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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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 $...
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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 ...
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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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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 ...