Landmark-Constrained Statistical Shape Analysis of Elastic Curves and Surfaces

Abstract

We present a framework for landmark-constrained elastic shape analysis of curves and surfaces. While most of statistical shape analysis focuses on either landmark-based or curve-based representations, we describe a new approach that is able to unify them. The described method has its roots in elastic shape analysis, where elastic metrics are used for comparisons and statistical analysis of shapes. The landmark information is incorporated into the shape representation via hard constraints in the re-parameterization group. We use the square-root velocity function representation for curves and square-root normal field representation for surfaces to greatly simplify the implementation of these methods. We give multiple complex examples from graphics and computer vision, wherein the landmark-constrained shape analysis framework is able to provide natural deformations between shapes as well as representative summaries, including the average and principal directions of variability, of samples of shapes.