arXiv subjects

Alexey Remizov

Publications and source records attributed to Alexey Remizov.

7 recordsLinked to original sources

arXiv2026

A Note on the Legendre Transformation

We present the Legendre transformation in a geometric way based on the procedure of the Legendrian lift. This approach allows us to understand some interesting properties of it, in particular, the reason for the appearance of singularities of dual curves. Also we consider application of the Legendre transformation to the Clairaut differential equation. Finally, we say a few words class of contact transformations and present an infinite group of contact transformations different from the Legendre transformation. Keywords: Legendre transformation, duality, contact transformation, contact structure, pedal curve, singular point, Clairaut equation

math.HO↗

arXiv2025

Remembering Igor R. Shafarevich: The Path to Freedom

2023 year marks the hundredth birth anniversary of prominent Russian mathematician and thinker Igor Rostislavovich Shafarevich (1923-2017). The article presents a selection of quotations from various works of him, devoted to the future of mathematics and science in general, with minor comments and explanations.

math.HO↗

arXiv2018

Highly corrupted image inpainting through hypoelliptic diffusion

We present a new image inpainting algorithm, the Averaging and Hypoelliptic Evolution (AHE) algorithm, inspired by the one presented in [SIAM J. Imaging Sci., vol. 7, no. 2, pp. 669--695, 2014] and based upon a semi-discrete variation of the Citti-Petitot-Sarti model of the primary visual cortex V1. The AHE algorithm is based on a suitable combination of sub-Riemannian hypoelliptic diffusion and ad-hoc local averaging techniques. In particular, we focus on reconstructing highly corrupted images (i.e. where more than the 80% of the image is missing), for which we obtain reconstructions comparable with the state-of-the-art.

cs.CV↗

arXiv2018

Cortical-inspired image reconstruction via sub-Riemannian geometry and hypoelliptic diffusion

In this paper we review several algorithms for image inpainting based on the hypoelliptic diffusion naturally associated with a mathematical model of the primary visual cortex. In particular, we present one algorithm that does not exploit the information of where the image is corrupted, and others that do it. While the first algorithm is able to reconstruct only images that our visual system is still capable of recognize, we show that those of the second type completely transcend such limitation providing reconstructions at the state-of-the-art in image inpainting. This can be interpreted as a validation of the fact that our visual cortex actually encodes the first type of algorithm.

cs.CV↗

arXiv2014

On the local and global properties of geodesics in pseudo-Riemannian metrics

The paper is a study of geodesic in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at such points that leads to a curious phenomenon: geodesics cannot pass through such a point in arbitrary tangential directions, but only in certain directions said to be admissible (the number of admissible directions is generically 1 or 3). Secondly, we study the global properties of geodesics in pseudo-Riemannian metrics possessing differentiable groups of symmetries. At the end of the paper, two special types of discontinuous metrics are considered.

math.DG↗

arXiv2014

Hypoelliptic diffusion and human vision: a semi-discrete new twist

This paper presents a semi-discrete alternative to the theory of neurogeometry of vision, due to Citti, Petitot and Sarti. We propose a new ingredient, namely working on the group of translations and discrete rotations $SE(2,N)$. The theoretical side of our study relates the stochastic nature of the problem with the Moore group structure of $SE(2,N)$. Harmonic analysis over this group leads to very simple finite dimensional reductions. We then apply these ideas to the inpainting problem which is reduced to the integration of a completely parallelizable finite set of Mathieu-type diffusions (indexed by the dual of $SE(2,N)$ in place of the points of the Fourier plane, which is a drastic reduction). The integration of the the Mathieu equations can be performed by standard numerical methods for elliptic diffusions and leads to a very simple and efficient class of inpainting algorithms. We illustrate the performances of the method on a series of deeply corrupted images.

math.AP↗

arXiv2010

On a class of vector fields with discontinuity of divide-by-zero type and its applications

We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three of them in the framework of differential geometry: singularities of geodesic flows in various singular metrics on surfaces.

math.DS↗