Hot answers tagged

17

All three transforms are inner product transforms, meaning the output is the inner product of a family of basis functions with a signal. The parametrization and form of the basis functions determine the properties of the transforms.The number of basis functions for a complete picture (i.e. a result that contains enough information to reconstruct the original ...


16

Imagine for one second, that you just plotted your daubechies-4 wavelet, as you can see here in red. Now imagine that you take this waveform in red, and simply do a cross-correlation with your signal. You plot that result. This will be the first row of your plot. This is scale-1. Next, you dilate your Daubechies-4 wavelet, (that is, you simply make it '...


11

These plots were helpful for me to understand, coming from a STFT background: The complex Morlet (sinusoidal) wavelet looks and behaves like the complex kernel of a STFT (since it's derived from the Gabor transform, a type of STFT). When you "slide it past" a signal of the same frequency, it matches, no matter the phase of the signal you're measuring, ...


10

The short-time Fourier transform doesn’t offer better analysis of data than the discrete Fourier transform, it offers a different kind of analysis. The DFT offers an exact decomposition of data to a frequency representation. The STFT offers an approximate decomposition to a time/frequency representation. Which is better depends on what you are after. The ...


10

As Mohammad stated already the terms Continuous Wavelet Transforms (CWT) and Discrete Wavelet Transforms (DWT) are a little bit misleading. They relate approximately as (Continuous) Fourier Transform (the math. integral transform) to DFT (Discrete Fourier Transform). In order to understand the details it is good to see the historical context. The wavelet ...


10

Remember that Wavelet Transforms are nothing but time-localized filtering/correlation operations. The wavelet transforms provide a unified framework for getting around the Heisenberg Uncertainly Principle that the Fourier Transform suffers from. So when you ask "what should my settings be for bandwidth, and center frequency", you are asking for filter ...


10

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, ...


9

Continuous wavelet transform is suitable for a scalogram because the analysis window can be sized and placed at any position. This flexibility allows for the generation of a smooth image in both the time in scale (analogous to frequency) directions. The continuous wavelet transform is a redundant transform because the analysis window can overlap. In fact ...


9

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

A very common yet unfortunate mis-conception in the field of wavelets has to do with the ill-coined terminology of "Continuous Wavelet Transforms". First thing's first: The Continuous Wavelet Transform, (CWT), and the Discrete Wavelet Transform (DWT), are both, point-by-point, digital, transformations that are easily implemented on a computer. The ...


8

If you're familiar with Fourier transforms, I think the bridge between the Fourier worlds and the wavelet worlds is the Gabor transform (a Gaussian-windowed STFT) and the complex Morlet wavelet transform. This is historically how they developed, too. They are basically the same thing, breaking down a signal into "blips" of complex sinusoids: But the time-...


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 ...


7

The HUP follows directly from the properties of the Fourier Transform, because time and frequency are orthogonal bases in which we can expand the co-efficient sequence of our signal. In fact all pairs of orthonormal bases will have some kind of Uncertainty Principle associated with them. In traditional Fourier analysis, the either the time axis or the ...


7

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) ...


6

Intuitively speaking, anything that is 'high frequency' is something that is 'rapidly changing in time'. Anything that is 'low frequency' is something that is 'slowly changing in time'. If you think about it, any time you have 'detail' in a signal or image, it means that you have, very quick, rapid variations in time or space. This then becomes the 'detail' ...


6

I think "Introduction to Wavelets and Wavelet Transforms: A Primer" by Sidney Burrus (et al.) is a very good and practical book. It is very clear, has exercises, and contains some Matlab programs. EDIT: I forgot to mention that this paper is also a very nice introduction to wavelets.


6

I don't think there is any difference. The documentation for dwt2 says Single-level discrete 2-D wavelet transform The dwt2 command performs a single-level two-dimensional wavelet decomposition... While the documentation for wavedec2 says Multilevel 2-D wavelet decomposition The difference is that dwt2 is single-level (produces a single A, H, ...


5

It looks like there is a problem with your scaling. The scaling for the DWT has the same interpretation as it does for the CWT. For the DWT, the scale of the analyzing function (wavelet) is increased using a dyadic scale (increasing by factors of 2) in non-overlapping time intervals. Increasing scale corresponds to narrower frequency distributions in ...


5

This is the example that i think is the best to understand Wavelet plot. Have a look at the image below, The Waveform (A) is our original Signal, Waveform (B) shows a Daubechies 20 (Db20) wavelet about 1/8 second long that starts at the beginning (t = 0) and effectively ends well before 1/4 second. The zero values are extended to the full 1 second. The ...


5

you can have a look at the LTFAT's wavelet module http://ltfat.sourceforge.net/doc/wavelets/index.php it runs in Matlab/Octave with backend written in C. It has fairly large database of wavelet filters and new ones can be added easily. What exactly do you mean by 2) Ability to run in parallel - VERY IMPORTANT Should the computation itself be somewhat ...


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

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)); ...


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

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 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

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 ...


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