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The square root velocity transform is a powerful tool for the efficient computation of distances between curves. Also, after factoring out reparametrisations, it defines a distance between shapes that only depends on their intrinsic…
Geometric frameworks for analyzing curves are common in applications as they focus on invariant features and provide visually satisfying solutions to standard problems such as computing invariant distances, averaging curves, or registering…
Motivated by applications from computer vision to bioinformatics, the field of shape analysis deals with problems where one wants to analyze geometric objects, such as curves, while ignoring actions that preserve their shape, such as…
This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curves modulo shape-preserving transformations. We summarize the…
Bayesian analysis of functions and curves is considered, where warping and other geometrical transformations are often required for meaningful comparisons. We focus on two applications involving the classification of mouse vertebrae shape…
We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
In this paper we are concerned with the approach to shape analysis based on the so called Square Root Velocity Transform (SRVT). We propose a generalisation of the SRVT from Euclidean spaces to shape spaces of curves on Lie groups and on…
In this paper we investigate a time dependent family of plane closed Jordan curves evolving in the normal direction with a velocity which is assumed to be a function of the curvature, tangential angle and position vector of a curve. We…
This article deals with flow of plane curves driven by the curvature and external force. We make use of such a geometric flow for the purpose of image segmentation. A parametric model for evolving curves with uniform and curvature adjusted…
In this paper we study the shape space of curves with values in a homogeneous space $M = G/K$, where $G$ is a Lie group and $K$ is a compact Lie subgroup. We generalize the square root velocity framework to obtain a reparametrization…
Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their…
We study evolution of a closed embedded plane curve with the normal velocity depending on the curvature, the orientation and the position of the curve. We propose a new method of tangential redistribution of points by curvature adjusted…
In many image analysis problems, the contours of objects carry important statistical information about shape. Such contours are typically affected by deformation variables including scaling, translation, rotation, and reparametrization.…
This paper puts forth a new formulation and algorithm for the elastic matching problem on unparametrized curves and surfaces. Our approach combines the frameworks of square root normal fields and varifold fidelity metrics into a novel…
In this article we investigate a system of geometric evolution equations describing a curvature driven motion of a family of 3D curves in the normal and binormal directions. Evolving curves may be subject of mutual interactions having both…
Past approaches for statistical shape analysis of objects have focused mainly on objects within the same topological classes, e.g., scalar functions, Euclidean curves, or surfaces, etc. For objects that differ in more complex ways, the…
We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions…
We minimize elastic energies on framed curves which penalize both curvature and torsion. We also discuss critical points using the infinite dimensional version of the Lagrange multipliers' method. Finally, some examples arising from the…
Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides comprehensive frameworks for simultaneous registration, deformation, and comparison of shapes. These methods achieve computational efficiency…