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 ...
- 8,815
27
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, ...
- 8,815
18
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 ...
- 2,359
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 (...
- 31.7k
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}...
- 6,080
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 ...
- 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 ...
- 8,815
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, ...
- 31.7k
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 ...
- 31.7k
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 <...
- 8,815
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 ...
- 2,028
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 ...
- 31.7k
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 \...
- 8,815
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 ...
- 8,815
6
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 ...
- 160
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 ...
- 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). ...
- 31.7k
5
votes
Is R suitable for digital signal processing
Since the bulk of R’s DSP capability comes from the signal package which was ported over from the open source project Octave (itself influenced by MATLAB), there's no intrinsic limitation of R.
What ...
- 1,770
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 ...
- 89k
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 ...
- 8,815
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 ...
- 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 (...
- 8,815
4
votes
Accepted
Daubechies wavelet transform
Looks like you need a general explanation of the discrete wavelet transform (DWT). DWT breaks a signal down into subbands distributed evenly in a logarithmic frequency scale, each subband sampled at a ...
- 13.5k
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} ...
- 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 ...
- 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 ...
- 31.7k
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$-...
- 31.7k
4
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) \...
- 141
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. ...
- 31.7k
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 ...
- 29.6k
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