Related papers: Antinorms and Radon curves
If the Radon transform of a compactly supported distribution $f \ne 0$ in $\mathbb R^n$ is supported on the set of tangent planes to the boundary $\partial D$ of a bounded convex domain $D$, then $\partial D$ must be an ellipsoid. As a…
Measuring the similarity of curves is a fundamental problem arising in many application fields. There has been considerable interest in several such measures, both in Euclidean space and in more general setting such as curves on Riemannian…
In this paper we consider the generalized Radon transform $\mathcal R$ in the plane. Let $f$ be a piecewise smooth function, which has a jump across a smooth curve $\mathcal S$. We obtain a formula, which accurately describes view aliasing…
Given a non-rational real space curve and a tolerance $\epsilon>0$, we present an algorithm to approximately parametrize the curve. The algorithm checks whether a planar projection of the space curve is $\epsilon$-rational and, in the…
We study the normal map for plane projective curves, i.e., the map associating to every regular point of the curve the normal line at the point in the dual space. We first observe that the normal map is always birational and then we use…
We study the integral transform over a general family of broken rays in $\mathbb{R}^2$. It is natural for broken rays to have conjugate points, for example, when they are reflected from a curved boundary. If there are conjugate points, we…
The radius of regularity sometimes spelled as the radius of nonsingularity is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being…
The theory of linear transports along paths in vector bundles, generalizing the parallel transports generated by linear connections, is developed. The normal frames for them are defined as ones in which their matrices are the identity…
This work introduces a novel and general class of continuous transforms based on hierarchical Voronoi based refinement schemes. The resulting transform space generalizes classical approaches such as wavelets and Radon transforms by…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a graph of density strictly exceeding 1/2, and…
This paper is the second part of a two-part paper investigating the structure and properties of dyadic polygons. A dyadic polygon is the intersection of the dyadic subplane $D^2$ of the real plane $R^2$ and a real convex polygon with…
A Bertrand (respectively, Mannheim) curve is a space curve whose principal normal line is the same as the principal normal (respectively, bi-normal) line of another curve. By definition, another curve is a parallel curve with respect to the…
We study reconstruction of an unknown function from its $d$-plane Radon transform on the flat $n$-torus when $1 \leq d \leq n-1$. We prove new reconstruction formulas and stability results with respect to weighted Bessel potential norms. We…
We consider skew ruled surfaces in the three-dimensional Euclidean space and some geometrically distinguished families of curves on them whose normal curvature has a concrete form. The aim of this paper is to find and classify all ruled…
Connections between integration along hypersufaces, Radon transforms, and neural networks are exploited to highlight an integral geometric mathematical interpretation of neural networks. By analyzing the properties of neural networks as…
In this paper we investigate the mapping properties in Lebesgue-type spaces of certain generalized Radon transforms defined by integration over curves.
A Radon-type transform called a cone transform that assigns to a given function its integral over various sets of cones has arisen in the last decade in the context of the study of Compton cameras used in Single Photon Emission Computed…
The `linear orbit' of a plane curve of degree $d$ is its orbit in $\P^{d(d+3)/2}$ under the natural action of $\PGL(3)$. In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane…
In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture in 1977 that an affine maximal graph of a smooth, locally uniformly convex function on two-dimensional Euclidean space…