Construction of a 4D Brain Atlas and Growth Model Using Diffeomorphic Registration - Longitudinal Registration and Shape Modeling - page 25

other applications, such as structural segmentation and morphometry studies,
templates with a greater level of anatomical detail are typically required.
A spatio-temporal atlas of high level of detail was created by Serag et al. [ 18 ]
using pairwise free-form deformations (FFDs) [ 17 ] and kernel regression with
time as dependent variable. The individual brain images are therefore mapped
into the atlas space using the inverses of time-dependent average transforma-
tions which were computed in [ 18 ] based on the arithmetic mean of pairwise
FFDs. This limits the atlas construction to small deformations between images
as otherwise invertibility is not guaranteed.
A natural choice of average to represent mean morphology is given by the
exponential map of the arithmetic mean of stationary velocity fields. The veloc-
ity fields are the generating elements of the Lie algebra [ 2 ] corresponding to the
one-parameter subgroup of diffeomorphisms. We therefore propose an alternative
approach to [ 18 ] based on a FFD model parameterized by a stationary veloc-
ity field that generates transformations with guaranteed invertibility. Our atlas
construction is related to the kernel based shape regression proposed by Davis
et al. [ 6 ] in that we also use a kernel method to regress a spatio-temporal tem-
plate from cross-sectional images and use a diffeomorphic registration. Davis et
al. utilize a groupwise template estimation [ 8 ] that minimizes a single objective
function to find both the template image and the transformations which relate
the individual to this mean image based on the sum of squared differences (SSD).
In contrast, we first obtain the transformations which map each anatomy into
a common atlas space and then compute the template image. This allows us to
use different (dis-)similarity measures for the decoupled optimization problems.
In particular, to deal with the wide MR intensity variations associated with
myelination and other processes during early brain development, we compute
all pairwise inter-subject transformations using an ecient diffeomorphic regis-
tration based on normalized mutual information (NMI). Given the one-to-one
correspondences between the anatomies of different subjects of similar ages, we
then estimate a mean image. We estimate the mean image such that it minimizes
the SSD of the observations in the coordinate system, which requires the least
residual deformation to explain the anatomical variability across all individuals.
Previous neonatal atlas construction methods [ 9 , 18 ] focused on the creation
of age-specific mean brain templates with corresponding tissue probability maps.
While these methods allow the generation of mean images at high temporal reso-
lution, the resulting atlas only consists of cross-sections of the growth process. A
deformable transformation model, which encodes the longitudinal changes that
occur during a given time interval, would enable the analysis of the biological
processes that underlie these changes based on the deformations.
Lately, generative models for the study of time series data and spatio-temporal
atlas building based on an extension of the large deformation diffeomorphic metric
mapping (LDDMM) registration have been proposed [ 15 , 19 ]. These models param-
eterize a time series of generally adult brain images by an initial image and ini-
tial momentum. Due to the significant age-related intensity variations, the time
series of neonatal mean brain images cannot be represented by a single deformed