There have been various developments in shape analysis in the last decade. We describe here some relationships of shape analysis with directional statistics. For shape, rotations are to be integrated out or to be optimized over whilst they are the basis for directional statistics. However, various concepts are connected. In particular, certain distributions of directional statistics have emerge...

We introduce an (equi-)affine invariant geometric structure by which surfaces that go through squeeze and shear transformations can still be properly analyzed. The definition of an affine invariant metric enables us to evaluate a new form of geodesic distances and to construct an invariant Laplacian from which local and global diffusion geometry is constructed. Applications of the proposed fram...

This tutorial covers a variety of methods for 3D shape matching and retrieval that are characterized by the use of a real-valued function defined on the shape (mapping function) to derive its signature. The methods are discussed following an abstract conceptual framework that distinguishes among the three main components of these class of shape matching methods: shape analysis, via the applicat...

We present a novel, log-radius profile representation for convex curves and define a new operation for combining the shape features of curves. Unlike the standard, angle profile-based methods, this operation accurately combines the shape features in a visually intuitive manner. This method have implications in shape analysis as well as in investigating how the brain perceives and generates curv...

We propose two differential geometric representations of planar shapes using: (i) direction functions and (ii) curvature functions, of their boundaries. Under either representation, planar shapes are treated as elements of infinite-dimensional shape spaces. Pairwise differences between the shapes are quantified using the lengths of geodesics connecting them on the shape spaces. We specify the g...

An af®ne-invariant (AI) deformable contour model for object matching, called AI-EigenSnake (AI-ES), is proposed in the Bayesian framework. In AI-ES, the prior distribution of object shapes is estimated from the sample data. This distribution is then used to constrain the prototype contour, which is dynamically adjustable in the matching process. In this way, large shape deformations due to the ...

The ubiquity of the Laplace-Beltrami operator in shape analysis can be seen by observing the wide variety of applications where it has been found to be useful. Here we demonstrate a small subset of such uses with their latest developments including a scale invariant transform for general triangulated meshes, an effective and efficient method for denoising meshes using Beltrami flows via high di...

This paper concerns the abstraction of sets of functions for use in abstract interpretation. The paper gives an overview of existing methods, which are illustrated with applications to shape analysis, and formalizes a new family of relational abstract domains that allows sets of functions to be abstracted more precisely than with known approaches, while being still machine-representable.

This paper introduces a novel shape representation which performs shape analysis in a hierarchical fashion using Gaussian and Laplacian pyramids. A background on hierarchical shape analysis is given along with a detailed explanation of the hierarchical method, and results are shown on natural contours. A comparison is performed between the new method and our proposed approach using Point Distri...