Elastic analysis of irregularly or sparsely sampled curves.

Area of Science:

• Statistics
• Computer Vision
• Data Analysis

Background:

• Analyzing curve data, such as movement paths or handwritten letters, often requires alignment (registration) to compare shapes irrespective of their parameterization.
• Existing methods for parameterization-invariant elastic distance analysis have limitations with real-world data that is irregularly or sparsely sampled.
• The square-root-velocity framework is a common approach but struggles with noisy or incomplete curve observations.

Purpose of the Study:

• To develop robust statistical methods for analyzing samples of curves in multiple dimensions, focusing on shape comparison.
• To enable accurate classification and clustering of curve data, particularly when observations are irregular or sparse.
• To provide a framework for computing smooth means and distances between curves that are invariant to parameterization.

Main Methods:

• Utilizing spline curves to model smooth or polygonal Fréchet means of open or closed curves with respect to elastic distance.
• Developing algorithms to approximate elastic distance for irregularly or sparsely observed curves by treating them as polygons.
• Demonstrating the identifiability of the spline model modulo parameterization.

Main Results:

• Successfully classified spirals from Parkinson's patients versus healthy controls using elastic distance to a mean spiral.
• Clustered sparsely sampled GPS tracks and computed smooth cluster means to identify new paths.
• Validated the proposed methods through simulations and implementation in the R-package "elasdics".

Conclusions:

• The proposed spline-based methods offer a robust and flexible approach for analyzing parameterization-invariant curve data, even with irregular or sparse sampling.
• These methods effectively handle real-world datasets, enabling meaningful classification and clustering of complex curve shapes.
• The R-package "elasdics" provides accessible tools for researchers to apply these advanced statistical techniques.