Image Processing Reference
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Figure 6.9: While standard NRSFM approaches assume that the shapes lie on a linear subspace, the
true manifold can be nonlinear. This manifold can be better approximated by locally smooth manifold
learning Rabaud and Belongie [ 2008 ]. Courtesy of V. Rabaud. © 2008 IEEE.
explicitly computed, which removes them from the variables to optimize. This is similar in spirit to
the formulation of Section for local deformation models, where the coefficients were directly
obtained from the mesh vertices. By assuming Gaussian noise over the measurements and over the
shape, the distribution over the measurements is also Gaussian. In this framework, NRSFM can be
formulated as maximizing the joint likelihood of the image measurements whose negative logarithm
can be written as
N f
w j
q T E j VV T
+ σ m I E j T
+ σ 2 I w j
E j
E j
L =
j = 1
N f
+ σ 2 I + N c N f ln ( 2 π) ,
E j VV T
+ σ m I E j T
j = 1
where w j is the vector containing the two rows of W associated to frame j , E j replicates d j R j
across the diagonal, with d j the scalar accounting for depth in frame j , V is the matrix whose k th
column contains the vectorized basis shape S k , and
q contains the vectorized mean shape. σ m and
σ are the Gaussian noise variance of the shape and of the measurements, respectively. This negative
log likelihood is minimized via an EM procedure, whose initialization is obtained using a rigid
structure from motion technique. A comparison of the results of the different algorithms proposed
in Torresani et al. [ 2008 ] and of other techniques on the shark data is shown in Fig. 6.7 . Fig. 6.8
depicts the robustness to the number of basis shapes of the same algorithms. As before, the error
is defined as the ratio between the 3D distance to ground-truth and the span of the true shape.
Note that the error obtained by the EM procedure of Torresani et al. [ 2008 ] is relatively stable with
respect to the number of basis shapes.
All the above-mentioned algorithms still rely on a linear subspace model to represent the
deformations of the object of interest. In practice, this only applies to relatively simple deformations,
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