4 research outputs found
Numerical inversion of SRNFs for efficient elastic shape analysis of star-shaped objects.
The elastic shape analysis of surfaces has proven useful in several application areas, including medical image analysis, vision, and graphics.
This approach is based on defining new mathematical representations of parameterized surfaces, including the square root normal field (SRNF), and then using the L2 norm to compare their shapes. Past work is based on using the pullback of the L2 metric to the space of surfaces, performing statistical analysis under this induced Riemannian metric. However, if one can estimate the inverse of the SRNF mapping, even approximately, a very efficient framework results: the surfaces, represented by their SRNFs, can be efficiently analyzed using standard Euclidean tools, and only the final results need be mapped back to the surface space. Here we describe a procedure for inverting SRNF maps of star-shaped surfaces, a special case for which analytic results can be obtained. We test our method via the classification of 34 cases of ADHD (Attention Deficit Hyperactivity Disorder), plus controls, in the Detroit Fetal Alcohol and Drug Exposure Cohort study. We obtain state-of-the-art results
Numerical inversion of SRNFs for efficient elastic shape analysis of star-shaped objects
The elastic shape analysis of surfaces has proven useful in several application areas, including medical image analysis, vision, and graphics. This approach is based on defining new mathematical representations of parameterized surfaces, including the square root normal field (SRNF), and then using the L2 norm to compare their shapes. Past work is based on using the pullback of the L2 metric to the space of surfaces, performing statistical analysis under this induced Riemannian metric. However, if one can estimate the inverse of the SRNF mapping, even approximately, a very efficient framework results: the surfaces, represented by their SRNFs, can be efficiently analyzed using standard Euclidean tools, and only the final results need be mapped back to the surface space. Here we describe a procedure for inverting SRNF maps of star-shaped surfaces, a special case for which analytic results can be obtained. We test our method via the classification of 34 cases of ADHD (Attention Deficit Hyperactivity Disorder), plus controls, in the Detroit Fetal Alcohol and Drug Exposure Cohort study. We obtain state-of-the-art results
Parametrizing Product Shape Manifolds by Composite Networks
Parametrizations of data manifolds in shape spaces can be computed using the
rich toolbox of Riemannian geometry. This, however, often comes with high
computational costs, which raises the question if one can learn an efficient
neural network approximation. We show that this is indeed possible for shape
spaces with a special product structure, namely those smoothly approximable by
a direct sum of low-dimensional manifolds. Our proposed architecture leverages
this structure by separately learning approximations for the low-dimensional
factors and a subsequent combination. After developing the approach as a
general framework, we apply it to a shape space of triangular surfaces. Here,
typical examples of data manifolds are given through datasets of articulated
models and can be factorized, for example, by a Sparse Principal Geodesic
Analysis (SPGA). We demonstrate the effectiveness of our proposed approach with
experiments on synthetic data as well as manifolds extracted from data via
SPGA