Questions tagged [wavelets]
For questions related to wavelets and wavelet theory.
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What does it mean for a Wavelet transform to “commute” with a translation?
I'm referencing this paper here: https://arxiv.org/pdf/1203.1513.pdf
Within this paper, it states that "A wavelet transform commutes with translations, and is therefore not translation invariant". ...
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15 views
Please help me understand this paper about Discrete Wavelet Transform!
I am trying to understand this paper says.
I have found this paragraph to be vague:
(a) Preprocessing Stage. The first step in this stage was to apply the
Discrete Wavelet Transform (DWT). This ...
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1answer
26 views
What information could be extracted from this Discrete Wavelet Transform of a time series data?
I have decomposed this time series data in 5 levels using Discrete Wavelet Transform Daubechies 4.
I am trying to understand what kind of information one can extract from applying DWT to a time ...
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9 views
What frequencies do the coefficients of a wavelet decomposition represent?
Every time I decompose a time series signal using Discrete Wavelet Transform I get a signal with half the frequency, right?
Suppose I have a time series signal sampled at 60 Hz.
The first discrete ...
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9 views
How exactly Wavelet Transform will be helpful to analyze a signal?
I have a time series noisy ECG signal data.
My whole intention is to get that data and extract the heartbeat signal from it and discover the BPM rate.
I have managed to remove the noise from it ...
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1answer
21 views
Is this the correct way to perform filtering using Wavelet decomposition?
My math is a little bit rusty. I am learning a lot of stuff recently and some stuff is not clear yet. I ask anybody answering this, to not answer using complex mathematical notation because like I ...
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1answer
28 views
How to proceed from the level 2 and up in a Discrete Wavelet Transform Daubechies kind 4
The wavelet elements of a Daubechies kind 4 are
$$(0.4830, 0.8365, 0.2241, -0.1294) $$
Let's say I have the following signal array
$$ [1, 2, 0, 4, 5, 6, 8, 10 ]$$
Doing a Daubechies 4 DWT of this ...
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32 views
The significant role of dual frames in the progress of Frame theory.
For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$
$$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
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1answer
71 views
Orthogonal wavelet transforms and the L2 norm
Question
I need to prove that:
$$
\text{arg min}_{\mathbf{x}} \dfrac{1}{2\sigma} \| \mathbf{x} - \mathbf{z} \|^2 + \lambda \| W \mathbf{x} \|_1 = W^{-1} \text{arg min}_{\mathbf{w}} \dfrac{1}{2\sigma}...
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34 views
Calculating the Mean Square Error (MSE) in Wavelet Denoising
I´m currently reading the paper (to be more precise: it´s a chapter from the book "Shearlets, Multiscale Analysis of Multivariate Data" by Kutyniok and Labate) "Image Processing Using Shearlets" by G....
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What is purpose of wavelet scaling function and how is it derived for e.g Haar wavelet or Dabuchies wavelet?
Scaling function is also called father wavelet. I understand concept of mother wavelet but not father wavelet. In the continuous wavelet transform there is no concept of scaling function but only when ...
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17 views
Eliminating minor crests/troughs in a tidal curve
I have some tidal data, I only have the turning points of the curve.
I want to eliminate the minor troughs/crests so I only get the high and low tide without the minor variations.
The curve
Is ...
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23 views
Wavelet analog of: fourier transform of derivative of a function is multiplication with polynomial
so I have a question concerning the continuous wavelet transform (please forgive me if this is something very simple however i couldn't seem to find any answer so far):
We know that for the Fourier ...
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1answer
36 views
What does it mean to multiply two sine waves together
Suppose I have a wave pool with two sources, A and B. The two sources both produce sine waves of the same frequency, and the sources are separated by a distance of the period of the wave.
So they ...
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37 views
Construction of wavelets
So far I've gone through the theory of construction of wavelets in finite dimensional case and also got a little bit idea on wavelets on the function space ($L^2(\mathbb R)$). It is clear that all the ...
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6 views
Do there exist discrete and self-orthogonal wavelets on non-cartesian grids?
Most 2D DWT:s that I know about are rather straight-forward separable 1D wavelets, for example Meyer, Daubechies famous maxflat, CDF as used in JPEG-2000, spline wavelets like Unser's and so on.
Has ...
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1answer
46 views
Classical Shearlet system forms a Parseval frame.
I have been reading about shearlets & frames and there is a part that I'd like help with. I have only studied the basics of functional analysis and wavelets.
First few definitions:
Classical ...
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25 views
comparison of wavelet coefficients
Consider a function the following spaces:
$$
\{f : \|(i\omega)k \hat f(\omega)\|_p ≤ 1, k ∈ N ∪ 0, p ∈ (1, ∞)\}.
$$
Denote by $ \psi^m_D$ an orthonormal Daubechies wavelet of order m. One can find ...
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17 views
Wavelet Analysis: A notation problem in this paper
Because of my research work, I am reading this paper: PDE net. But I ran into a notation problem: In definition 2.1, I do not understand how $\textbf{q}[ \textbf{k} ]$ yields a real number. Therefore, ...
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28 views
Encryption using Wavelet transform
I need to know about an article, book or other reference that deals with encryption using Fourier transform and the Wavelet.
(I plan to use matlab or other recommended software.)
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14 views
difference between frequency localization and space localization in wavelet analysis.
I understood the mathematical definition of frequency localization and space localization but I'm not clear pictorially,say for example in order to focus on movement of the hand in cricket we can ...
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59 views
Solving the following linear ODE by a numerical method
ODE:
$$y'(x)+3y(x)=1$$
Initial condition: $y(0)=0$
We know that the exact solution is:
$y \left( t \right) =1/3-1/3\,{{\rm e}^{-3\,t}}.$
My Objective:
I want to solve the ODE by Legendre wavelets ...
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30 views
The integration of Legendre functions
We know the integration of Legendre wavelet function is $\int_{0}^{T}\Psi(s)ds=P.\Psi(t)$. We can find the matrix $P$ as follows.
My question: I want to learn how to find Matrix $P$. I can' t ...
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78 views
Solving 2nd order ODE with variable coefficients
ODE:
$$X''(t)+A(t)X'(t)+B(t)X(t)=F(t),$$
IC's: $$X(0)=U_1, $$
$$X'(0)=U_2$$
where
$X(t)=[X_1 X_2...X_n]^T$ and $U_1, U_2,F(t)$ are $n\times1$ matrices,
$A(t), B(t)$ are $n\times n$ matrices.
...
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25 views
Help to understand wavelet basis on an interval proposed by Cohen, Daubechies and Vial.
I am reading Albert Cohen, Ingrid Daubechies, Pierre Vial. Wavelets on the Interval and Fast Wavelet Transforms, 1993.
In this paper wavelet bases on an interval (for, example on space $L^2[0,1]$) ...
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22 views
Wavelets for preconditioning in MATLAB
I have been trying to understand the Wavelet transform in order to use it as a precondioner for ill-conditioned linear problems of the form $A\vec{x}=\vec{b}$.
I have come across this paper that is ...
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8 views
Rapid evaluation of Daubechies Scaling function
Let the Daubechies 4-tap scaling function $\phi\in C_{0}([0,3])$ be defined by
\begin{align}
\phi(x) &= \frac{1+\sqrt{3}}{4} \phi(2x) + \frac{3+\sqrt{3}}{4}\phi(2x-1) + \frac{3-\sqrt{3}}{4} \phi(...
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47 views
how is DFT the change of basis operator?
I understand that DFT are the coefficients when we write a vector z with respect to the Fourier basis.But the following statements are giving me a vague picture about the idea but not very clear,they ...
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16 views
Amplitude of derivatives approximated by continuous wavelet transform
I do have a question regarding the meaning of the amplitude of an approximated derivative derived from a continuous wavelet transform. From what is known, applying the CWT (Haar wavelet) to a signal ...
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15 views
Generic “wavelet lifting” style matrix factorizations?
As context I will present the wavelet lifting scheme. In matrix terms it is a factorization of convolution that splits into two parts:
Predict ($P$) step.
Update ($U$) step.
$$S=\begin{bmatrix}I&...
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11 views
Invertibility conditions of signal processed by filter banks
Professor Strang, crediting Professor Daubechies, concludes that the conditions for invertibility of a filtered signal are summarized in
$$\begin{align}
F_0(\omega)H_0(\omega) - H_0(\omega+\pi)F_0(\...
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41 views
How to calculate the wavelet frame of Mexican Hat function
I was confused these days when I was learning wavelet frame through the famous Daubechies's book - Ten lectures on wavelets. Specially, I only obtained parts of her listed frame bounds results in ...
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1answer
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Wavelets for signal modulation
How does the Continuous Wavelet Transform handle signal modulation? for instance if an external 8-year period influenced the amplitude of an annual period. i.e. every 8 years the amplitude of the ...
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1answer
41 views
Which Daubechies D6 wavelet to choose?
I was trying to calculate coefficients for D6 wavelet as an exercise. When I did factorization part (where you find polynomial $L(e^{i\omega})$ such that $L(e^{i\omega})L(e^{-i\omega}) = Q(cos \omega)$...
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1answer
37 views
dilation operator on $L^2({\mathbb{R}})$ is continuous
Prove the statement let $D:\mathbb{R}^+\rightarrow L^2(\mathbb{R})$ defined by $D(a)=f_a$ and $f_a(x)=\frac{1}{\sqrt{a}}f(\frac{x}{a})$, where $f\in L^2(\mathbb{R})$ then the mapping $D$ is ...
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23 views
Wavelet (Haar Basis) Decomposition in nonparametric regression
Given vectors $Y = (y_1,.....,y_n), X = (x_1,.....,x_n)$ where $n=2^J$ and $x_i \in [0,1]$, I am told we have a wavelet decomposition of $Y$ in the form
$$Y = W\Theta$$
where $\Theta$ is a vector ...
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36 views
Radial Basis Fn vs. Wavelets (not Neural Networks)
I am interested in parameterizing a surface without a mesh. One technique used in the field of optics is to use Radial Basis Functions (e.g. Gaussians).
From a naive point of view, the decomposition ...
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20 views
Blind Signal Separation for sparse sources
Assume we have $N$ measurements $z_1, ..., z_N \in \mathbb{R}^{n_z} $ that generated by
$$ z_i = M v_i + e_i $$
where $v_i \in \mathbb{R}^{n_v}$, $n_v < n_z$ and $e_i$ an error sampled from ...
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0answers
22 views
Why wavelet transform converts functions into disperse signals?
I am studying wavelet transformations and I have read that this type of transformations converts almost any type of functions (including music or images) into disperse signals. I would like to know ...
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19 views
Boundedness of Fourier transforms for Battle-Lemarie wavelets
Let $\phi$ and $\psi$ be the scaling function and wavelet for the Battle-Lemarie wavelet of order $n$. Is it true that
$$
\sup_{x \in \mathbb{R}} |\hat\phi(x)| < \infty,
$$
$$
\sup_{x \in \mathbb{...
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22 views
Finding a function that fullfills certain conditions
The exercise one is about triadic wavelet decomposition. A triad I think is a group of three elements corresponding somehow with each other. There was nothing about it on the slides.
We have to find $...
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73 views
Shearlets - Understanding Anisotropic Features
For $\psi \in L^2(\mathbb R^2)$, the continuous shearlet system $SH(\psi)$ is defined by $SH(\psi) = \{\psi_{a,s,t} = T_t D_{A_a}D_{S_s}\psi:a>0,s\in \mathbb R, t \in \mathbb R^2$},
where $D_{A_a}...
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35 views
Legendre wavelets for solving Differential Equations
Can Legendre (or Haar) wavelets be used for Solving Linear Systems of Differential Equations with variable coefficients?
If the answer is yes, could you suggest some good references (book, article ...
3
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1answer
136 views
Building a matrix from a wavelet.
I am currently working with wavelets and computer vision, but I want to understand them in a more mathematical way. I know that the object in my study is defined by
$$\Phi_{(s, l)}(x) = \frac{1}{\...
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24 views
Does a “chirp”-like generalization of the Gabor or Morlet wavelets definitions exist in the lit somewhere?
Please forgive me for predicating this on the definition of the continuous Fourier Transform preferred by most electrical engineers:
$$ X(f) \triangleq \mathscr{F} \Big\{ x(t) \Big\} \triangleq \int\...
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Signal Processing Question Wavlets
Let $\displaystyle f(x)= \begin{cases} 4x^2-4x+1 & 0\le x\le 1, \\ 0 & \text{otherwise}. \end{cases} $
a) Find the approximation $f_{-3}$ of $f \in V_{-3}$ using the Haar scaling function
...
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38 views
Functions with Fourier transform support $[-\pi,\pi]$
What kind of functions, $f \in L^2(\mathbb{R})$ are such that $\operatorname{supp} \hat{f} \subset [-\pi,\pi]$?
I am trying to prove that the set $\{V_j\}_{j \in \mathbb{Z}}$ consisting of functions $...
0
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1answer
63 views
compressive sensing and biorthogonal wavelet matrix
I want to use compressive sensing to reconstruct an image from fewer samples. My problem is with Psi matrix which I want to be Biorthogonal wavelet coefficients but I don't know how to define it. I ...
0
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1answer
19 views
Describe a signal presence in a time series
I am using the continuous wavelet transform to extract periodic components of an original signal. I'm looking for a simple, digestible way to describe how 'much' of the original signal is comprised of ...
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1answer
43 views
Is wavelet noise reduction just removing the higher frequency coefficients?
I read some tutorials in noise reduction using wavelets, and they seem to be too simple.
With Fourier transforms, there is a distinction between types of noise, and some attempts to estimate the ...
