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

${\mathbb{L}^2}$

norm to compare their shapes. Past work is based on using the pullback of the

${\mathbb{L}^2}$

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.