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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 interpretations are provided.
If unfamiliar with CWT, I recommend this tutorial. SSWT is implemented in MATLAB as wsst, and in Python, ssqueezepy. (-- All answer code)
...
12
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 of the image smoothed by a Gaussian kernel is identical to the image convolved with the Laplace of the Gaussian kernel. This convolution can be further expanded, ...
10
I think it is kind'a similar to soft and hard thresholding using in wavelet de-noising. Have you come across this topic? pywt has already an in-built function for this purpose. Please take a closer look at this code and try to play with it:
import pywt
import matplotlib.pyplot as plt
import numpy as np
ts = [2, 56, 3, 22, 3, 4, 56, 7, 8, 9, 44, 23, 1, 4, 6,...
9
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 (tinnitus)
easiness of the mathematical formulation for the analysis part
possibility to discretize it or allow fast implementations
existence of a suitable stable ...
9
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
$$||\Psi_{a,t}(\tau)|| = \int_\mathbb{R} \frac{1}{|a|}\Psi(\frac{\tau-t}{a})\Psi^*(\frac{\tau-t}{a})\mathrm{d}\tau$$
Set $\tau' = \frac{\tau-t}{a} \implies d\tau' = d\tau / ...
8
It is important to define the time and frequency widths $\Delta_t$ and $\Delta_{\omega}$ of a signal before discussing any special forms of the uncertainty principle. There is no unique definition of these quantities. With appropriate definitions it can be shown that only the Gaussian signal satisfies the uncertainty principle with equality.
Consider a ...
8
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 non-stationary features of data, and they both have merits or drawbacks, depending on their parameters and signal's properties. STFT is typically analyzing ...
7
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 sense of the variety and which one is used by Pywavelets (which does not provide much theory or explanation in its documentation).
In order to grasp the meaning ...
7
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, robust to noise, and stable against time-warping deformations - proving useful in many classification tasks and attaining SOTA on limited datasets.
Core results ...
6
The other answers are good, but I thought that I would try to give a more intuitive/visual answer since I am an intuitive/visual guy.
The picture below is the plot of two tones that are almost the same frequency. One tone is plotted in red, and the other in blue.
I generated the picture in Matlab with the following code:
tone1 = sin(2*pi*.05 * (0:99));
...
6
Both JPEG and JPEG 2000 use the change of basis compression type.
Namely, we transform the data into a different representation assuming in this representation the number of parameters needed to describe to data is lower.
Or to the least, most of the information is gathered within few parameters.
Now, if you look at the energy level of the DCT coefficients ...
6
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 time domain, you will need choose what is best for you, you will find some advantages and disadvantages of each.
Time Domain:
In Time Domain you can try some ...
5
The short-time Fourier transform is generally a redundant transformation, usually implemented with the same subsampling over every frequency. If the window is well chosen, it is complete: you can invert it and recover any initial signal.
Since it is redundant and complete, it has many perfect inverses. It can be implemented and understood using more ...
5
If you approximate the Fourier transform
$$X(f)=\mathcal F(x)(f)=\int_{-\infty}^\infty x(t)\,e^{-2\pi j\,ft}\,dt$$
by the discrete Fourier transformation for by sampling on the time segment $[-T,T]$ as
$$X(f_n)\approx \sum_{k=-N}^{N-1} x(k\tau)\,e^{-2\pi j\,f_nk\tau}\,\tau=s[n]\,\tau$$
with $T=N\tau$, $f_n=n/(N\tau)=n/N*f_s=n/T$, $n=-N,...,N-1$, $s$ the ...
5
complex morlet was added Aug 10, 2007
ricker and cwt were added Sep 20, 2011
There's no indication that cwt is meant to be compatible with morlet. As cwt docstring says:
Wavelet function, which should take 2 arguments. ... second is a width parameter, defining the size of the wavelet (e.g. standard deviation of a gaussian).
The morlet function takes 4 ...
5
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 you have picked up on, are ecosystem preferences. We learned MATLAB in university, picked up numpy/scipy/sklearn at work, so R isn't the first weapon of choice.
...
5
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 taking the square root of the sum of the squares of all the results), you will get a value - call it "E1".
Now, let's take another signal which is still all ...
5
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 localization because of their infinite duration. The localization of a function depends on its spread in time and frequency. A complex exponential has zero spread in ...
5
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). Traditionally, the Ricker wavelet is the 1D version. The Marr wavelet or the Mexican hat are names given in the context of 2D image decompositions, you can consider ...
5
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 decomposition is not "expansive", i.e. the number of samples and of coefficients is the same.
wavelet filter coefficients are not re-scaled, as happens in some ...
5
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 naturally to more complex signals.
Band of lines concentrated about a maximum in CWT, and a perfect line in ssqueezed right about f=8. Next, the phase ...
4
Wavelets are ideal for localized events. The Fourier Transform represents a function as a sum of sines and cosines, neither of which are localized. The spectrogram does keep some time information, at the expense of frequency resolution
In your case, the signal is not localized at all. The spectrogram smears your 15 Hz band over several Hz, as it captures ...
4
A transform being linear has very little to do with its ability to analyze linear or nonlinear systems.
The wavelet transform $W[s(t)]$ of a signal $s(t)$ is linear because $$W[a s_1(t) + b s_2(t)]=a W[s_1(t)]+b W[s_2(t)]$$ for real or complex $a$ and $b$.
The signal you're analyzing is just a signal, it has no concept of linearity. However, if you try to ...
4
Shannon entropy is a way of measuring the degree of unexpectedness or unpredictability of a random variable. For example rolling a die has higher entropy than flipping a coin because the die has more possible outcomes making it harder to predict. Same goes for a biased coin versus a fair coin.
4
It's the uncertainty principle.
http://en.wikipedia.org/wiki/Uncertainty_principle#Harmonic_analysis
4
The Conjugation is part of the definition of the Convolution as an Inner Product operation.
So you wrote the operation correctly.
Usually people uses "Real" Wavelets hence no need for that.
Anyhow, the definition is an inner product (Projecting the function onto the base of the wavelets) and it requires the conjugation operation.
4
Indeed, that's the Heisenberg Uncertainty Principle - you can't have both very good frequency and time resolution. You always have to sacrifice something. In case of Short Time Fourier Transform it's straightforward, but for wavelets are being 'squeezed', which is changing their frequency resolution. Figure below describes more than a thousand words:
EDIT:
...
4
Wavelets are not key to denoising. There are different ways to denoise an image, for example in the original signal domain or in the transform domain (i.e. Fourier or wavelet). Wavelets work best for additive noise, where the noise is random & not correlated in time.
wavelet_denoise (float *fimg[3], unsigned int width,
unsigned int height, float ...
4
JPEG is far simpler. It divides the image into 8x8 pixel blocks, and processes each using a Discrete Cosine Transform. The results are quantised and then encoded. The quality is fixed by the encoder.
JPEG2000 uses a 2D wavelet function, the output of which is four "images", each a quarter the size of the original. One of those is actually an image, ...
4
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 rate proportional to the frequencies in that band. The traditional Fourier transformation has no time domain resolution at all, or when done using many short ...
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