Acceleration of the frame algorithm

Gröchenig, Karlheinz

doi:10.1109/78.258077

Cited by 110 publications

(75 citation statements)

References 20 publications

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“…As shown in [18], log ρ ≈ 2 √ κ−1 when κ is large, and hence in order to have ρ N alg < ǫ/2, then we must have N alg > log 2 ǫ log 1 ρ ≈ √ κ−1 2 (log 2 + log 1 ǫ ). Furthermore, each iteration (after the first, when the inner products are given) of the frame algorithm requires the calculation of inner products when applying the frame operator.…”

Section: Cost Comparisonmentioning

confidence: 99%

“…In fact, there are usually many different ways of writing out the frame expansion of a function since the frame functions (as well as the dual frame functions) are not linearly independent. Furthermore, there is no closed form equation for constructing the inverse frame operator S −1 , which makes the theoretical result in (18) or (19) computationally infeasible. For this reason, several methods have been developed to numerically construct a finite-dimensional frame approximation, which will be further discussed in Section 4.…”

Section: Fourier Framesmentioning

confidence: 99%

“…Notice also that knowledge of the frame bounds is important, as the convergence of the algorithm is sensitive to the ratio A/B, with A/B ≈ 0 resulting in very slow convergence and A/B ≈ 1 resulting in fast convergence. Finally, several accelerated frame algorithms, developed in [18], can vastly improve the convergence of the frame algorithm when the frame bounds are poor. The (Chebyshev) accelerated frame algorithm [18, Thm.…”

Section: Fourier Framesmentioning

confidence: 99%

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Detection of Edges from Nonuniform Fourier Data

Gelb

Hines

2011

J Fourier Anal Appl

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Abstract. Edge detection is important in a variety of applications. While there are many algorithms available for detecting edges from pixelated images or equispaced Fourier data, much less attention has been given to determining edges from nonuniform Fourier data. There are applications in sensing (e.g. MRI) where the data is given in this way, however. This paper introduces a method for determining the locations of jump discontinuities, or edges, in a one-dimensional periodic piecewise-smooth function from nonuniform Fourier coefficients. The technique employs the use of Fourier frames. Numerical examples are provided.

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Section: Cost Comparisonmentioning

confidence: 99%

Section: Fourier Framesmentioning

confidence: 99%

Section: Fourier Framesmentioning

confidence: 99%

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Detection of Edges from Nonuniform Fourier Data

Gelb

Hines

2011

J Fourier Anal Appl

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“…Good values for the frame bounds are essential for estimates of the speed of convergence in algorithms involving frames; cf. [33]. Recently, Theorem 3.3 has been used to construct Riesz bases for weighted L 2 -spaces consisting of solutions to certain Sturm-Liouville problems; cf.…”

Section: Frames and Riesz Bases 281mentioning

confidence: 99%

Frames, Riesz bases, and discrete Gabor/wavelet expansions

Christensen

2001

Bull. Amer. Math. Soc.

106

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Abstract. This paper is a survey of research in discrete expansions over the last 10 years, mainly of functions in L 2 (R). The concept of an orthonormal basis {fn}, allowing every function f ∈ L 2 (R) to be written f = cnfn for suitable coefficients {cn}, is well understood. In separable Hilbert spaces, a generalization known as frames exists, which still allows such a representation. However, the coefficients {cn} are not necessarily unique. We discuss the relationship between frames and Riesz bases, a subject where several new results have been proved over the last 10 years. Another central topic is the study of frames with additional structure, most important Gabor frames (consisting of modulated and translated versions of a single function) and wavelets (translated and dilated versions of one function). Along the way, we discuss some possible directions for future research.

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“…Moreover, in order to achieve the optimal convergence rate a precise knowledge of the frame bounds is required. The conjugate-gradient method, described for frames by Gröchenig in [7], is an improved algorithm for inverting a frame operator that does not require estimates of the frame bounds.…”

Section: Theorem 1 (Frame Algorithm) Let S Be a Self-adjoint Operatomentioning

confidence: 99%

Stable Filtering Schemes with Rational Dilations

Johnson

2007

J Fourier Anal Appl

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Abstract. The relationship between multiresolution analysis and filtering schemes is a well-known facet of wavelet theory. However, in the case of rational dilation factors, the wavelet literature is somewhat lacking in its treatment of this relationship. This work seeks to establish a means for the construction of stable filtering schemes with rational dilations through the theory of shift-invariant spaces. In particular, principal shift-invariant spaces will be shown to offer frame wavelet decompositions for rational dilations even when the associated scaling function is not refinable. Moreover, it will be shown that such decompositions give rise to stable filtering schemes with finitely supported filters, reminiscent of those studied by Kovačević and Vetterli.

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Acceleration of the frame algorithm

Cited by 110 publications

References 20 publications

Detection of Edges from Nonuniform Fourier Data

Detection of Edges from Nonuniform Fourier Data

Frames, Riesz bases, and discrete Gabor/wavelet expansions

Stable Filtering Schemes with Rational Dilations

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