Graphics Reference
In-Depth Information
˜
and the illumination direction
s
, changing
p
uv
to
p
uv
according to (
5.7
) hardly
changes the resulting relative pixel grey values. The corresponding Lambertian
reflectance is given by
ρ
sin
(c
t
θ
i
)
≈
ρ
sin
θ
i
with
ρ
=
ρ/c
t
, resulting in a new
value
ρ
for the surface albedo but almost unchanged relative intensities through-
out the image. The new gradient
p
uv
along with the new albedo
ρ
is still a near-
π/
2, which also remains true if
R(p,q)
contains
higher order terms in cos
θ
i
. Hence, the iterative update rule (
3.24
) is expected to
converge quickly.
4. The iterative update rule (
3.24
) is initialised with
p
(
0
)
optimum configuration if
θ
i
=
p
uv
, and the iteration
procedure is started. After execution, the iteration index
m
is incremented (
m
←
m
uv
1), and the surface profile
z
(m)
+
is computed by numerical integration of
p
uv
uv
and
q
uv
.
5. Steps 2, 3, and 4 are repeated until the average change of the surface profile falls
below a user-defined threshold
Θ
z
,i.e.until
(z
(m)
z
(m
−
1
)
1
/
2
u,v
<Θ
z
.Inall
)
2
−
uv
uv
described experiments a value of
Θ
z
=
0
.
01 pixel is used.
As long as the lateral pixel resolution in metric units is undefined, the depth profile
z
uv
is computed in pixel units. However, multiplying these values by the lateral pixel
resolution on the reconstructed surface (e.g. measured in metres per pixel) readily
yields the depth profile in metric units.
5.2.2 Accounting for the Detailed Shadow Structure in the Shape
from Shading Formalism
This section describes how the detailed shadow structure rather than the average
depth difference derived from shadow analysis is incorporated into the shape from
shading formalism. The depth difference
(z)
(s)
shadow
along shadow line
s
amounts
to
shadow
=
u
(s)
−
u
(s
i
+
1
tan
μ
shadow
.
(z)
(s)
(5.8)
e
In the reconstructed profile, this depth difference is desired to match the depth dif-
ference
z
u
(s
i
,v
(s)
(5.9)
obtained by the shape from shading algorithm. Therefore, it is useful to add a
shadow-related error term to (
3.21
), leading to
z
u
(s)
e
,v
(s)
−
(z)
(s)
sfs
=
(z)
(s)
2
S
(z)
(s)
shadow
sfs
−
e
=
e
s
+
λe
i
+
ηe
z
with
e
z
=
.
(5.10)
|
u
(s)
−
u
(s)
i
|+
1
e
s
=
1
The scene is supposed to be illuminated from either the right-hand or the left-hand
side. The depth difference
(z)
(s)
sfs
can be derived from the surface gradients by
means of a discrete approximation of the total differential
dz
∂z
∂z
=
∂x
dx
+
∂y
dy
.
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