Image Processing Reference
In-Depth Information
Figure 4.2:
Reconstruction results for a very flexible plastic sheet. In spite of the many creases, the overall
shape is correctly recovered up to small errors due to erroneous correspondences. © 2007 IEEE.
[
2005
],
Sim and Hartley
[
2006
] that expressed the minimization of the true reprojection error as a
Second Order Cone Programming (SOCP) problem
Boyd and Vandenberghe
[
2004
]. In its general
form, an SOCP can be written as
f
T
x
minimize
x
(4.3)
c
i
x
subject to
A
i
x
+
b
i
2
≤
+
d
i
,
1
≤
i
≤
m,
where
f
is the vector that defines the objective function,
A
i
is a matrix,
b
i
and
c
i
are vectors, and
d
i
is a scalar. Problems of this type are convex and, thus, have a unique minimum that can be found
very effectively using available packages such as SeDuMi
Sturm
[
1999
]. Furthermore, SOCP can
be used to formulate problems more general than linear programming, quadratic programming and
quadratically-constrained quadratic programming.
For the specific case of deformable surface reconstruction, minimizing the reprojection error
can be expressed as
minimize
γ,
x
γ
(4.4)
subject to
(
P
1
−
u
i
P
3
)
h
i
,(
P
2
−
v
i
P
3
)
h
i
2
≤
γ
P
3
˜
q
i
,
1
≤
i
≤
N
c
,
q
i
=
q
i
,
1
T
where
P
k
contains the
k
th
line of the projection matrix, and
˜
is the vector of ho-
mogeneous coordinates of the 3D point matching the
i
th
q
i
is obtained from the
vertex coordinates
x
and barycentric coordinates.
γ
is an additional slack variable that encodes the
maximum reprojection error for all feature points.
While a solution of the above problem minimizes the true reprojection error, it still is under-
constrained. This is why temporal consistency was introduced in
Salzmann
et al.
[
2007a
], to prevent
the orientation of mesh edges from varying excessively from one frame to the next, as illustrated in
Fig.
4.3
. This can be expressed as additional SOCP constraints of the form
feature point.
˜
2
≤
λl
i,j
,
v
t
+
1
i
v
j
−
v
i
v
t
+
1
j
−
+
l
i,j
(4.5)
v
j
−
v
i
2
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