Data-Driven, Soft Alignment of Functional Data Using Shapes and Landmarks

TL;DR

This paper introduces a soft alignment method for functional data that incorporates landmark information, improving upon existing Fisher-Rao based approaches by reducing over-alignment and effectively combining shape and landmark data.

Contribution

It extends the Fisher-Rao and SRVF framework to include landmarks, enabling a balanced alignment that considers both functions and landmarks without requiring exact overlay.

Findings

The method reduces over-alignment of noise.

It effectively incorporates landmark information.

Demonstrates superior performance in practical scenarios.

Abstract

Alignment or registration of functions is a fundamental problem in statistical analysis of functions and shapes. While there are several approaches available, a more recent approach based on Fisher-Rao metric and square-root velocity functions (SRVFs) has been shown to have good performance. However, this SRVF method has two limitations: (1) it is susceptible to over alignment, i.e., alignment of noise as well as the signal, and (2) in case there is additional information in form of landmarks, the original formulation does not prescribe a way to incorporate that information. In this paper we propose an extension that allows for incorporation of landmark information to seek a compromise between matching curves and landmarks. This results in a soft landmark alignment that pushes landmarks closer, without requiring their exact overlays to finds a compromise between contributions from functions and landmarks. The proposed method is demonstrated to be superior in certain practical scenarios.