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...

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...

Shape optimization based on the shape calculus is numerically mostly performed by means of steepest descent methods. This paper provides a novel framework to analyze shapeNewton optimization methods by exploiting a Riemannian perspective. A Riemannian shape Hessian is defined yielding often sought properties like symmetry and quadratic convergence for Newton optimization methods.

Shape memory alloys are typical temperature-sensitive metallic functional materials due to superelasticity and shape recovery characteristics. The conventional shape memory effect involves the formation and deformation of thermally induced martensite and its reverse transformation. The shape recovery process usually takes place over a temperature range, showing relatively low temperature-sensit...

In this dissertation we develop and apply models of shape and models of image intensities (appearance models) in object-based image processing tasks. We make contributions in three areas of interest: constructing novel flexible models of shape and of image intensities, using these models to extract object boundaries from images, and analyzing differences between groups of shapes from given, ext...

Knowledge about the biological variability of anatomical objects is essential for statistical shape analysis and discrimination between healthy and pathological structures. This paper describes a novel approach that incorporates variability of an object population into the generation of a characteristic 3D shape model. The proposed shape representation is a coarse-scale sampled medial descripti...