Harmonic Embeddings for Linear Shape Analysis

Abstract

We present a novel representation of shape for closed contours in ?2 or for compact surfaces in ?3 explicitly designed to possess a linear structure. This greatly simplifies linear operations such as averaging, principal component analysis or differentiation in the space of shapes when compared to more common embedding choices such as the signed distance representation linked to the nonlinear Eikonal equation. The specific choice of implicit linear representation explored in this article is the class of harmonic functions over an annulus containing the contour. The idea is to represent the contour as closely as possible by the zero level set of a harmonic function, thereby linking our representation to the linear Laplace equation. We note that this is a local represenation within the space of closed curves as such harmonic functions can generally be defined only over a neighborhood of the embedded curve. We also make no claim that this is the only choice or even the optimal choice within the class of possible linear implicit representations. Instead, our intent is to show how linear analysis of shape is greatly simplified (and sensible) when such a linear representation is employed in hopes to inspire new ideas and additional research into this type of linear implicit representations for curves. We conclude by showing an application for which our particular choice of harmonic representation is ideally suited.