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K/
2
p
u
+
k,v
+
l
u
0
,v
0
=−
F
σ
DEM
(k, l)
∂z
DEM
∂x
L/
2
∂e
DEM
∂p
l
−
u
+
k,v
+
uv
k
=−
K/
2
l
=−
L/
2
K/
2
2
u
0
,v
0
L/
2
F
σ
DEM
(k, l)
∂p
u
+
k,v
+
l
∂p
uv
×
(5.39)
k
=−
K/
2
l
=−
L/
2
for the pixel at
(u
0
,v
0
)
. Equation (
5.39
) corresponds to a correlation operation. The
terms of the sum over
u
and
v
in (
5.39
) are zero except for
u
0
=
u
+
k
and
v
0
=
v
+
l
.
Omitting the zero terms in (
5.39
) yields
K/
2
u
0
,v
0
=−
K/
2
L/
2
L/
2
∂e
DEM
∂p
F
σ
DEM
(
−
u
0
,
−
v
0
)
F
σ
DEM
(k, l)
i
=−
K/
2
j
=−
L/
2
k
=−
K/
2
l
=−
L/
2
p
u
0
+
k
+
i,v
0
+
l
+
j
2
.
∂z
DEM
∂x
×
j
−
(5.40)
u
0
+
k
+
i,v
0
+
l
+
In an analogous manner, it follows that
v
0
)
K/
2
u
0
,v
0
=−
K/
2
L/
2
L/
2
∂e
DEM
∂q
F
σ
DEM
(
−
u
0
,
−
F
σ
DEM
(k, l)
i
=−
K/
2
j
=−
L/
2
k
=−
K/
2
l
=−
L/
2
q
u
0
+
k
+
i,v
0
+
l
+
j
2
.
∂z
DEM
∂y
×
j
−
(5.41)
u
0
+
k
+
i,v
0
+
l
+
Hence, the terms
∂e
DEM
/∂p
and
∂e
DEM
/∂q
can be determined based on a subse-
quent correlation and convolution operation as long as the function
f
σ
DEM
is a linear
filter, allowing a computationally efficient implementation of the iterative update
rule (
5.38
).
The described algorithm is not only able to perform a three-dimensional surface
reconstruction but also to estimate the non-uniform albedo map
ρ
uv
,evenifonlya
single image is available. For this purpose, Grumpe et al. (
2011
) propose an embed-
ding of (
5.38
) into an iterative scheme as follows.
1. Initialise the iteration index
m
=
0 and set the surface
z
uv
(
0
)
to the initial surface,
z
DEM
z
PHCL
e.g.
z
uv
(
0
)
=
or
z
uv
(
0
)
=
(see below).
uv
uv
2. Determine the incidence angle
θ
(m)
i
and the emission angle
θ
(m)
for each pixel
e
based on
z
uv
(m)
and the corresponding surface gradients.
3. Compute the non-uniform surface albedo
ρ
(m)
by determining a pixel-wise solu-
uv
tion of
R
uv
(ρ
(m)
uv
,θ
(m)
,θ
(m)
,α)
=
I
uv
with respect to
ρ
(m)
uv
.
e
i
4. Set the albedo of the next iteration to
ρ
(m
+
1
)
ρ
(m)
=
g
σ
(m)
ρ
∗
uv
, where
g
σ
(m)
ρ
is a
uv
Gaussian low-pass filter of half width
σ
(m)
' denotes a convolution.
5. Apply the iterative update rule according to (
5.38
) in order to determine the sur-
face
z
uv
(m
and '
∗
ρ
+
1
)
.
6. Set
σ
(m
+
1
)
to a value smaller than
σ
(m)
.
ρ
ρ
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