PerformingNon-Rigid Structure fromMotion - Deformable Surface 3D Reconstruction from Monocular Images - page 64

6.2.1 FROMBASIS SHAPESTOBASISTRAJECTORIES
As mentioned in Chapter 5 , while most NRSFM approaches represent the shape with a linear
subspace models, some recent works have proposed different formulations. Among them, the method
of Akhter et al. [ 2008 ] introduced an alternative approach to enforcing temporal consistency by
formulating NRSFM in trajectory space. In other words, instead of reconstructing the whole shape
at each time instant, the trajectory over the whole sequence of each 3D point is estimated.
To this end, the usual shape basis is replaced by a trajectory basis, as shown in Fig. 6.4 .
More specifically, the x −

T

, y − , and z − trajectories of a 3D point q i =[ x i ,y i ,z i ] in N f frames

=[ x i , ··· ,x N f =[ y i , ··· ,y N f =[ z i , ··· ,z N f

T , t i T , and t i

are defined as t i

T , respectively.
Assuming that these trajectories can be described as a linear combination of N t basis trajectories θ k ,
this yields

] ] ]

i i i

N t N t N t

a i,k θ k , t i a i,k θ k , t i a i,k θ k ,

t i

= = = (6.12)

k = 1 k = 1 k = 1

where a i,k , a i,k , and a i,k are the x − coefficients for point i and basis trajectory k , and θ k

is an N f -dimensional vector. Given this formulation, NRSFM can be re-written as the factorization
problem

, y − , and z −

⎡ ⎤

a 1 , 1 a N c , 1

···

⎡ ⎤ ⎣ . . . ⎦

β 1

⎣ ⎦

a 1 ,N t a N c ,N t

β 1

···

⎡ ⎤

a 1 , 1 a N c , 1

β 1

···

R 1

. . . .

⎣ ⎦

W = ... , (6.13)

R N f

a 1 ,N t a N c ,N t β N f

···

β N f a 1 , 1 a N c , 1 ···

β N f

. . .

a 1 ,N t a N c ,N t

···

θ 1 , ,θ N t ] contains the j th element of all θ k , and W is the same measurement

matrix as before. W is then factorized into and , and the resulting corrective transform is
estimated by ensuring that the rotation matrices are orthonormal. In practice, the Akhter et al.
[ 2008 ] method assumes that the basis trajectories θ k are known and can be generated from the
Discrete Cosine Transform. While this might seem restrictive, it was shown to generalize to many
different trajectories. The results are compared to those of Torresani et al. [ 2008 ] and Xiao et al.
[ 2004b ]inFig. 6.5 . Note that the reconstructions of Akhter et al. [ 2008 ] correspond more closely
to what is depicted by the images.

where β j =[ ···