Tight framelets and fast framelet filter bank transforms on manifolds

Abstract

Tight framelets on a smooth and compact Riemannian manifold M provide a tool of multiresolution analysis for data from geosciences, astrophysics, medical sciences, etc. This work investigates the construction, characterizations, and applications of tight framelets on such a manifold M. Characterizations of the tightness of a sequence of framelet systems for L 2 pMq in both the continuous and semi-discrete settings are provided. Tight framelets associated with framelet filter banks on M can then be easily designed and fast framelet filter bank transforms on M are shown to be realizable with nearly linear computational complexity. Explicit construction of tight framelets on the sphere S 2 as well as numerical examples are given.Keywords: tight framelets, affine system, compact Riemannian manifold, quadrature rule, filter bank, FFT, fast spherical harmonic transform, Laplace-Beltrami operator, unitary extension principle 2010 MSC: 42C15, 42C40, 42B05, 41A55, 57N99, 58C35, 94A12, 94C15, 93C55, 93C95 Introduction and motivationIn the era of information technologies, the rapid development of modern high-tech devices, for example, a super computer, PC, smart phone, wearable and VR/AR device, is driven internally by Moore's Law [55] which contributes to the exponential growth of the computational power, while externally stimulated by the tremendous need of both the public and individual parties in processing massive data from finance, economy, geology, bio-information, cosmology, medical sciences and so on. It has been noticed that Moore's Law is slowing down due to the constrains of the physical law [19] but the volume of data is dramatically increasing. Dealing with Big Data is becoming a crucial part of an individual person, party, government and country.Real-world data often inherit high-dimensionality such as data from a surveillance system, seismology, climatology. High-dimensional data are typically concentrated on a low-dimensional manifold [60,67], for instance, the sphere in remote sensing and CMB data [6], more complex surfaces in brain imaging [68], and discrete graph data from social and traffic networks [61]. Analysis and learning tools on manifolds hence play an increasingly important role in machine learning and statistics.The key to successful manifold learning lies in that data on a manifold may exhibit high complexity on one hand while they are highly sparse at a certain domain via an appropriate multiscale representation system on the other hand. Sparsity within such representations, stemming from computational harmonic analysis, enables efficient analysis and processing of high-dimensional and massive data.Multiresolution analysis in general are designed for data in the Euclidean space R d , d ě 1, for example, a signal in R, an image in R 2 and a video in R 3 . Multiscale representation systems in R d including wavelets, framelets, curvelets, shearlets, etc., which are capable of capturing the sparsity of data, have been well-developed and widely used, see e.g. [7,11,14,17,21,49,50]. The core of the...

The unitary extension principle on locally compact abelian groups

“…We also note that the approach in the paper heavily uses the LCA group structure. We would like to point the attention of the reader to a different generalization of the UEP, taking place on smooth and compact Riemannian manifolds; see [24]. We also mention that there is a growing literature on wavelet analysis on the p-adic numbers; see [1] and the references therein.…”

Analysis of Framelet Transforms on a Simplex

“…In the paper, we only consider the framelet transforms for one framelet system with starting level 0. The results can be generalized to a sequence of framelet systems as [26,14], which will allow one more flexibility in constructing framelets.…”

“…Here, the sequences a and b n are said to be low-pass (mask) and high-pass (mask) respectively. We introduce the continuous and semi-discrete framelets on the simplex following the construction and notation of [26,7]. The continuous framelets on the simplex T 2 are, for j ∈ N 0 , (…”

“…If the framelets are tight on the simplex, a function in L 2 (T 2 ) can be represented using the framelet coefficients. The following property as a consequence of [26,Theorem 2.4] shows that the tightness of framelets is equivalent to a multiscale representation of framelets of a level by lower levels.…”

“…Multiresolution analysis on a simplex T 2 in R 2 has many applications such as in numerical solution of PDEs and computer graphics [9,10,12]. In this paper, we construct framelets (or a framelet system) on T 2 , following the framework of [26], and give the transforms of coefficients for framelets.…”