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Title: Shape Analysis of Functional Data with Elastic Partial Matching
Elastic Riemannian metrics have been used successfully for statistical treatments of functional and curve shape data. However, this usage suffers from a significant restriction: the function boundaries are assumed to be fixed and matched. Functional data often comes with unmatched boundaries, {\it e.g.}, in dynamical systems with variable evolution rates, such as COVID-19 infection rate curves associated with different geographical regions. Here, we develop a Riemannian framework that allows for partial matching, comparing, and clustering functions under phase variability {\it and} uncertain boundaries. We extend past work by (1) Defining a new diffeomorphism group G over the positive reals that is the semidirect product of a time-warping group and a time-scaling group; (2) Introducing a metric that is invariant to the action of G; (3) Imposing a Riemannian Lie group structure on G to allow for an efficient gradient-based optimization for elastic partial matching; and (4) Presenting a modification that, while losing the metric property, allows one to control the amount of boundary disparity in the registration. We illustrate this framework by registering and clustering shapes of COVID-19 rate curves, identifying basic patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods.
Award ID(s):
1953087 1955154
Publication Date:
Journal Name:
IEEE Transactions on Pattern Analysis and Machine Intelligence
Page Range or eLocation-ID:
1 to 1
Sponsoring Org:
National Science Foundation
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