Intensity-Based and Polarisation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 165

Fig. 3.5 Definition of the
goniometer angles γ 1 and γ 2

over the image. It is straightforward to determine the surface normal n , the inci-
dence angle θ i , and the emission angle θ e from the goniometer angles γ 1 and γ 2
and the vectors s and v . For each configuration of goniometer angles, five images
are acquired through a linear polarisation filter at orientation angles ω of 0 ◦ ,45 ◦ ,
90 ◦ , 135 ◦ , and 180 ◦ . Due to the encountered wide range of reflected intensities, a
high dynamic range image is synthesised from four low dynamic range images ac-
quired with different exposure times. For each filter orientation ω , an average pixel
pixel grey value over an image area containing a flat part of the sample surface is
computed. A sinusoidal function of the form ( 3.52 ) is then fitted to the measured
pixel pixel grey values. The filter orientation Φ for which maximum intensity is ob-
served corresponds to the polarisation angle, and the polarisation degree is readily
obtained from the sinusoidal fit according to D p = I v /I c (cf. Sect. 3.4.1 ). In prin-

ciple, three measurements would be sufficient to determine the three parameters I c ,
I v , and Φ , but the fit becomes less noise-sensitive when more measurements are
used.
As is apparent from the previous discussion, no accurate physically motivated
model for the polarisation properties of rough metallic surfaces is available. Hence,
d'Angelo and Wöhler ( 2005a ) fit a phenomenological model, here a polynomial
in terms of the goniometer angles γ 1 and γ 2 , to the measured values of the polar-
isation angle and degree. The polarisation angle is represented by an incomplete
third-degree polynomial of the form

d Φ γ 1 γ 2 + e Φ γ 2 , R Φ (γ 1 ,γ 2 ) = a Φ + b Φ γ 2 + c Φ γ 1 γ 2 + (3.60)

which is antisymmetric in γ 2 , and R Φ (γ 1 , 0 ) const, corresponding to copla-

nar vectors n , s , and v . In an analogous manner, the polarisation degree is repre-
sented by an incomplete polynomial of the form

= a Φ =