PerformingNon-Rigid Structure fromMotion - Deformable Surface 3D Reconstruction from Monocular Images - page 78

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.