Multiresolution analysis and adaptive estimation on a sphere using stereographic wavelets

2018/09/05 by Bogdan Ćmiel, Karol Dziedziul, Natalia Jarzębkowska · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Image Fusion Techniques #Artificial intelligence #Computer science #Computer vision #Discrete wavelet transform #Economics #Estimation #Geometry #Image and Signal Denoising Methods #Management #Mathematics #Multiresolution analysis #Statistical and numerical algorithms #Stereographic projection #Wavelet #Wavelet transform #math.ST #msc:42C40 #stat.TH

paper · pdf · doi:10.1016/j.na.2018.08.003

published in Nonlinear Analysis 179, 41-71 (Elsevier BV)

openalex publication_date 2018/09/05 · openalex created_date 2018/09/27 · arxiv created 2018/10/11 · arxiv updated 2018/10/15 · openalex updated_date 2026/08/05

Abstract

We construct an adaptive estimator of a density function on d dimensional unit sphere S^d (d ≥ 2 ), using a new type of spherical frames. The frames, or as we call them, stereografic wavelets are obtained by transforming a wavelet system, namely Daubechies, using some stereographic operators. We prove that our estimator achieves an optimal rate of convergence on some Besov type class of functions by adapting to unknown smoothness. Our new construction of stereografic wavelet system gives us a multiresolution approximation of L²(S^d) which can be used in many approximation and estimation problems. In this paper we also demonstrate how to implement the density estimator in S² and we present a finite sample behavior of that estimator in a numerical experiment.

Citations

• Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities
• Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds
• Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds