Related papers: The continuous Procrustes distance between two sur…
The pose of a rigid object is usually regarded as a rigid transformation, described by a translation and a rotation. However, equating the pose space with the space of rigid transformations is in general abusive, as it does not account for…
Modern time series analysis requires the ability to handle datasets that are inherently high-dimensional; examples include applications in climatology, where measurements from numerous sensors must be taken into account, or inventory…
This paper rigorously solves the challenging problem of recognizing periodic patterns under rigid motion in Euclidean geometry. The 3-dimensional case is practically important for justifying the novelty of solid crystalline materials…
The matching distance is a computationally tractable topological measure to compare multi-filtered simplicial complexes. We design efficient algorithms for approximating the matching distance of two bi-filtered complexes to any desired…
Finding the point in an algebraic variety that is closest to a given point is an optimization problem with many applications. We study the case when the variety is a Fermat hypersurface. Our formula for its Euclidean distance degree is a…
A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first…
This paper addresses to the problem of finding the (minimum) Euclidean distance between two linear varieties. This problem is, usually, solved minimising a target function. We propose a novel approach: to use the Moore-Penrose generalised…
The most useful data mining primitives are distance measures. With an effective distance measure, it is possible to perform classification, clustering, anomaly detection, segmentation, etc. For single-event time series Euclidean Distance…
Many geometric optimization problems can be reduced to finding points in space (centers) minimizing an objective function which continuously depends on the distances from the centers to given input points. Examples are $k$-Means, Geometric…
In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by…
The computational complexity of some depths that satisfy the projection property, such as the halfspace depth or the projection depth, is known to be high, especially for data of higher dimensionality. In such scenarios, the exact depth is…
Many applications using large datasets require efficient methods for minimizing a proximable convex function subject to satisfying a set of linear constraints within a specified tolerance. For this task, we present a proximal projection…
In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time.…
This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology…
A novel algorithm is proposed for quantitative comparisons between compact surfaces embedded in the three-dimensional Euclidian space. The key idea is to identify those objects with the associated surface measures and compute a weak…
High fidelity representation of shapes with arbitrary topology is an important problem for a variety of vision and graphics applications. Owing to their limited resolution, classical discrete shape representations using point clouds, voxels…
Distance measuring is a very important task in digital geometry and digital image processing. Due to our natural approach to geometry we think of the set of points that are equally far from a given point as a Euclidean circle. Using the…
Detecting poorly textured objects and estimating their 3D pose reliably is still a very challenging problem. We introduce a simple but powerful approach to computing descriptors for object views that efficiently capture both the object…
Proximal distance algorithms combine the classical penalty method of constrained minimization with distance majorization. If $f(\boldsymbol{x})$ is the loss function, and $C$ is the constraint set in a constrained minimization problem, then…
There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types…