We develop a multi-scale model of shape based on a volumetric representation of solids in 3D space. A signed energy function (SEF) derived from the model is designed to quantify the magnitude of regional shape changes that correlate well with local shrinkage and expansion. The methodology is applied to the analysis of longitudinal morphological data representing hippocampal volumes extracted from one-year repeat magnetic resonance scans of the brain of 381 subjects collected by the Alzheimer's

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... isease Neuroimaging Initiative. We first establish a strong correlation between the SEFs and hippocampal volume loss over a one-year period and then use SEFs to characterize specific regions where hippocampal atrophy over the one-year period differ significantly among groups of normal controls and subjects with mild cognitive impairment and Alzheimer's disease.