The Theory of Multiresolution Analysis Frames and Applications to Filter Banks

Benedetto, John J.; Li, Shidong

doi:10.1006/acha.1997.0237

Cited by 347 publications

(272 citation statements)

References 37 publications

(59 reference statements)

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“…Examples with maximum stopband attenuation for given stopbands are shown in Figures (3) and (4). We see that the stopband attenuation of the new bi-filter frequency responses are indeed greater than that of CDF examples of the same length.…”

Section: 13-tap Pffs Dual-filters (Solid) Versus Cdf 17-tap Filter mentioning

confidence: 88%

“…The construction of PFFS has typically two (non-symmetrical) directions. One corresponds to finding the left inverse V from a given U ; the other relates to finding the U from a given V , all according to equation (4). We refer to [26] for details of the constructions.…”

Section: Theorem 11 ([26]mentioning

confidence: 99%

“…From the sub-band processing point of view, since these nice properties are demanded in practical applications [27], [29], [4], [18], [30], [28], [22], [11], [3] we now understand the need to bring some information in the high-pass band (W 0 ) back to the low-pass band to off-set some of the draw backs due to a biorthogonal dual that is not quite as nice. This has to be done in such a way that the perfect reconstruction principle is not violated.…”

Section: Construction Of Biorthogonal Wavelets and Filters Via Pffsmentioning

confidence: 99%

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Parametric Optimization of Biorthogonal Wavelets and Filterbanks via Pseudoframes for Subspaces

Hoffman

2012

Excursions in Harmonic Analysis, Volume 2

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Abstract. We present constructions of biorthogonal wavelets and associated filter banks with optimality using pseudoframes for subspaces (PFFS). PFFS extends the theory of frames in that pseudoframe sequences need not reside within the subspace of interest. In particular, when PFFS is applied to biorthogonal wavelets the underlying flexibility presents opportunities to incorporate optimality, regularity as well as perfect reconstruction into one parametric design approach. This approach reduces certain filter optimization problems to optimization over a free parameter. While past constructions can be reproduced, results with additional optimality are also obtained and presented here with numerical examples. Filter coeffiecients are provided along with graphs are provided and Matlab implementation code is available upon request from the authors.

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Section: 13-tap Pffs Dual-filters (Solid) Versus Cdf 17-tap Filter mentioning

confidence: 88%

Section: Theorem 11 ([26]mentioning

confidence: 99%

Section: Construction Of Biorthogonal Wavelets and Filters Via Pffsmentioning

confidence: 99%

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Parametric Optimization of Biorthogonal Wavelets and Filterbanks via Pseudoframes for Subspaces

Hoffman

2012

Excursions in Harmonic Analysis, Volume 2

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“…A sufficient condition for {V j , φ} to be an FMRA was given in [4], Theorem 4.6: Note the similarity between Theorem 5.4 and the results for an MRA; cf. [50], Th.…”

Section: Frame Multiresolution Analysismentioning

confidence: 90%

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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“…Frame multiresolution analysis (FMRA) was introduced by Benedetto and Li [4]. The purpose of the theory is to construct frames for L 2 (R) of the form {D 2 j T k ψ} j,k∈Z .…”

Section: Frame Multiresolution Analysismentioning

confidence: 99%

Untitled

2001

Bull. Amer. Math. Soc.

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The Theory of Multiresolution Analysis Frames and Applications to Filter Banks

Cited by 347 publications

References 37 publications

Parametric Optimization of Biorthogonal Wavelets and Filterbanks via Pseudoframes for Subspaces

Parametric Optimization of Biorthogonal Wavelets and Filterbanks via Pseudoframes for Subspaces

Frames, Riesz bases, and discrete Gabor/wavelet expansions

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