Graphics Reference
In-Depth Information
1. As the regarded applications deal with relatively smooth and flat surfaces, the
initial values
p
(
0
)
uv
and
q
(
0
)
uv
in (
3.24
) are set to zero. The iteration index
m
is set
to
m
0. The surface gradients are then computed according to (
3.24
). Subse-
quently, the initial surface profile
z
(m)
=
uv
is reconstructed by numerical integration
of the obtained values of
p
uv
and
q
uv
according to Jiang and Bunke (
1997
)as
described in Sect.
3.2.3
.
2. The average depth difference
(z)
ave
shadow
based on shadow analysis is given by
S
u
(s)
e
u
(s
i
+
1
tan
μ
shadow
.
1
S
(z)
ave
shadow
=
−
(5.5)
s
=
1
u
(s)
u
(s)
i
In (
5.5
) the effective shadow length is given by
(
|
−
|+
1
)
, such that the
e
shadow length is 1 if
u
(s)
i
u
(s
e
; i.e. a single pixel along the direction of incident
light lies in the shadow. The corresponding depth difference
(z)
ave
sfs
=
given by the
shape from shading analysis is obtained by
S
z
m
u
(s)
,v
(s)
−
z
m
u
(s)
,v
(s)
.
1
S
(z)
ave
sfs
=
(5.6)
e
i
s
=
1
At the pixels marking the ridge that casts the shadow, denoted by
(u
(s
i
,v
(s)
)
or
(u
(s
e
,v
(s)
)
depending on the direction of illumination, the surface gradient
p
in
the horizontal image direction corresponds to tan
μ
shadow
. These values are kept
constant throughout the following steps.
3. Assuming that the scene is illuminated exactly in the horizontal image direc-
tion, the shape from shading analysis cannot yield reliable information about the
surface gradient
q
uv
in the vertical image direction. Especially for small illu-
mination angles
μ
, changing
q
uv
for a certain surface element does not signifi-
cantly change the angle
θ
i
between the corresponding surface normal
n
and the
direction of illumination
s
, which results in a small value of
∂R/∂q
in (
3.24
)
for Lambertian reflection. Once the initial values of
q
uv
are small, they remain
small during the iteration process according to (
3.24
). The angle
θ
i
is mainly
governed by the surface gradient
p
uv
in the horizontal image direction. Hence,
for all surface elements the angle
θ
i
=
−
θ
i
between the respective surface
element (not its surface normal) and the illumination direction
s
is multiplied by
a constant factor
c
t
such that
(z)
ave
π/
2
shadow
=
(z)
ave
sfs
. For small values of
q
uv
,the
horizontal surface gradient
p
uv
is replaced by the new value
p
uv
according to
˜
cot
μ
c
t
θ
i
and
π
2
−
π
2
+
1
p
uv
θ
i
=
p
uv
=−
˜
+
μ
±
arctan
.
(5.7)
In (
5.7
) the plus sign is used for
p
uv
<
0 and the minus sign for
p
uv
>
0. The
surface is tilted away from the light source if
c
t
<
1 and towards the light source
if
c
t
>
1, as illustrated in Fig.
5.12
b. The value of
c
t
necessary to adjust the
value of
(z)
ave
sfs
to that of
(z)
ave
shadow
is determined by means of the bisection
method. This procedure has the strong advantage that for small illumination an-
gles
μ
, small surface gradients
p
uv
, and thus small angles
θ
i
between the surface
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