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In-Depth Information
Fig. 3.2
Definition of the
surface normal
n
,the
illumination direction
s
,the
viewing direction
v
, the phase
angle
α
, the polar angle
θ
i
and azimuth angle
φ
i
of
incidence, and the polar angle
θ
e
and azimuth angle
φ
e
of
emission. The vectors
n
,
s
,
and
v
are generally not
coplanar, such that
α
≤
θ
i
+
θ
e
map of the surface visible in the image (cf. Sect.
3.2.2.2
). This approach is termed
shape from shading (Horn,
1986
,
1989
; Horn and Brooks,
1989
). An overview of
the variety of techniques to infer a height map from the determined surface gradients
is provided in Sect.
3.2.3
.
3.2.1 The Bidirectional Reflectance Distribution Function (BRDF)
Prior to the detailed description of intensity-based three-dimensional surface recon-
struction methods we introduce the basic radiometric quantities. The light power in-
cident on the apparent surface, measured in W m
−
2
, is termed irradiance. The light
quantity emitted into a specific direction by a surface is measured in W m
−
2
sr
−
1
and is termed radiance. The normalisation to unit solid angle is necessary due to the
generally direction-dependent emission from the surface (Horn,
1986
).
According to Horn (
1986
), the radiance received by the imaging device is de-
pendent on both the viewing direction and the direction of the incident light. The
illumination direction
s
is given by the 'zenith angle' or 'polar angle'
θ
i
and the 'az-
imuth angle'
φ
i
of incidence, while the direction
v
into which the reflected light is
emitted is denoted by the zenith angle
θ
e
and the azimuth angle
φ
e
of emission (cf.
Fig.
3.2
). At this point, the 'bidirectional reflectance distribution function' (BRDF)
f(θ
i
,φ
i
,θ
e
,φ
e
)
is introduced, which denotes the radiance
L
surf
of a surface viewed
from the direction given by
(θ
e
,φ
e
)
divided by the irradiance
E
illum
due to illumi-
nation from the direction given by
(θ
i
,φ
i
)
. Accordingly, the BRDF is defined as
dL
surf
(θ
e
,φ
e
)
dE
illum
(θ
i
,φ
i
)
=
dL
surf
(θ
e
,φ
e
)
L
illum
(θ
i
,φ
i
)
cos
θ
i
dΩ
i
f(θ
i
,φ
i
,θ
e
,φ
e
)
=
(3.6)
(Nicodemus et al.
1977
;Dorseyetal.
2008
). In (
3.6
),
dL
surf
(θ
e
,φ
e
)
denotes the
differential radiance as perceived by the detector
dE
illum
(θ
i
,φ
i
)
=
L
illum
cos
θ
i
dΩ
i
the differential irradiance received from the light source situated in the direction
given by
(θ
i
,φ
i
)
(cf. Fig.
3.3
),
L
illum
(θ
i
,φ
i
)
the radiance of the light source, and
dΩ
i
the differential solid angle under which the regarded surface element appears
from the position of the light source (Nicodemus et al.
1977
). Horn (
1986
) states
(regarding the Lambertian case) that the factor cos
θ
i
originates from geometrical
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