Analysis of shape data: From landmarks to elastic curves

Abstract

Proliferation of high-resolution imaging data in recent years has led to substantial improvements in the two popular approaches for analyzing shapes of data objects based on landmarks and/or continuous curves. We provide an expository account of elastic shape analysis of parametric planar curves representing shapes of two-dimensional (2D) objects by discussing its differences, and its commonalities, to the landmark-based approach. Particular attention is accorded to the role of reparameterization of a curve, which in addition to rotation, scaling and translation, represents an important shape-preserving transformation of a curve. The transition to the curve-based approach moves the mathematical setting of shape analysis from finite-dimensional non-Euclidean spaces to infinite-dimensional ones. We discuss some of the challenges associated with the infinite-dimensionality of the shape space, and illustrate the use of geometry-based methods in the computation of intrinsic statistical summaries and in the definition of statistical models on a 2D imaging dataset consisting of mouse vertebrae. We conclude with an overview of the current state-of-the-art in the field.

Using a Riemannian elastic metric for statistical analysis of tumor cell shape heterogeneity

“…The elastic metric has been the object of several theoretical and computational developments that primarily focused on curves (for a recent overview, see [4]), with applications to various kinds of biological shapes, including tumor images from MRI (but no cell shapes), plant leafs, or protein backbones [5,6,23]. More recent studies have applied the elastic metric to cell shapes [15,7,17,11] for classification [15,7], dimensionality reduction [11] and regression with metric learning [17].…”