Approximations of the diffeomorphic metric and their applications in shape learning

In neuroimaging studies based on anatomical shapes, it is well-known that the dimensionality of the shape information is much higher than the number of subjects available. A major challenge in shape analysis is to develop a dimensionality reduction approach that is able to efficiently characterize anatomical variations in a low-dimensional space. For this, there is a need to characterize shape variations among individuals for N given subjects. Therefore, one would need to calculate (2(N)) mappings between any two shapes and obtain their distance matrix. In this paper, we propose a method that reduces the computational burden to N mappings. This is made possible by making use of the first- and second-order approximations of the metric distance between two brain structural shapes in a diffeomorphic metric space. We directly derive these approximations based on the so-called conservation law of momentum, i.e., the diffeomorphic transformation acting on anatomical shapes along the geodesic is completely determined by its velocity at the origin of a fixed template. This allows for estimating morphological variation of two shapes through the first- and second-order approximations of the initial velocity in the tangent space of the diffeomorphisms at the template. We also introduce an alternative representation of these approximations through the initial momentum, i.e., a linear transformation of the initial velocity, and provide a simple computational algorithm for the matrix of the diffeomorphic metric. We employ this algorithm to compute the distance matrix of hippocampal shapes among an aging population used in a dimensionality reduction analysis, namely, ISOMAP. Our results demonstrate that the first- and second-order approximations are sufficient to characterize shape variations when compared to the diffeomorphic metric constructed through (2(N)) mappings in ISOMAP analysis.