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to substantial morphological differences during short periods of time, partially
homogeneous image regions, and smooth surfaces that lack gradient information.
This limits the applicability of established motion estimation methods [ 5 ]and
uniform priors during regularization. In this paper we propose a locally linear
regularization that allows for spatial weighting of the temporal smoothing of
resulting deformations.
The availability of a number of consecutive observations enables various solu-
tions to the aperture problem. The rich mathematical formalism of diffeomor-
phism groups [ 6 ] provides a basis for many of these. For example, a theory of
linear least-squares geodesic regression using the initial-momentum representa-
tion of diffeomorphisms [ 7 ] has been proposed in [ 8 - 10 ]. The authors in [ 11 ]
use a vector momenta formulation to the same aim, while optimization of an
acceleration-based model has been proposed in [ 12 ] in order to obtain smooth
deformations between consecutive observations. The time-varying velocity field
representation of diffeomorphisms [ 13 ] has been used for higher-order formula-
tions such as spline interpolation [ 14 ] and kernel regression [ 15 , 16 ]. Adding a
temporal component to the registration problem by smoothing the deformations
to an explicit [ 17 , 18 ] or implicit [ 19 ] common reference frame has also been
proposed. Due to their ecient computation and useful mathematical proper-
ties, diffeomorphisms generated by Stationary Velocity Fields (SVF) [ 20 ]are
commonly used in this setting. With the notable exception of [ 21 ], previously
proposed regression and smoothing approaches rely on defining a common ref-
erence space for their operation. This can become increasingly di cult in the
presence of large deformations, where regularization necessary to render the reg-
istrations tractable further shapes the resulting deformation fields. Also, the
assumption of linearity can prove too strong in these cases. While theoretically
established in a general sense [ 22 ], computation of higher-order models has so
far only been applied to lower-dimensional shape representations such currents
[ 23 ], which require a-priori segmentation of the structures of interst.
In [ 24 ], Lorenzi et al. proposed a method for estimating smooth longitudi-
nal deformations using regression of pair-wise registrations encoded using SVFs.
This requires a common reference space for all deformations and thus registra-
tion of a baseline to all later time-points. Instead, we propose to only consider a
local temporal neighborhood for the temporal smoothing of the resulting defor-
mations. This also enables us to increase the performance of our method by
defining a local weighting of the temporal smoothing.
We briefly review the foundations of image registration in the log-euclidean
framework for diffeomorphisms [ 20 ] in Section 2 . We then show how its mathe-
matical properties can be used to construct a spatially adaptive smoothing prior
for serial image registration. We evaluate our method on a simplified synthetic
model of cortical folding and a publicly available data set of fetal brain develop-
ment [ 25 ] in Section 3 .
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