Fully discrete needlet approximation on the sphere

Wang, Yu Guang; Gia, Quôc Thông Lê; Sloan, Ian H.; Womersley, Robert S.

doi:10.1016/j.acha.2016.01.003

Cited by 35 publications

(44 citation statements)

References 39 publications

(75 reference statements)

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“…The fact that the lack of polynomial-exactness of the quadrature for framelets leads to noticeably worse approximation errors was also observed in [45]. The dependence of L 2 approximation errors of the tight framelets on smoothness of function space is consistent with that of the filtered approximation on S 2 , see [51,56,65,69]. Example 4.2 (Multiple high-pass filters).…”

Section: Numerical Examplesmentioning

confidence: 60%

Tight framelets and fast framelet filter bank transforms on manifolds

Wang

Zhuang

2020

Applied and Computational Harmonic Analysis

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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...

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Section: Numerical Examplesmentioning

confidence: 60%

Tight framelets and fast framelet filter bank transforms on manifolds

Wang

Zhuang

2020

Applied and Computational Harmonic Analysis

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“…We state some known results for needlets and needlet approximations in this section, which we will use later in the paper. All results can be found in [26,27,40].…”

Section: Needlets and Filtered Operatorsmentioning

confidence: 96%

“…is a spherical polynomial of degree at most 2 J − 1. As in [40], we introduce the filter H related to the needlet filter h:…”

Section: Needlets and Filtered Operatorsmentioning

confidence: 99%

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Needlet approximation for isotropic random fields on the sphere

Gia

Sloan

Wang

et al. 2017

Journal of Approximation Theory

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In this paper we establish a multiscale approximation for random fields on the sphere using spherical needlets -a class of spherical wavelets. We prove that the semidiscrete needlet decomposition converges in mean and pointwise senses for weakly isotropic random fields on S d , d ≥ 2. For numerical implementation, we construct a fully discrete needlet approximation of a smooth 2-weakly isotropic random field on S d and prove that the approximation error for fully discrete needlets has the same convergence order as that for semidiscrete needlets. Numerical examples are carried out for fully discrete needlet approximations of Gaussian random fields and compared to a discrete version of the truncated Fourier expansion. polynomial of degree 2 j − 1, associated with a positive weight quadrature rule with degree of precision 2 j+1 − 1. Main results.Needlets form a multiscale tight frame for square-integrable functions on S d , see [26,27]. The resulting tight frame has good approximation performance for random fields on the sphere.Needlet decomposition for random fields. For a random field T on S d , the (semidiscrete) needlet approximation of order J for J ∈ N 0 is defined as( 1.1) Let ⌈·⌉ be the ceiling function. Given 1 ≤ p < ∞ and d ≥ 2, the needlet approximation given by (1.1) of a ⌈p⌉-weakly isotropic random field T on S d converges to T in L p Ω × S d . (See Theorem 3.4, and see Section 2 for the definition of n-weakly isotropic, n ∈ N.)In Theorem 3.9, we also prove that when p is a positive integer the needlet approximation on the set of p-weakly isotropic random fields is bounded in the L p Ω × S d sense.Let d ≥ 2. Let T be a 2-weakly isotropic random field satisfying ∞ ℓ=1 A ℓ ℓ 2s+d−1 < ∞. Here A ℓ is the angular power spectrum of T , see Section 4.1. Then T (ω) is in the Sobolev space W s 2 (S d ) P-almost surely (or P-a. s.). (This fact was proved in [19, Section 4] and [3]; we restate this result in Corollary 4.4.)Semidiscrete needlet approximation. In Theorems 4.6 and 4.7, we prove that the semidiscrete needlet approximation V need J (T ) has the following mean and pointwise approximation errors. Let s > 0.( 1.2)The needlet approximation V need J (T ; ω) has the following L 2 (S d )-error bound: P-a. s.,where the constants c in (1.2) and (1.3) depend only on d, s, h and κ. Fully discrete needlet approximation. For implementation, we discretise the needlet coefficient (T (ω), ψ jk ) L 2 (S d ) = S d T (ω, x)ψ jk (x) dσ d (x) by a positive weight numerical integration rule,This then gives the fully discrete needlet approximation for the random field T :In Theorems 4.10 and 4.11, we prove that for s > d/2, the fully discrete needlet approximation V need J,N (T ) has the same mean and pointwise error orders as those for the semidiscrete needlet approximation V need J (T ), cf. (1.2)-(1.3). Let Q N be a discretisation quadrature exact for polynomials of degree 3 · 2 J−1 − 1. Then 4) and P-a. s.5) (ii) for p = ∞, lim J→∞ E T − V need J (T ) L∞(S d ) = 0.Remark. By the Fubini theorem, (3.7) implies that the nee...

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“…Rather the aim is to find sequences of point sets which are at least computationally spherical t-designs, have a low number of points and are geometrically well-distributed on the sphere. Such point sets provide excellent nodes for numerical integration on the sphere, as well as hyperinterpolation [56,40,59] and fully discrete needlet approximation [65]. These methods have a requirement that the quadrature rules are exact for certain degree polynomials.…”

Section: Introductionmentioning

confidence: 99%

Efficient Spherical Designs with Good Geometric Properties

Womersley

2018

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

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Dedicated to Ian H. Sloan on the occasion of his 80th birthday in acknowledgement of his many fruitful ideas and generosity.Abstract Spherical t-designs on S d ⊂ R d+1 provide N nodes for an equal weight numerical integration rule which is exact for all spherical polynomials of degree at most t. This paper considers the generation of efficient, where N is comparable to (1 + t) d /d, spherical t-designs with good geometric properties as measured by their mesh ratio, the ratio of the covering radius to the packing radius. Results for S 2 include computed spherical t-designs for t = 1, ..., 180 and symmetric (antipodal) t-designs for degrees up to 325, all with low mesh ratios. These point sets provide excellent points for numerical integration on the sphere. The methods can also be used to computationally explore spherical t-designs for d = 3 and higher.

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Fully discrete needlet approximation on the sphere

Cited by 35 publications

References 39 publications

Tight framelets and fast framelet filter bank transforms on manifolds

Tight framelets and fast framelet filter bank transforms on manifolds

Needlet approximation for isotropic random fields on the sphere

Efficient Spherical Designs with Good Geometric Properties

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