On the stable sampling rate for binary measurements and wavelet reconstruction

Hansen, Anders C.; Thesing, Laura

doi:10.1016/j.acha.2018.08.004

Cited by 51 publications

(31 citation statements)

References 49 publications

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“…Once the boundary functions have been evaluated they can be saved and used over and over at a similar cost to the interior basis functions. The use of the proposed boundary functions as part of a wavelet transform can easily be extended to more dimensions, [12]. First, if the signal is not already discrete, sample it on a regular grid.…”

Section: Resultsmentioning

confidence: 99%

“…As noted in [15] this construction of wavelets on an interval is associated with a multiresolution analysis. Furthermore [12] argues that the construction can be extended to wavelet functions.…”

Section: Derivation Of Boundary Wavelet Scaling Functionsmentioning

confidence: 99%

“…-Evaluate 10-Evaluate ⟨x , J,k ⟩ according to (12) -Evaluate ⟨x l , ⟩ according to (14) Similarly, the function FourierBoundaryWavelets evaluates equation 15for a given scale J and = {0, … , a − 1} . This also results in 2a boundary functions, but involves a few additional steps.…”

Section: Package Overviewmentioning

confidence: 99%

“…-Evaluate 15-Evaluate ⟨x , J,k ⟩ according to (12) -Evaluate ⟨x l , ⟩ according to (14) -Find the Fourier transform of on the interval according to (16) -Evaluate the Fourier transform of using (17) -Evaluate the Fourier transform of the window using (18)…”

Section: Package Overviewmentioning

confidence: 99%

“…We have chosen to use 12 steps of the cascade algorithm to sample the basis functions and therefore we need test signals of length 2 12 . With the chosen dataset we can make 332 disjoint test signals where the samples in each signal is the interval [n2 12 , (n + 1)2 12 ) . Furthermore we have chosen to use the Daubechies 3 and 5 wavelets at a scale of 7.…”

Section: Comparison To Classical Wavelet Approachmentioning

confidence: 99%

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Orthonormal, moment preserving boundary wavelet scaling functions in Python

Holm¹,

Arildsen²,

Nielsen³

et al. 2020

SN Appl. Sci.

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In this paper we derive an orthonormal basis of wavelet scaling functions for L 2 ([0, 1]) motivated by the need for such a basis in the field of generalized sampling. A special property of this basis is that it includes carefully constructed boundary functions and it can be constructed with arbitrary smoothness. This construction makes assumptions about the signal outside the interval unnecessary. Furthermore, we provide a Python package implementing this wavelet decomposition. Wavelets defined on a bounded interval are widely used for signal analysis, compression, and for numerical solution of differential equations. We show that for many cases using the basis that we derive results in smaller error than the commonly used alternative.

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Section: Resultsmentioning

confidence: 99%

Section: Derivation Of Boundary Wavelet Scaling Functionsmentioning

confidence: 99%

Section: Package Overviewmentioning

confidence: 99%

Section: Package Overviewmentioning

confidence: 99%

Section: Comparison To Classical Wavelet Approachmentioning

confidence: 99%

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Orthonormal, moment preserving boundary wavelet scaling functions in Python

Holm¹,

Arildsen²,

Nielsen³

et al. 2020

SN Appl. Sci.

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Non-uniform Recovery Guarantees for Binary Measurements and Infinite-Dimensional Compressed Sensing

Thesing

Hansen

2021

J Fourier Anal Appl

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Due to the many applications in Magnetic Resonance Imaging (MRI), Nuclear Magnetic Resonance (NMR), radio interferometry, helium atom scattering etc., the theory of compressed sensing with Fourier transform measurements has reached a mature level. However, for binary measurements via the Walsh transform, the theory has long been merely non-existent, despite the large number of applications such as fluorescence microscopy, single pixel cameras, lensless cameras, compressive holography, laser-based failure-analysis etc. Binary measurements are a mainstay in signal and image processing and can be modelled by the Walsh transform and Walsh series that are binary cousins of the respective Fourier counterparts. We help bridging the theoretical gap by providing non-uniform recovery guarantees for infinite-dimensional compressed sensing with Walsh samples and wavelet reconstruction. The theoretical results demonstrate that compressed sensing with Walsh samples, as long as the sampling strategy is highly structured and follows the structured sparsity of the signal, is as effective as in the Fourier case. However, there is a fundamental difference in the asymptotic results when the smoothness and vanishing moments of the wavelet increase. In the Fourier case, this changes the optimal sampling patterns, whereas this is not the case in the Walsh setting.

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