This paper introduces a new approach for the joint alignment of a large
collection of segmented images into the same system of coordinates while
estimating at the same time an optimal common coordinate system. The atlas
resulting from our group-wise alignment algorithm is obtained as the hidden
variable of an Expectation-Maximization (EM) estimation. This is achieved
by identifying the most consistent label across the collection of images at each
voxel in the common frame of coordinates.
In an iterative process, each subject is iteratively aligned with the current
probabilistic atlas until convergence of the estimated atlas is reached. Two
different transformation models are successively applied in the alignment
process: an affine transformation model and a dense non-rigid
deformation field. The metric for both transformation models is the mutual
information that is computed between the probabilistic atlas and each subject.
This metric is optimized in the affine alignment step using a gradient based
stochastic optimization (SPSA) and with a variational approach to estimate the
non-rigid atlas to subject transformations.
A first advantage of our method is that the computational cost increases
linearly with the number of subjects in the database. This method is therefore
particularly suited for a large number of subjects. Another advantage is that,
when computing the common coordinate system, the estimation algorithm identifies
weights for each subject on the basis of the typicality of the segmentation.
This makes the common coordinate system robust to outliers in the population.
Several experiments are presented in this paper to validate our atlas
construction method on a population of 80 brain images segmented into 4 labels
(background, white and gray matters and ventricles). First, the 80 subjects were
aligned using affine and dense non-rigid deformation models. The results are
visually assessed by examining how the population converges closer to
a central tendency when the deformation model allows more degrees of freedom
(from affine to dense non-rigid field). Second, the stability of the atlas
construction procedure for various sizes of population was investigated by
starting from a subset of the total population which was incrementally
augmented until the total population of 80 subjects was reached. Third, the
consistency of our group-wise reference (hidden variable of the EM algorithm)
was also compared to the choice of an arbitrary subject for a subset of 10
subjects. According to William's index, our reference choice
performed favorably. Finally, the performance of our algorithm was quantified on
a synthetic population of 10 subjects (generated using random B-Spline
transformations) using a global overlap measure for each label. We also measured
the robustness of this measure to the introduction of noisy subjects in the