Functional Representations of Images for Statistical Learning: Shapes, Surfaces, and Beyond

This is a invited talk given at ICSA-Canada Chapter 2026 Symposium.

Abstract:

In this presentation, we introduce a novel perspective on image analysis that emphasizes objects and their shapes rather than individual pixels. By moving away from pixel-based representations and analyzing images as collections of structured objects characterized by color, contour, and geometry, we develop a framework that is more interpretable, less sensitive to resolution changes, and inherently parsimonious. We first describe how contours can be represented using coordinate functions and expanded in a Fourier basis. This representation provides a principled solution to the alignment problem and enables the use of a broad range of tools from functional data analysis, including functional principal component analysis and functional regression, for shape-based inference. We then discuss extensions to more complex images containing multiple interacting objects. We illustrate the proposed framework through statistical learning tasks such as sampling, classification, and clustering on real image datasets. Finally, we extend this perspective to the representation of entire images as smooth surfaces via basis expansions. This surface-based functional representation offers a flexible and low-dimensional description of image structure, with promising applications in settings such as brain imaging, where capturing smooth spatial variation and geometric structure is essential.

You can download the slides here later.