A Locally Linear Method for Enforcing Temporal Smoothness in Serial Image Registration - Longitudinal Registration and Shape Modeling - page 16

Fig. 3. Synthetic cortical folding sequences

Fig. 4. Endpoints of anisotropic synthetic cortical folding sequences

dimension consisting of 20 images of dimension 128 128 each, starting from

a common reference shape with varying symmetry of the gyrification, the end-
points of which are shown in Figure 4 .

×

Accuracy and Shape of Deformations. In order to assess the accuracy of
our method, we compute the residual between the observed S t ( x ) ,t ∈ (1 ... 20)

and the images S 1 ( ˆ 1 ⓦ···ⓦˆ t ( x )) obtained by deforming S 1 to the corresponding
time-point. We compare concatenation of independent pair-wise registrations 3 to
those obtained using the proposed uniform ( 3 ) and locally adaptive ( 4 ) temporal
consistency priors.
In all synthetic experiments, nearest neighbour interpolation is used to com-
pute the deformed images S ( ˆ t ( x )). The registration error is thus given as count
of mislabeled pixels. Results for the single-sulcus and two-sulci experiments are
showninFigure 5 as box-plots over all anisotropic gyrification sequences and
summarized as averages over all experiments in Table 1 .
In both the single-sulcus (Figure 5a ) and two-sulci (Figure 5b ) experiments,
both simple temporal smoothing and the spatially adaptive prior outperform
the naive pair-wise registration. The adaptive method outperforms the uniform
prior in the first frames of the simulated sequences, whereas the performance of
the uniform prior is better overall.
We use the norm of the gradients of the Jacobi Determinant of the velocity
fields

,x ∈ C in order to assess spurious motion of the obtained
deformations. Figure 6 shows the results for the two sets of synthetic experi-
ments. Surprisingly, the spatially adaptive methods gives the smoothest overall
deformation in both cases. The effect of accounting for temporal coherence in
the serial registrations is noticeable, especially in the second half of the pairwise
sulcification experiment by the strong reduction of outliers in motion complex-
ity. The spatially uniform temporal regularization on the other hand does not
reduce the overall motion complexity.

∇ det( v t ( x ))

3 In practice, direct use of the composition of registrations computed from consecu-
tive images lead to the propagation of matching errors resulting from finite image
resolution and optimization time. We therefore initialize every registration between
S 1 ( φ 1 ◦ φ 2 ◦···◦φ t− 1 )and S t with that between S t− 1 and S t . Note that this does

not correspond to registering all S t to S 1 .