Edge detection with polynomial frames on the sphere

IF 0.6 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory Pub Date : 2026-05-01 Epub Date: 2025-12-26 DOI:10.1016/j.jat.2025.106279

Frederic Schoppert

引用次数: 0

Abstract

In a recent article, we have shown that a variety of localized polynomial frames, including isotropic as well as directional spherical systems, are suitable for detecting jump discontinuities that lie along circles on the sphere. More precisely, such edges can be identified in terms of their position and orientation by the asymptotic decay of the frame coefficients in an arbitrary small neighborhood. In this paper, we will extend these results to discontinuities which lie along general smooth curves. In particular, we prove upper and lower estimates for the frame coefficients when the analysis function is concentrated in the vicinity of such a singularity. The estimates are given in an asymptotic sense, with respect to some dilation parameter, and they hold uniformly in a neighborhood of the smooth curve segment under consideration.

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球面上多项式帧的边缘检测

在最近的一篇文章中，我们已经证明了各种局部多项式框架，包括各向同性和定向球系统，适合于检测沿球体上的圆的跳跃不连续。更准确地说，这样的边缘可以通过任意小邻域内框架系数的渐近衰减来识别它们的位置和方向。在本文中，我们将这些结果推广到沿一般光滑曲线的不连续点。特别地，我们证明了当分析函数集中在奇异点附近时框架系数的上估计和下估计。给出了关于膨胀参数的渐近估计，它们在所考虑的光滑曲线段的邻域内一致成立。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.