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Overview
The short answer is that they have the maximum number of vanishing moments for a given support (i.e number of filter coefficients). That's the "extremal" property which distinguishes Daubechies wavelets in general. Loosely speaking, more vanishing moments implies better compression, and smaller support implies less computation. In fact, the ...
15
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 '...
14
L1 norm minimization (compressed sensing) can do a relative better job than conventional Fourier denoising in terms of preserving edges.
The procedure is to minimize an objective function
$$
|x-y|^2 + b|f(y)|
$$
where $x$ is the noisy signal, $y$ is the denoised signal, $b$ is the regularziation parameter, and $|f(y)|$ is some L1 norm penalty. ...
13
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 ...
12
As I understand it, the normalization is because the Haar wavelet conserves energy of the signal. In that, when you take signal from one domain to another, you aren't supposed to add energy to it, (although conceivably you might lose energy).
The normalization is just a way to ensure that the energy of your Haar-transformed signal in the Haar-domain has the ...
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 Gabor wavelet is basically the same thing. It's apparently another name for the Modified Morlet wavelet. Quoting from Wavelets and Signal Processing:
[The Modified Morlet wavelet] does not satisfy the admissibility
condition but is nonetheless commonly used. Sometimes this wavelet is
called the "Gabor wavelet," but that term is improper because ...
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
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
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 ...
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
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
You can consider anisotropic diffusion. There are many methods based on this technique. Generally spoken, it is for images. It is an adaptive denoising method which aims to smooth non-edge parts of an image, and preserve edges.
Also, for Total variation minimization, you can use this tutorial. Authors provide MATLAB code also. They recognize the problem as ...
7
1) Create a 50 Hz sinusoid and then simply add it to your ECG signal. You can control the power of the 50 Hz noise by multiplying the sinusoid by some gain factor (can be less than or more than 1) before you add it to the ECG.
2) I'm not familiar with the Welch periodogram, but if it displays the power spectral density then it should do fine. I would just ...
7
A moment is a generalization of the notion in physics of moment of a (point) mass about an axis being the product of the mass and the distance from the axis.
For a continuous random variable $X$ with probability density function $f(x)$, the $n$-th moment is
$$m_n = \int_{-\infty}^\infty x^n f(x)\,\mathrm dx.$$
The zero-th moment is $1$ (the area under ...
6
Simply speaking both the const-Q-transform and the Gabor-Morlet wavelet-transform are just continuous wavelet transforms. Or, more precisely, approximations thereof, as there will always be discretization issues in real applications.
A property of wavelet transforms is that they have build in the constant Q-factor property, or in other words logarithmic ...
6
Chaohuang has a good answer, but I will also add that one other method that you can use would be via the Haar Wavelet Transform, followed by wavelet co-efficient shrinkage, and an Inverse Haar Transform back to the time-domain.
The Haar wavelet transform decomposes your signal into co-efficients of square and difference functions, albeit at different ...
6
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 ...
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, ...
6
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 ...
6
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, ...
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
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 ...
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