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34 votes
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Synchrosqueezing Wavelet Transform explanation?

Synchrosqueezing is a powerful reassignment method. To grasp its mechanisms, we dissect the (continuous) Wavelet Transform, and how its pitfalls can be remedied. Physical and statistical ...
OverLordGoldDragon's user avatar
28 votes
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Wavelet Scattering explanation?

Wavelet Scattering is an equivalent deep convolutional network, formed by cascade of wavelets, modulus nonlinearities, and lowpass filters. It yields representations that are time-shift invariant, ...
OverLordGoldDragon's user avatar
9 votes

Synchrosqueezing Wavelet Transform explanation?

Low-level intuition can be obtained by inspecting the phase transform, visually. Answer complements and is complemented by this one. (-- Answer code) We consider a pure sinusoidal tone; ideas extend ...
OverLordGoldDragon's user avatar
8 votes
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Continuous Wavelet Transform vs Discrete Wavelet Transform

On the one hand with the DWT, only a restricted choice of wavelets is available: those that implement 2-band perfect reconstruction (Daubechies, Symmlets, Coiflets, Spline). They are non-redundant, ...
Laurent Duval's user avatar
8 votes
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Advantage of STFT over wavelet transform

Wavelet transforms and short-term/short-time Fourier transforms are broad names for classes of transformations that are not totally distinct and may overlap (pun intended). Both can be efficient for ...
Laurent Duval's user avatar
8 votes
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How to validate a wavelet filterbank (CWT)?

Wavelets isn't just sampling with scales from some min to max - but it is what many implementations do, including scipy and <...
OverLordGoldDragon's user avatar
7 votes
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Wavelet Scattering properties & implementation?

Scattering overview provided in this answer. Computational structure Fig 4, Deep Scattering Spectrum In steps: (First order begins) $x$ convolves with $\psi1_i$ --> $W1_i$ Modulus, $W1_i \...
OverLordGoldDragon's user avatar
7 votes
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Why does a signal with constant frequency have spots that changes colors at a specific value of scale (and so frequency) in the scalogram?

Re: real part There are oscillations because that's what the wavelet transform is - a decomposition into zero-mean, localized oscillations. CWT is convolution (rather, cross-correlation) of signal ...
OverLordGoldDragon's user avatar
5 votes

Continuous Wavelet Transform vs Discrete Wavelet Transform

The CWT & DWT implementations differ in how they discretize the scale parameter used to stretch or shrink copies of the basic wavelet. The finer grain scale parameter in the CWT can be useful for ...
ruoho ruotsi's user avatar
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5 votes
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Wavelet "center frequency" explanation? Relation to CWT scales?

The "frequency" in "center frequency" does refer to Fourier frequency - rate of sinusoidal oscillation - but measures (and interpretations) can vary depending on the wavelet. Said ...
OverLordGoldDragon's user avatar
4 votes
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Inverse Continuous Wavelet Transform derivation?

Summary: the dual wavelet's role is analogous to that of $e^{j\omega t}$; it undoes the wavelet's convolving with the signal (integrated inner product). The main intricacy's indeed in normalization; ...
OverLordGoldDragon's user avatar
4 votes

Continuous Wavelet Transform vs Discrete Wavelet Transform

Fundamentally: DWT is orthogonal, CWT is redundant. Former packs the most information per sample, latter spreads out its decomposition. As will be explained: CWT yields vastly superior analysis ...
OverLordGoldDragon's user avatar
4 votes

Advantage of STFT over wavelet transform

STFT is frequency-shift equivariant - same absolute shift has same effect on representation regardless of original frequency${}^1$: $$ \hat x(\omega) \rightarrow \hat x(\omega - c) \Leftrightarrow \...
OverLordGoldDragon's user avatar
3 votes

Wavelet "center frequency" explanation? Relation to CWT scales?

The document “The Continuous Wavelet Transform: A Primer” by Luís Aguiar-Conraria and Maria Joana Soares, which you are referring to, says it well: this [standard] inverse relation between scale and ...
Laurent Duval's user avatar
3 votes
Accepted

Scalograms in python

scipy's cwt is primitive and error prone; below is via ssqueezepy.cwt: Code: Note that if you seek to code the wavelet yourself,...
OverLordGoldDragon's user avatar
3 votes

Why does a synchrosqueezed wavelet transform show oscillating behavior?

This answer delves deeper into low-level aspects of the phase transform to better understand the wavy phenomenon; complements main answer. 2. How wavy is w? Recall,...
OverLordGoldDragon's user avatar
3 votes
Accepted

Why does a synchrosqueezed wavelet transform show oscillating behavior?

This was interesting to figure out. The key lies in the phase transform, and how CWT interacts with own derivative upon insufficient component separation. Relevant are, and I'll be answering, the ...
OverLordGoldDragon's user avatar
3 votes
Accepted

How is wavelet center frequency computed?

Short version: DFT's bin indices are input length-dependent; "center frequency" is measured relative to the function generating the wavelet. Generated length can vary, so must be accounted ...
OverLordGoldDragon's user avatar
3 votes
Accepted

Joint Time-Frequency Scattering explanation?

JTFS is an extension of Wavelet Scattering that exploits time-frequency structure, adding sensitivity to frequency-dependent time shifts, invariance to frequency transposition, and stability against ...
OverLordGoldDragon's user avatar
3 votes

Inverse of wavelet transform modulus gives poor results

Inverting CWT modulus The problem is known as phase retrieval, and for |CWT|, the transform is proven invertible to within a global phase shift, $e^{j\omega_0}$, in ...
OverLordGoldDragon's user avatar
2 votes

Wavelet Transform and STFT

Accepted answer is wrong: DWT (actually CWT) plot y-axis must read frequency not scales; the two are inversely related. CWT and STFT aren't equivalentlish-ly similar as suggested; the same plot ...
OverLordGoldDragon's user avatar
2 votes
Accepted

Wavelet Transform and STFT

In the STFT, you apply windowing and Fourier transform on the signal using sliding patches and then combine the resulting transforms, which will help you eventually end up with a uniform time/...
Husrev's user avatar
  • 67
2 votes
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How is wavelet time & frequency resolution computed?

The key is units, and understanding wavelet behavior in context of application (in this case CWT). Full implementations for all discussed here is available at squeezepy. This answer assumes analytic ...
OverLordGoldDragon's user avatar
2 votes

STFT with varying window lengths (like the continuous wavelet transform)

This is sometimes done when creating a log frequency scaled spectrogram (which might better match human time versus pitch perception). One issue with using multiple sizes of STFTs it that this results ...
hotpaw2's user avatar
  • 35.3k
2 votes

What is the importance of the translational invariance of the CWT?

CWT is translation-invariant in feature sense: translating a pattern translates its representation but not modify it. In coefficient sense, it is translation equivariant: shift signal $\Leftrightarrow$...
OverLordGoldDragon's user avatar
2 votes
Accepted

What is the importance of the translational invariance of the CWT?

First, translational invariance makes signal processing somehow independent on the time origin of the recording. The term "invariance" might be debated, as done here: What is the difference ...
Laurent Duval's user avatar
2 votes
Accepted

What exactly is meant by "translation invariant dictionaries/wavelets"?

It's rather translation equivariant: $$ \text{CWT}_{s, t}x(t - t_0) = \text{CWT}_{s, t - t_0}x(t) \tag{1} $$ and $$ \langle x(t−t_0),\psi(t) \rangle= \langle x(t), \psi(t+t_0)\rangle \tag{2} $$ That ...
OverLordGoldDragon's user avatar
2 votes

Advantage of STFT over wavelet transform

In response to the reply on unifying the STFT and CWT (I can't comment yet): I keep a recoded version of ARSS over here https://github.com/LydiaMarieWilliamson/ARSS though it will undergo rebasing and ...
Lydia Marie Williamson's user avatar
2 votes

Calculating signal power from Continuous Wavelet Transform in MATLAB

For the DWT, the energy is preserved only when the discrete wavelet is orthogonal, since orthogonality allows $L^2$-norm isometry. When it is not, like for biorthogonal wavelets, the relationship ...
Laurent Duval's user avatar
2 votes

Does Fast Continuous Wavelet Transform (fCWT) have theory-supported novelty or just simply a computation optimization?

I've modestly reviewed the paper. I'm skeptical of its speedups and implementation accuracy. It includes time of sampling the wavelets in benchmarks, which is valid, but arguably the main use case is ...
OverLordGoldDragon's user avatar

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