Image Processing Reference

In-Depth Information

Combining these two equations for a single triangle moving rigidly in
N
f
frames results in the

system of equations

l
1
T
l
1

d
T
d

−
2
d
T
l
1

+

=

0

.

l
N
f
T
l
N
f

d
T
d

2
d
T
l
N
f

−

+

=

0
,

(6.24)

where

⎡

⎤

⎡

⎤

⎡

⎤

p
2
−

p
1

2

2

2

q
2
−

q
1

1

−

1

−

1

⎣

⎦

⎣

⎦
,
l
j

p
3
−

p
2

⎣

⎦
.

2

d

=

q
3
−

q
2

=

2

,
=

−

11

−

1

(6.25)

2

2

p
1
−

p
3

q
1
−

q
3

2

2

−
1

−
11

Since the quadratic term in
d
is the same in all the equations, it can easily be eliminated. This yields

a linear system of equations in
d
, which can be solved in closed-form. From
d
, the 3D triangle can be

reconstructed up to a depth sign flip and a global depth ambiguity. These ambiguities are then solved

by accounting for all rigidly moving triangles in the images. The non-rigid triangles are discarded

based on their reprojection error. A strength of this approach is that it can handle topology changes, as

when the sheet of paper depicted by Fig.
6.16
is being torn in two. A potential limitation that it shares

with the
Ecker
et al.
[
2008
],
Perriollat
et al.
[
2010
],
Salzmann
et al.
[
2008a
] methods discussed in

Section
4.2.2.1
, which also rely on 3D distance constraints, is that the Euclidean distances between

triplets of surface points does not truly remain constant when the surface deforms.The approximation

is only valid when the curvature of the triangle linking them is small, which means that the points

cannot be too distant from each other.

In short, there has been a number of exciting recent advances in NRSFM that are now

departing from its early formulations
Bregler
et al.
[
2000
],
Ullman
[
1983
]. These new techniques

are starting to produce results and appear to be more robust to noise and able to handle much larger

deformations than before. This indicates that reliable solutions to this problem might be found in

spite of its complexity.

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