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

Search WWH ::

Custom Search