Bayesian Parameter Estimation and Segmentation in the Multi-Atlas Random Orbit Model

<div><p>This paper examines the multiple atlas random diffeomorphic orbit model in Computational Anatomy (CA) for parameter estimation and segmentation of subcortical and ventricular neuroanatomy in magnetic resonance imagery. We assume that there exist multiple magnetic resonance image (MRI) atlases, each atlas containing a collection of locally-defined charts in the brain generated via manual delineation of the structures of interest. We focus on maximum a posteriori estimation of high dimensional segmentations of MR within the class of generative models representing the observed MRI as a conditionally Gaussian random field, conditioned on the atlas charts and the diffeomorphic change of coordinates of each chart that generates it. The charts and their diffeomorphic correspondences are unknown and viewed as latent or hidden variables. We demonstrate that the expectation-maximization (EM) algorithm arises naturally, yielding the likelihood-fusion equation which the a posteriori estimator of the segmentation labels maximizes. The likelihoods being fused are modeled as conditionally Gaussian random fields with mean fields a function of each atlas chart under its diffeomorphic change of coordinates onto the target. The conditional-mean in the EM algorithm specifies the convex weights with which the chart-specific likelihoods are fused. The multiple atlases with the associated convex weights imply that the posterior distribution is a multi-modal representation of the measured MRI. Segmentation results for subcortical and ventricular structures of subjects, within populations of demented subjects, are demonstrated, including the use of multiple atlases across multiple diseased groups.</p></div

Example of subcortical segmentations from single- and multi-atlas LDDMM approaches.

<p>Panel <b>A</b> shows the automated segmentation results of two subjects using single-atlas LDDMM, while panel <b>B</b> shows the segmentation results for the same subjects using multi-atlas LDDMM approach.</p

Example slices for a comparison of multi-atlas LDDMM, STAPLE, and Spatial STAPLE.

<p>Three representative 2-D slices of three structures near medial temporal regions – the amygdala, the hippocampus, and the ventricle in both hemispheres obtained respectively from manual delineation (top row), likelihood-fusion via multi-atlas LDDMM (2nd row), STAPLE (3rd row), and Spatial STAPLE (bottom row).</p

The likelihood contribution of each atlas averaged over the atlas structures.

<p>The quantity was obtained from the 28 different atlases (columns) for each structure (row).</p

Depiction of the variability within different single atlases.

<p>Scatterplot of Dice overlaps of automated segmentations of sixteen different structures of one subject from 6 different atlases using single atlas LDDMM.</p

Depiction of two charts and the associated diffeomorphisms chosen to illustrate the interpretation.

<p>The charts are related via diffeomorphic coordinate transformations as depicted in the figure, in which points X, Y in the hippocampus chart and the amygdala chart are compared using the forward and inverse mappings. In our paper the charts are manually delineated structures including the amygdala, caudate, hippocampus, putamen, thalamus, lateral ventricle, the 3rd ventricle, and the 4th ventricle.</p

Mean and standard deviations of Dice overlaps computed across the fifteen subjects in the second group for each structure for comparisons of STAPLE, Spatial STAPLE, and likelihood-fusion via multi-atlas LDDMM.

<p>Bold typesetting indicates that the Dice overlap ratio obtained from the corresponding is statistically significant higher than that of other methods (<i>p<0.05</i>).</p

The average Dice overlaps between manual volume and the automated volume measured over the fourteen datasets of the first group for each structure for comparisons of STAPLE, Spatial STAPLE, and likelihood-fusion via multi-atlas LDDMM.

<p>Bold typesetting indicates that the Dice overlap ratio obtained from the corresponding is statistically significant higher than that of other methods (<i>p<0.05</i>).</p

A comparison of segmentation accuracy between single-atlas LDDMM and multi-atlas LDDMM.

<p>Panels <b>A</b>, <b>B</b>, <b>C</b> respectively show the mean Dice overlaps and the standard deviations of the sixteen structures obtained from single-atlas LDDMM (red) and likelihood-fusion via multi-atlas LDDMM (green) for the three different groups.</p

The convex weighting function normalized over each structure.

<p>For each structure, we color-coded the quantity that is depicted for each atlas (column) and each of the 16 structures (rows).</p