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The main drawback of this variational approach is the large number of iterations
necessary until convergence is achieved. Frankot and Chellappa (
1988
) and Sim-
chony et al. (
1990
) propose methods to enforce integrability of the gradient field
by performing a least-mean-squares adjustment of an integrable gradient field to
the measured, generally non-integrable gradient field. In the approach of Frankot
and Chellappa (
1988
), the basic step consists of computing the Fourier transform
of (
3.28
), which yields
iω
u
P
ω
u
ω
v
+
iω
v
Q
ω
u
ω
v
Z
ω
u
ω
v
=−
,
(3.33)
ω
u
+
ω
v
where
i
=
√
−
1, and
P
ω
u
ω
v
,
Q
ω
u
ω
v
, and
Z
ω
u
ω
v
are the Fourier transforms of
p
uv
,
q
uv
, and
z
uv
. The corresponding depth map
z
uv
is readily obtained by computing
the inverse Fourier transform of
Z
ω
u
ω
v
. This concept is extended by Simchony et
al. (
1990
) to solving the Poisson equation
2
z
f
with the Dirichlet boundary
condition that
z
is known on the boundary of the surface.
The Fourier-based approach is extended by Wei and Klette (
2004
) to a 'strong
integrability' error term that additionally takes into account the differences between
the second-order derivatives of
z
uv
and the first-order derivatives of
p
uv
and
q
uv
according to
∇
=
∂
2
z
∂x
2
∂p
∂x
2
∂
2
z
∂y
2
∂q
∂y
2
,
δ
u,v
e
strong
int
=
e
int
+
uv
−
+
uv
−
uv
uv
(3.34)
where
δ
is a weight factor and the 'weak integrability' error term
e
int
is given
by (
3.25
). According to Wei and Klette (
2004
), minimisation of the error term (
3.34
)
is performed based on a Fourier transform in the continuous image domain, which
yields the expression
i(ω
u
+
δω
u
)P
ω
u
ω
v
+
i(ω
v
+
δω
v
)Q
ω
u
ω
v
ω
u
+
Z
ω
u
ω
v
=−
(3.35)
ω
v
+
δ(ω
u
+
ω
v
)
for the Fourier-transformed depth map
Z
ω
u
ω
v
. Again, the height map
z
uv
is obtained
by computing the inverse Fourier transform of
Z
ω
u
ω
v
.
Agrawal et al. (
2005
) propose an algebraic approach to the reconstruction of
height from gradients which exploits the information contained in the 'curl' of
the given non-integrable vector field, denoted by
∂p/∂y
∂q/∂x
. The curl of
a vector field denotes the deviation from integrability. The method determines
a residual gradient field which is added to the measured gradients
p
uv
and
q
uv
such that an integrable gradient field results. Agrawal et al. (
2005
) show that the
method of Simchony et al. (
1990
) yields an integrable gradient field, the divergence
∂p/∂x
−
∂q/∂y
of which is identical to that of the measured, non-integrable gradi-
ent field. In principle, the residual gradient field can be obtained by solving a set of
linear equations, which is, however, ill-posed for shape from shading applications
since the curl tends to be nonzero for nearly all pixels. Hence, the number of un-
known residual gradient values exceeds the number of pixels by about a factor of
+
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