Intensity-Based and Polarisation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 145

=

function are approximated by N 2 terms proportional to powers of cos θ r .The

coefficients σ n denote the strength of the specular components relative to the diffuse
component, while the exponents m n denote their widths. Generally, all introduced
phenomenological parameters depend on the phase angle α . The angle θ r is defined
according to ( 3.11 ), such that our phenomenological reflectance model only depends
on the incidence angle θ i , the emission angle θ e , and the phase angle α .Likethe
Phong model, the specular BRDF according to ( 3.14 ) does not fulfil the Helmholtz
reciprocity condition ( 3.7 ), but it provides a useful and numerically well-behaved
description of the reflectance behaviour of the kind of specularly reflecting rough
metallic surfaces regarded in Chap. 6 in the context of industrial quality inspec-
tion.
In the scenario of remote sensing (cf. Chap. 8 ), a very powerful physically mo-
tivated BRDF model for planetary regolith surfaces is introduced by Hapke ( 1981 ,
1984 , 1986 , 1993 , 2002 ). For three-dimensional surface reconstruction, however, it
isshownbyMcEwen( 1991 ) that a very good approximation of the true reflectance
behaviour over a wide range of incidence and emission angles and surface orienta-
tions is obtained by the phenomenological lunar-Lambert law

ρ 2 L(α) L(α)

cos θ e + 1

1
cos θ i +

f LL (ρ, θ i ,θ r ,α) = − (3.15)

with L(α) as a phase angle-dependent empirical factor determined by McEwen
( 1991 ). A more detailed discussion of the Hapke model and the lunar-Lambert
BRDF is given in Chap. 8 .

3.2.2 Determination of Surface Gradients

3.2.2.1 Photoclinometry

Photoclinometric and shape from shading techniques take into account the geomet-
ric configuration of camera, light source, and the object itself, as well as the re-
flectance properties of the surface to be reconstructed. We always assume parallel
incident light and an infinite distance between camera and object as proposed by
Horn ( 1986 , 1989 ), which is a good approximation in the application scenario of
industrial quality inspection (cf. Chap. 6 ) and nearly exactly fulfilled in the remote
sensing scenario (cf. Chap. 8 ). Under these conditions, the intensity I uv of the image
pixel located at (u, v) amounts to

I uv = R I (ρ, n , s , v ). (3.16)

According to Horn ( 1986 ), ( 3.16 ) is termed the 'image irradiance equation'. Here, ρ
is the surface albedo, n the surface normal, v the direction to the camera, s the direc-
tion of incident light, and R I the reflectance map. The reflectance map indicates the
relationship between surface orientation and brightness, based on information about
surface reflectance properties and the distribution of the light sources that illuminate