Questions tagged [wavelets]

For questions related to wavelets and wavelet theory.

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21 views

Pros/Cons to using Spectral and Diffusive Graph Wavelets

As I understand, there are two major methods of constructing wavelets on graphs. Spectral wavelets, from David K Hammond et. al, and diffusive wavelets from Coifman and Maggioni. I can't quite parse ...
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33 views

Haar analysis decay rate

Question Prove the partial Haar sum $P_M(f_N)$, defined by $P_M(f_N)=\sum_{0\leq j < M} Q_j(f_N)+P_0(f_N)$, decays exponentially in $M: ||f_N-P_M(f_N)||_{L^2([0,1))} = 2^{-M/2}$, where $f_N=2^N\...
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35 views

Fourier transform following an decay rate of $1/\sqrt{M}$

The Question Show that if we view $f_N = 2^N\chi_{[0,2^{-N}]}$ as a periodic function on $[0,1),$ with $M^{th}$ partial Fourier sum $S_M(f_N)(x)=\sum_{|m|\leq M}\hat{f_N}(m)e^{2\pi imx},$ then the ...
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29 views

Values of an orthonormal wavelet expansion

Suppose that $\varphi$ is the mother scaling function of some orthonormal wavelet family. Let $$g(x) = 2^{j/2} \sum_{k \in \mathbb{Z}} c_k \varphi(2^j x - k)$$ be an expansion of some function $f$ in ...
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18 views

Normalization of function from [A,B] to [0,1]

While working on wavelets, I came across wavelets defined on the interval $[0,1]$. But to sole a problem defined on another interval, I have to first convert the problem function to the required ...
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1answer
88 views

Why is Strömberg Wavelet section in 2018 book an amost exact copy of 2016 Wikipedia article? [closed]

Does anyone know why Section 2.10 Strömberg Wavelet of the 2018 book Wavelet: Analysis and Methods seems to be an almost exact copy of the 2016 Wikipedia article of the same name? There are some small ...
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1answer
84 views

How is Stromberg Wavelet S orthogonal to Simple tents?

$\newcommand{\inprod}[2]{\left\langle{#1}\,|\,{#2}\right\rangle}$ $\newcommand{\eqd}{\triangleq}$ $\newcommand{\setn}[1]{{\left\{{#1}\right\}}}$ $\newcommand{\setu}{\cup}$ $\newcommand{\brp}[1]{{\left(...
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43 views

Are wavelets analogous to convolutional filters?

I'm trying to understand shearlets and wavelets and this video gave me the impression that they are similar to CNN filters, in the sense that they extract/detect spatially localized features via ...
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1answer
22 views

Wave packets and duration of wave packet

Let $T=\frac{1}{30}$ be the time it takes for the wave to complete one cycle. That is, T is the time period. Then the duration of the wave packet is $\frac{10}{T}$. Now my question is, how does one ...
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1answer
26 views

Multidimensional signal synchronization

I am trying to find ways of synchronizing multidimensional waves gotten from the brain. For one dimensional signals, I used Cross correlation and was able to synchronise, but for multi dimensional ...
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46 views

DFT but using non-sinusoidal periodic waveforms.

Is there an established way of decomposing a discrete periodic (complex) signal into a sum of non-sinusoidal periodic waveforms (eg square, triangle, and sawtooth)? For my use case the input waveform ...
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1answer
27 views

How to determine the integral limits of continuous wavelet transform?

Now I have one function $f(x)$, and its support interval is $x\in[T_0,T_1]$. The support interval of the mother wavelet $\psi(x)$ is $x\in[-\Delta,+\Delta]$. And we also know that the continuous ...
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29 views

What functions are not representable by a Discrete Wavelet Transform?

If I take the discrete wavelet transform of a function $f: \mathbb{R} \to \mathbb{R}$ I get a countable set of coefficients, $\omega_{k,l}$, $k,l \in \mathbb{Z}$ where $$ f(x) = \sum_{k=-\infty}^{\...
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29 views

Mathematical Functions That Generate Waves

I know that $ \sin(x), \cos(x), \csc(x)\text{(cosec(x), I think)}, \sec(x)$ and such functions are capable of generating waves. Also, $\varphi(x)\sin(x)$ or $\varphi(x)cos(x)$ and such functions (...
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Using wavelet decompositions to solve Problems in additive combinatorics

Fourier analysis is used in additive combinatorics as a way to detect structure in sets(Roth’s theorem, Szemeredi’s Theorem, Erdos-Szemeredi conjecture, Green-Tao Theorem etc). In particular(from what ...
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1answer
49 views

Relationship between autocorrelation function and wavelet coefficient

the autocorrelation function can be represented using the spectral density in Fourier space. Is there a similar relationship between the autocorrelation function and the coefficient in the wavelet ...
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1answer
33 views

Construction of Compact Boundary-Corrected Scaling Functions

I'm trying to construct the (left) boundary-corrected Daubechies scaling function, following the article A. Cohen, I. Daubechies, P. Vial. "Wavelets on the Interval and Fast Wavelet Transforms&...
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16 views

Help understanding this passages about Fourier transforms

Let $$\Psi_{J,K,\theta}(x,y)=\frac{1}{\sqrt{J}}\Psi\left(\frac{\cos(\theta)x+\sin(\theta)y-K}{J}\right)$$ Let $$\mathcal{T}[f](J,K,\theta)=\int_{\mathbb{R}^2}\overline{\Psi_{J,K,\theta}(x,y)}f(x,y)\:...
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18 views

Filter a signal using Haar Wavelets

Given $s = [1, 2, 1, 0, −1, 0, 1, 0, 2, 0, 1, 1, 0, −2, −1, 1]$, samples from a continuous signal with sampling interval $\Delta t = 4ms$, filter $s$ eliminating all frequencies higher than $40$Hz ...
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14 views

Analytical Wavelets

I'm trying to understand the difference between a Time-frequency analysis done with standard wavelet, and another done with analytical wavelet. If I take as a signal a function $f \in L^2(\mathbb{R})$ ...
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14 views

Why V2 is not dense in L2(R)?

v2 contains all of constant Piecewise function having compact supports with possible break on rational points denominators of 4.
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1answer
30 views

What is the orthonormal transform matrix of Daubechies 4 wavelet?

I read following paper : [2019, Emilie Chouzenoux, A Proximal Interior Point Algorithm with Applications to Image Processing]. problem The problem is constructed as Proximal interior point method, ...
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1answer
26 views

What is the wavelet decomposition operator matrix?

I read following paper. [2019, Emilie Chouzenoux, A Proximal Interior Point Algorithm with Applications to Image Processing] Wavelet operator The problem is constructed as Proximal interior point ...
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7 views

wavelet packet transform and lifting scheme

so the lifting scheme is basically an alternative to performing the dwt with several advantages. But here are three questions which I did not find an answer to: is it possible to use lifting also ...
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40 views

Wavelets vs Fourier Transforms

It's supposed that computing a spectrogram for a signal is faster using wavelets. However, wavelets need to be applied to the signal for every time step and frequency, giving the implementation a time ...
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19 views

How to solve following wave equation?

Use the CTCS scheme with X2-X1=1/3 and t2-t1=1/5 to solve the wave equation Utt(x,t)=Uxx(x,t) with the boundry conditions: U(0,t)=U(1,t)=0 0<=t<=0.6 U(x,0)=(3^1/2)sin(pi....
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29 views

Show that Haar-wavelet is a wavelet.

In my textbook for wavelets I found an example which shows that the Haar-Wavelet $H$ is indeed a wavelet. So I need to show $0 < 2\pi \int_\mathbb{R} \frac{|\widehat{H}(\omega)|^2}{|\omega|} d\...
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18 views

Overview of Current Applications of Wavelets

Pretty soft question, but what are the current applications of wavelets? I'm familiar with the typical example of signal/image compression and this seems to be what many books on the topic focus on, ...
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9 views

VisuShrink and SureShrink fail with non-iid noise

I have a statistics/functional analysis question. I heard that VisuShrink and SureShrink can only work when the noise is iid. Would there be a proof for this? Thank you.
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33 views

Something like fourier transform

Iam a physics student, I don't cover any proper course in harmonic analysis, On the way of studying wavelets I make some guesses, But I don't know they are true or not. They are marked as 1 and 2 1) ...
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19 views

Can Group of affine transformations generate CWT?

I know group of translations in momentum space generate fourier transform. Because in fourier transform we decompose a function as a sum different frequency components $e^{ikx} $ What group of ...
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12 views

Differentiable wavelet family for $L^2(\mathbb{R}^d)$?

A family of functions $\psi^{1}, ..., \psi^M \in L^2(\mathbb{R}^d)$ is called a wavelet-family if \begin{equation} \left\{\psi^i_{j,k}(x) = 2^{\frac{vj}{2}} \psi^i\left(2^jx - k \right) \middle| j \...
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1answer
21 views

Normalized wavelets

i am studying a class of functions called ridgelets, i am looking to prove that they are normalized. Let $\varphi$ be a Real smooth function with sufficient decay and vanishing mean such that $$\int_{...
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1answer
34 views

Wavelet Analysis of non-stationary time series

In this paper: A Practical Guide to Wavelet Analysis, I read that "The wavelet transform can be used to analyze time series that contain nonstationary power at many different frequencies." I also ...
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12 views

Energy of a short time fourier transform involving father wavelets

Lots of definitions here. My definition for the fourier transform is $$\widehat{f}(w)=\int_{-\infty}^{\infty}f(t)e^{-iwt}dt$$ $1$) I must show that $$\int_{-\infty}^{\infty}|Sf(\mu, \varepsilon)|^...
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18 views

Wavelet admissible constant for Morlet wavelet (or Gabor wavelet)

In Continuous wavelet transform, we found the definitions of continuous wavelet transform (CWT) and reconstruction formula to recover the original signal $x(t)$; for the latter: $$x(t)=C_{\psi }^{{-1}}...
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26 views

Can we design a basis or frame of discrete wavelets for spherical harmonics on hexagonal grids?

Introduction: Spherical harmonics are functions which are often very useful in science and engineering when trying to model things which obey some kind of spherically symmetric differential equation, ...
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27 views

What is the $\Phi$ and $\Psi$ in Wavelets analysis?

I am studying wavelets and somenthing that catch my attention is the existance of a so caled scaling function $\Phi$. Let's take the Haar wavelet. I do not understand why is the scalling function ...
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1answer
78 views

Complete family of smooth, orthogonal functions with compact support exists?

are there families of function that are:$\def\R{\mathbb R}$ Smooth, i.e. $C^\infty(\R\to\R)$, and Complete, i.e. they can point-wise approximate* any piece-wise continuous function $\R\to\R$ almost ...
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1answer
18 views

Whats the $L_2$ Norm in relation to Wavelets and functions?

I have read that in order for a function to be a wavelet, it needs to fufull the L2 Norm property. But I don't know what that is and there wasn't an explanation either. I know theres a L2 norm in ...
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1answer
38 views

Do we have $| \langle f, \psi \rangle_{L^2} | \le C_j \| f \|_{\mathcal{C}^2} \| \psi \|_{1}$?

For $j,k \in \mathbb{Z}$, $m \in \mathbb{Z}$ and $c > 0$ define $$ \psi_{\lambda}(x) := \psi_{j,k,m}(x_1,x_2) := 2^{3j/4} \psi(S_k A_{2^j} x - cm), $$ where $\psi \in L^2(\mathbb{R}^2)$ and $$ A_{2^...
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1answer
27 views

Proof that family of reciprocal exponential “wobbles” are everywhere differentiable?

The other day (for purpose of modelling temporary "wobbles") I investigated $$f(t): t\to \exp\left(\frac{N\pi i}{N\pi|t|+1} \right), \forall t\in \mathbb R, N\in \mathbb N$$ I suppose it shall be ...
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1answer
31 views

Calculate SNR if I have the noise?

I am looking for ways to calculate signal-to-noise ratio (SNR). As I understand it, this measure is often used when you have a separated clean signal and noisy signal, and can thus measure the power ...
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1answer
27 views

Uniform convergence for bounded variational function

Let $f_{\xi} = f*h_{\xi}$, where $h_{\xi}(t) = \frac{1}{\pi}\frac{sin(\xi x)}{x}$. Suppose $f$ has a bounded variation $||f||_V < +\infty$ and that it is continuous in a neighborhood of $t_0$. ...
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37 views

Shearlets: Why are parabolic scaling matrices $A_a := \text{diag}(a, \sqrt{a})$ called parabolic

For continuous shearlet systems the parabolic scaling matrices $$A_a := \begin{pmatrix} a & 0 \\ 0 & \sqrt{a}\end{pmatrix}, \quad a > 0$$ are very important. Why are they called parabolic? ...
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33 views

Fourier transform on the scaling equation

Given the scaling equation of an multiresolution analysis, namely $$\varphi=\sqrt{2}\sum\limits_{k\in \mathbb{Z}}^{}h_k\varphi(2\cdot -k)$$ in $L²(ℝ)$-sense. Does this equation also hold on the ...
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41 views

Multiresolution Analysis scaling function

Given a multiresolution analysis $(V_j)_{j\in \mathbb{Z}}$ in $L²(ℝ)$ with scaling function $\varphi \in L²(ℝ)$. As $(2^{1/2}\varphi(2t-k))_{k\in \mathbb{Z}}$ is an orthonormal basis of $V_1$ and $\...
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1answer
35 views

Can one construct a Discrete Wavelet Transform for many separate high pass bands?

A Discrete Wavelet Transform is usually designed with one mother and father wavelet which are generated by a sequence of convolutions of discrete FIR filters. The mother wavelet has $$\int \psi(t)dt =...
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1answer
103 views

Twice continuously differentiable wavelets with compact support

Do there exists wavelets such that both mother wavelet and the scaling function have compact support and they are both twice continuously differentiable?
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42 views

Wavelets in Fourier domain

Given a wavelet family $\psi_{s, a}$ generated by translations and dilations of a mother wavelet $\psi$ $$ \psi_{s, a}(x)=\frac{1}{s} \psi\left(\frac{x-a}{s}\right) $$ we can show a wavelet ...

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