Singular geodesic coordinates for representing diffeomorphic maps in computational anatomy, with application to the morphometry of early Alzheimer's disease in the medial temporal lobe

Abstract

In this work we develop novel algorithms for building one to one correspondences between anatomical forms by providing a sparse representation of dense registration information. These sparse parameterizations of complex high dimensional data allow robustness in the face of noise and anomalies, and a platform for inference that is effective in the face of multiple comparisons. We review background in the theory of generating smooth, invertible transformations (the diffeomorphism group), and build our parameterization as a function supported on surfaces bounding anatomical structures of interest. We show how dimensionality can be reduced even further and still provide a rich family of mappings using principal component analysis or Laplace Beltrami eigenfunctions supported on the surface. We develop algorithms for surface matching and image matching within this model, and demonstrate the desired robustness by working with published large neuroimaging datasets that include many low quality examples. Finally we turn to addressing challenges associated with some specific data types: images with multiple labels, and longitudinal data. We use the mapping tools developed to draw conclusions about the progression of early Alzheimer's disease in the medial temporal lobe