Skip to main content
34 votes
Accepted

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
29 votes
Accepted

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
19 votes
Accepted

What Is the Difference between Difference of Gaussian, Laplace of Gaussian, and Mexican Hat Wavelet?

Laplace of Gaussian The Laplace of Gaussian (LoG) of image $f$ can be written as $$ \nabla^2 (f * g) = f * \nabla^2 g $$ with $g$ the Gaussian kernel and $*$ the convolution. That is, the Laplace ...
Cris Luengo's user avatar
  • 2,700
9 votes
Accepted

Which transform most closely mimics the human auditory system?

In designing such transformations, one should take into account competing interests: fidelity to the human auditory system (that varies with people), including non-linear or even chaotic aspects (...
Laurent Duval's user avatar
9 votes

Factor $|a|^{-1/2}$ in definition of mother wavelets

My answer is for real scale $a$ and the fact that wavelet transform is usually defined in $L_2$ with norm $$||\Psi(\tau)|| = \int_\mathbb{R} \Psi(\tau)\Psi^*(\tau)\mathrm{d}\tau $$ So $$||\Psi_{a,t}...
AlexTP's user avatar
  • 6,605
9 votes

Discrete wavelet transform; how to interpret approximation and detail coefficients?

Wavelet transforms can be more difficult to interpret than FFT at face value due to the various representations, nomenclature and output formats. I had to study more than 15 resources to get a good ...
khuang834's user avatar
  • 191
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
Accepted

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
Accepted

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
Accepted

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

What's the difference between the Gabor and Morlet wavelets?

For those looking for a compact description, I found this nice docstring while inspecting the kymatio GitHub repository: A Morlet filter is the sum of a Gabor ...
fr_andres's user avatar
  • 170
7 votes
Accepted

Slow Down Music Playing While Maintaining Frequency

Yeah some of us can do it, you can speed up or slow down without affect the pitch, some guys call this applications of Time Stretch, there different ways to do it, you can do in frequency domain or ...
ederwander's user avatar
  • 2,038
7 votes
Accepted

Is there an equivalent of Parseval's theorem for wavelets?

Yes indeed! In theory as long as the wavelet is orthogonal, the sum of the squares of all the coefficients should be equal to the energy of the signal. In practice, one should be careful that: the ...
Laurent Duval's user avatar
7 votes
Accepted

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
Accepted

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
6 votes
Accepted

Shift invariant in wavelet

Let's say you have a signal which is all zeros except for a spike at one point where x(8)=1 (total N=32, for example). If you perform the DWT on this signal and then calculate the total energy (by ...
BLOKDAK's user avatar
  • 113
6 votes

What Is the Difference between Difference of Gaussian, Laplace of Gaussian, and Mexican Hat Wavelet?

The Ricker wavelet, the (isotropic) Marr wavelet, the Mexican hat or the Laplacian of Gaussians belong to be the same concept: continuous admissible wavelets (satisfying certain conditions). ...
Laurent Duval's user avatar
6 votes

What Is the Difference between Difference of Gaussian, Laplace of Gaussian, and Mexican Hat Wavelet?

Let's see how DoG approximates LoG for the 2D case (for an image, e.g.). By derivative theorem of convolution (by associativity and commutativity), $$\nabla^2[f(x, y) \ast G(x, y)] = \nabla^2 G(x, y) \...
Sandipan Dey's user avatar
5 votes
Accepted

"Fourier Transform can localize signals in frequency domain, but not in time domain." -- What does it mean in layman's terms?

In the Fourier transform, the basis functions are complex exponentials. These functions are perfectly localized in the frequency domain, i.e., they exist at one frequency, but they have no time ...
Matt L.'s user avatar
  • 90.5k
5 votes
Accepted

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
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
  • 1,770
5 votes
Accepted

What is "spin" for the 2D (separable) Morlet?

cause it s p i n Explanation, ground up When there's complex numbers, there's rotation. Recall, multiplying by $e^{j\theta}$ rotates a number by $\theta$ radians: and since $|e^{jX}|=1$ for any (...
OverLordGoldDragon's user avatar
4 votes

How to Map CWT to Synchrosqueezed wavelet transform?

Let me explain the intuition briefly. The authors of the paper you've cited assume that the signal $x(t)$ can be written in the form \begin{align*} x(t) &= \sum_{k=1}^K a_k(t) \exp(2\pi\mathrm{i} ...
Alex's user avatar
  • 41
4 votes

Slow Down Music Playing While Maintaining Frequency

The tool/theory you describe is really a large area of research in music technology, broadly called audio time-scale modification. A large component of this field is how you might prevent audible ...
Speedy's user avatar
  • 436
4 votes

"Fourier Transform can localize signals in frequency domain, but not in time domain." -- What does it mean in layman's terms?

To localize here means: to find where the signal is mostly concentrated, and with what precision. This could be either in the time or the frequency domain. An answer could be: the signal's center of ...
Laurent Duval's user avatar
4 votes
Accepted

Difference(relation?) between filter banks and wavelet decomposition

If we stick to the linear version and discrete versions of filter banks and wavelets, filter banks represent the generic tool, and wavelets can be implemented as a specific instance of iterated $2$-...
Laurent Duval's user avatar
4 votes
Accepted

What are the known use cases for different wavelet families?

Answer 0: ask yourself if you really need wavelets Say yes. Let us concentrate on 2-band real discrete wavelet first. JPEG 2000 is a special case, where CDF9/7 and 5/3 biorthogonal wavelets are used. ...
Laurent Duval's user avatar
4 votes
Accepted

Bandpass filter to get EEG frequency bands?

As the frequency bands are simple frequency ranges, I wonder if I can use several bandpass filters to get them (instead of using WPT / FFT)? Sure! That's how it's usually done! Is there any ...
Marcus Müller's user avatar
4 votes

Audio time stretching, without pitch shifting

The phase vocoder was first implemented by Flanagan and Golden[1] using analog filters bank, later Portnoff apply the same concept digitally using FFT. The phase vocoder use the difference of ...
ederwander's user avatar
  • 2,038
4 votes
Accepted

What is the difference between Constant-Q Transform and Wavelet Transform and which is better

A Constant Q transform is a variation on the DFT. In other words, it is a type of wavelet transform. I only have a casual understanding of both types of transforms myself, so take what I'm saying with ...
Lowell Camp's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible