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Rouy and Tourin (
1992
) propose a numerical solution for this formulation of
the shape from shading problem. More recently, Kimmel and Sethian (
2001
)have
extended the shape from shading approach based on an eikonal equation to the
more general scenario of a Lambertian surface observed under oblique illumi-
nation. They introduce a numerical approximation of an equation of the form
∇
z(x,y)
=
f(x,y)
according to
max
D
−
x
uv
z,
0
2
+
max
D
−
y
uv
z,
0
2
D
+
y
D
+
x
f
uv
,
uv
z,
−
uv
z,
−
=
(3.38)
where
z
uv
=
z(uΔx,vΔy)
is the discretised version of
z(x,y)
defined on the pixel
raster,
D
−
x
uv
z
=
(z
uv
−
z
u
−
1
,v
)/Δx
,
D
+
x
uv
z
=
(z
u
+
1
,v
−
z
uv
)/Δx
, and the same anal-
ogously for the direction parallel to the vertical image axis (
y
direction). Kimmel
and Sethian (
2001
) state the proof by Rouy and Tourin (
1992
) that the shape from
shading problem can be solved based on this numerical scheme, and they empha-
sise that while the solution
z
is built up, small
z
values have an influence on larger
z
values but not vice versa. Hence, Kimmel and Sethian (
2001
) suggest to initialise
all
z
values to infinity except at the local minima, where they are initialised with the
correct depth value. If only a single minimum point exists, the surface is initialised
with
z
uv
=∞
except for the minimum. If the absolute value of the minimum is
unknown (which is commonly the case), it is set to zero. According to Kimmel and
Sethian (
2001
), an update step for
z
uv
then corresponds to
z
1
=
min
(z
u
−
1
,v
,z
u
+
1
,v
),
z
2
=
min
(z
u,v
−
1
,z
u,v
+
1
)
2
(z
1
+
z
2
+
2
f
uv
−
(3.39)
z
2
)
2
)
(z
1
−
if
|
z
1
−
z
2
|
<f
uv
z
uv
=
min
(z
1
,z
2
)
+
f
uv
otherwise
.
Kimmel and Sethian (
2001
) state that the computational complexity of this scheme
is between
(N
2
)
depending on the surface, where
N
is the number of
pixels. Furthermore, they utilise the fast marching method by Sethian (
1999
), which
relies on the principle that the solution for
z
is built up from small values towards
large values ('upwinding') and on an efficient scheme for taking into account the
updated
z
values. This approach yields a computational complexity of better than
O
O
(N)
and
O
(N
log
N)
. The proposed scheme immediately yields the shape from shading so-
lution in the zero phase angle case.
In the more general scenario of a Lambertian surface illuminated by a light source
located in a direction which is not identical to that of the camera, the reflectance map
is given by the scalar product
I(x,y)
=
·
s
n
, where the albedo corresponds to unity,
and
1. Kimmel and Sethian (
2001
) assume that the image is rotated
such that the value of
s
y
is set to zero. However, the described numerical solution
scheme for the eikonal equation cannot be used directly to reconstruct the surface.
Hence, they transform the observed image into the coordinate system of the light
source, leading to the expression
s
=
n
=
1
p
2
q
2
˜
+˜
=
x,y)
2
−
1
(3.40)
I(
˜
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