Nonparametric Estimation of Means on Hilbert Manifolds and Extrinsic Analysis of Mean Shapes of Contours

TL;DR

This paper develops a nonparametric extrinsic approach for estimating means on Hilbert manifolds, with applications to shape analysis of planar contours in digital images, including hypothesis testing and computational comparisons.

Contribution

It introduces a novel extrinsic mean estimation method on Hilbert manifolds for shape analysis, with a focus on digital image contours, and provides computational cost comparisons.

Findings

Proposed a one-sample hypothesis test for means on Hilbert manifolds.

Defined extrinsic mean shapes of planar contours via embedding into Hilbert-Schmidt operators.

Compared computational efficiency with existing shape analysis methods.

Abstract

Motivated by the problem of nonparametric inference in high level digital image analysis, we introduce a general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample spaces. To perform inference on these means, we appeal to the concept of neighborhood hypotheses from functional data analysis and derive a one-sample test. We then consider analysis of shapes of contours lying in the plane. By embedding the corresponding sample space of such shapes, which is a Hilbert manifold, into a space of Hilbert-Schmidt operators, we can define extrinsic mean shapes of planar contours and their sample analogues. We apply the general methods to this problem while considering the computational restrictions faced when utilizing digital imaging data. Comparisons of computational cost are provided to another method for analyzing shapes of contours.