance function measured for raw forged iron
at a phase angle of α 75 ◦ (as shown in Sect. 3.2.2 , for the materials regarded

here the reflectance function corresponds to the BRDF multiplied by cos θ i ). We
define an analytical form for the reflectance for which we perform a least-mean-
squares fit to the measured reflectance values, depending on the incidence angle θ i
and the angle θ r between the specular direction r and the viewing direction v (cf.
Fig. 3.4 a):

=

ρ 1

.

N

( cos θ r ) m n
cos θ i

f spec
N

(ρ, θ i ,θ r ,α) = + σ n · (3.14)

n = 1

For θ r > 90 ◦ only the diffuse component is considered. The reflectance measure-
ment is performed for a small part of the surface, for which the albedo ρ can be
assumed to be constant. The shapes of the specular components of the reflectance