Phase Retrieval for the Cauchy Wavelet Transform

Mallat, Stéphane; Waldspurger, Irène

doi:10.1007/s00041-015-9403-4

Cited by 64 publications

(69 citation statements)

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“…But we know that the reconstruction of a function from the modulus of its wavelet transform is not stable to noise [Mallat and Waldspurger, 2014]. So we do not hope the difference between f and f rec to be small.…”

Section: A Experimental Settingmentioning

confidence: 99%

“…It has been established in [Mallat and Waldspurger, 2014] that, when reconstruction is unique, it is also stable, in the sense that the reconstruction operator is continuous (although it may not be uniformly continuous 2016b]), it was shown that the reconstruction operator actually enjoyed a stronger property of local stability (the stability constants depend on the wavelet family; reconstruction is more stable when the wavelets are more redundant, that is, when the overlap between their Fourier supports is large).…”

Section: Well-posedness Of the Inverse Problemmentioning

confidence: 99%

“…From [Mallat and Waldspurger, 2014], we know that the reconstruction operator is not uniformly continuous: the reconstruction error (13) can be small (the modulus of the wavelet transform is almost exactly reconstructed), even if the error on the signal (14) is not small (the difference between the initial signal and its reconstruction is large). We show that this phenomenon can occur for all classes of signals, but is all the more frequent when the wavelet transform has a lot of small values, especially in the low frequency components.…”

Section: Stability Of the Reconstructionmentioning

confidence: 99%

“…In the case of the scalogram (or spectrogram), a theoretical study of the well-posedness, for relatively general wavelets, seems out of reach. However, we can reasonably conjecture that the inverse problem is well-posed for generic wavelet families.Indeed, it has been proven in several cases that reconstruction is unique, as soon as the sampling rate is high enough (for Cauchy wavelets [Mallat and Waldspurger, 2014] and for real band-limited wavelets [Alaifari et al, 2016a]; in the case of the spectrogram, it is known, for general windows, that at least almost all signals are uniquely determined [Jaganathan et al, 2016]). By contrast, there is no case in which uniqueness is known not to hold, except for families which exhibit a severe lack of redundancy (Shannon wavelets, for example).…”

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Phase retrieval for wavelet transforms

Waldspurger

2017

IEEE Trans. Inform. Theory

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Abstract-This article describes a new algorithm that solves a particular phase retrieval problem, with important applications in audio processing: the reconstruction of a function from its scalogram, that is, from the modulus of its wavelet transform.It is a multiscale iterative algorithm, that reconstructs the signal from low to high frequencies. It relies on a new reformulation of the phase retrieval problem, that involves the holomorphic extension of the wavelet transform. This reformulation allows to propagate phase information from low to high frequencies.Numerical results, on audio and non-audio signals, show that reconstruction is precise and stable to noise. The complexity of the algorithm is linear in the size of the signal, up to logarithmic factors. It can thus be applied to large signals.Index Terms-Phase retrieval, scalogram, iterative algorithms, multiscale method I. INTRODUCTIONThe spectrogram is an ubiquitous tool in audio analysis and processing, eventually after being transformed into melfrequency cepstrum coefficients (MFCC) by an averaging along frequency bands. A very similar operator, yielding the same kind of results, is the modulus of the wavelet transform, sometimes called scalogram.The phase of the wavelet transform (or the windowed Fourier transform in the case of the spectrogram) contains information that cannot be deduced from the single modulus, like the relative phase of two notes with different frequencies, played simultaneously. However, this information does not seem to be relevant to understand the perceptual content of audio signals [Balan et al., 2006;Risset and Wessel, 1999], and only the modulus is used in applications. To clearly understand which information about the audio signal is kept or lost when the phase is discarded, it is natural to consider the corresponding inverse problem: to what extent is it possible to reconstruct a function from the modulus of its wavelet transform? The study of this problem mostly begun in the early 80's [Griffin and Lim, 1984;Nawab et al., 1983].On the applications side, solving this problem allows to resynthesize sounds after some transformation has been applied to their scalogram. Examples include blind source separation [Virtanen, 2007] or audio texture synthesis [Bruna and Mallat, 2013].The reconstruction of a function from the modulus of its wavelet transform is an instance of the class of phase retrieval problems, where one aims at reconstructing an unknown signal x ∈ C n from linear measurements Ax ∈ C m , whose phase has been lost and whose modulus only is available, |Ax|. These problems are known to be difficult to solve.The author is with CNRS and CEREMADE, Université Paris Dauphine (e-mail: waldspurger@ceremade.dauphine.fr).Two main families of algorithms exist. The first one consists of iterative algorithms, like gradient descents or alternate projections [Fienup, 1982;Gerchberg and Saxton, 1972]. In the case of the spectrogram, the oldest such algorithm is due to Griffin and Lim [Griffin and Lim, 1984]. These methods are simp...

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Section: A Experimental Settingmentioning

confidence: 99%

Section: Well-posedness Of the Inverse Problemmentioning

confidence: 99%

Section: Stability Of the Reconstructionmentioning

confidence: 99%

mentioning

confidence: 99%

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Phase retrieval for wavelet transforms

Waldspurger

2017

IEEE Trans. Inform. Theory

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“…And it is an ideal tool for signal time-frequency analysis and processing [7,8] . Mallat proposed the Mallat fast wavelet decomposition and reconstruction algorithm based on multi-resolution analysis in 1987 [9] , improving the computing speed of wavelet transform enormously. Since it was discovered in 1990s that network traffic is multi-scale, wavelet transform has been introduced in the research of network traffic analysis because wavelet transform is taken for an effective way of analyzing and describing fractal characteristics.…”

Section: Wavelet Transform Analysismentioning

confidence: 99%

A Traffic Classification Method Based on Wavelet Spectrum of Scatter Factor and Improved K-means

Fei¹,

Wang²,

He³

et al. 2017

dtetr

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Based on the problem that supervised machine learning requires labeled samples and fails to identify unknown traffic, the author innovatively integrates wavelet transform and K-means algorithm of unsupervised machine learning by combining the advantage of wavelet transform in solving multi-fractal network traffic and proposes a traffic identification method based on wavelet spectrum of scatter factor and improved K-means. This method represents each stream sequence with wavelet spectrum of scatter factor, which is taken as the input of clustering algorithm. The author carries out a cluster analysis with GA K-means algorithm. The experimental result suggests that this method has an obvious superiority in stability and accuracy of classification.

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Phase retrieval of finite Blaschke projection

Zhou

2020

Math Methods in App Sciences

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Phase retrieval by Fourier measurements is a classical application in coherent diffraction imaging, and the modified Blaschke products (MBPs) are the generalization of linear Fourier atoms. Motivated by this, we investigate the phase retrieval modeled as to reconstruct scriptPfalse(ffalse)=∑k=0∞⟨f,Bfalse{a0,a1,…,akfalse}⟩Bfalse{a0,a1,…,akfalse} by the intensity measurements false{false|⟨f,Bk1⟩false|,false|⟨f,Bk2⟩false|,false|⟨f,Bk3⟩false|:k≥1false}, where f lies in Hardy space ℋ2false(scriptDfalse) such that f(a0)=0, Bfalse{a0,a1,…,akfalse} and Bki are all the finite MBPs. We establish the condition on Bki such that scriptPfalse(ffalse) can be determined, up to a unimodular scalar, by the above measurements. A byproduct of our result is that the instantaneous frequency of the target can be exactly reconstructed by the above intensity measurements. Moreover, a recursive algorithm for the phase retrieval is established. Numerical simulations are conducted to verify our result.

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Phase Retrieval for the Cauchy Wavelet Transform

Cited by 64 publications

References 16 publications

Phase retrieval for wavelet transforms

Phase retrieval for wavelet transforms

A Traffic Classification Method Based on Wavelet Spectrum of Scatter Factor and Improved K-means

Phase retrieval of finite Blaschke projection

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