Exact reconstruction with directional wavelets on the sphere
Y. Wiaux, J. D. McEwen, P. Vandergheynst, O. Blanc
备注：
22 pages, 2 figures. Version 2 matches version accepted for publication in MNRAS. Version 3 (identical to version 2) posted for code release announcement - "Steerable scale discretised wavelets on the sphere" - S2DW code available for download at http://www.mrao.cam.ac.uk/~jdm57/software.html
本文提出了一种新的球面上带限信号的方向小波分析和精确重构方法。它代表了Antoine & Vandergheynst（1999）和Wiaux et al.（2005）开发的小波形式主义的演变。小波在球面上任意点的平移及其固有旋转仍然通过连续的三维旋转来定义。通过一种新的核扩张，直接在调和空间中定义小波的扩张，该核扩张是对已有调和扩张的一种改进。首先确定了一族适合于这种核扩张的具有紧调和支集的因子分解可转向函数。尺度离散小波形式主义，然后派生，依赖于这种膨胀。分析尺度的离散性质允许带限信号的精确重建。最后给出了相应的精确多分辨率算法，并对实现进行了测试。该形式主义是感兴趣的，特别是在小波中的稀疏扩展的球体上的信号的去噪或反卷积。在天体物理学中，它特别适用于识别宇宙微波背景（CMB）数据中的局部方向特征，例如拓扑缺陷的印记，特别是宇宙弦，并在与其他信号分量分离后进行重建。
英文摘要：
A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of the wavelet formalism developed by Antoine & Vandergheynst (1999) and Wiaux et al. (2005). The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation is firstly identified. A scale discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background (CMB) data, such as the imprint of topological defects, in particular cosmic strings, and for their reconstruction after separation from the other signal components.
AstroParticule et Cosmologie, CNRS and Université Paris 7（粒子物理与宇宙学，法国国家科学研究中心与巴黎第七大学）；Laboratoire Probabilités et Modèles Aléatoires, CNRS and Universités Paris 6-7（概率与随机模型实验室，法国国家科学研究中心与巴黎第六至七大学）；Laboratoire du Traitement et de la Communication de l’Information, CNRS and Télécom Paris（信息与通信处理实验室，法国国家科学研究中心与巴黎电信学院）；Laboratoire Paul-Painlevé, Université Lille-1（保罗·潘勒韦实验室，里尔第一大学）
