Questions tagged [wavelets]

For questions related to wavelets and wavelet theory.

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Prove that $W_f(a, b) = \frac{1}{|a|} \int_\mathbb{R} f(x) \overline{\phi({\frac{x-b}{a})}} \ \text{dx}$

We define the wavelet transform of a function $f\in L^2$ is a function $W_f : \mathbb{R}^2 \rightarrow \mathbb{R}$ is given by $$W_f(a, b) = \frac{1}{|a|} \int_{-\infty}^\infty f(x) \overline{\phi({\...
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Representing rectangular function with Gaussian series

Is it possible to represent the rectangular function with an infinite series of scaled Gaussian functions? https://en.wikipedia.org/wiki/Rectangular_function Thank you, Alex
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Cauchy's theorem with a non closed curve

I'm reading Wavelets - tools for science and technology by Jaffard, Meyer and Ryan and I have a problem with a statement of section 10.4.1. They say that there is a form of Cauchy's theorem that says :...
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How to apply a 2D wavelet-packet decomposition?

I'm reading No-reference image quality assessment using statistical wavelet-packet features Pattern Recognition Letters and at the top right of page 3 they talk about "After computing G and L, a ...
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Concatenate 2D-Discrete Wavelet Transform Horizontal,vertical and diagonal coefficients to generate new 2D image features [closed]

I am working on a classification task and I used 2D-DWT as a feature extractor. I want to ask about more details why I can concatenate 2D-DWT coefficients to generate image of features. I am thinking ...
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Show that $f_n$ converges uniformly to $f$ on $[0, 1]$

Show that $f_n$ converges uniformly to $f$ on $[0, 1]$. Please help me. I have no idea.
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Prove that harmonic wavelets $\psi (2^mt -k)$ and harmonic scaling function $\phi (t-k)$ are orthogonal

I want to prove that harmonic wavelets $\psi (2^mt -k) = \frac{\text{exp}\{4\pi i (2^mt-k)\}-\text{exp}\{2\pi i (2^mt -k)\}}{2\pi i (2^mt-k)}$ and harmonic scaling function $\phi (t-k) = \frac{\text{...
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What is dyadic sampling in the context of a wavelet transform?

In Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges the authors introduce to the reader (page 24) the notion of wavelet transforms as a way of having multiscale representations (...
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How to select a threshold when denoising financial data using sym4 wavelets?

I need to denoise series of financial prices using wavelets. I am using python to do this task and in particular the pywt.threshold function which requires hard value for the threshold. This is my ...
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1D Interpolating subdivision for lifting schemes

I am looking into wavelet lifting methods first introduced by Swelden, and explained in this paper: Build your own wavelets at home. In this paper (in chapter 2 specifically), they discuss ...
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Multiresolution analysis, change of integral and summation and convergence a.e.

Let $(V_n, \phi)$ be multiresolution analysis with scaling function $\phi$ and $V_n \subset L_2(\mathbb{R})$. I need an explanation why $\phi$ can be written in following way (taken from Bachman's ...
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constant function representation by Haar wavelet

Let say our interval of interest is $A = [0,1]$ and scaling function $\phi(x) = 1$ on $A$. The Haar wavelet is defined as $\psi_{j,k}(x)$ from mother wavelet $\psi(x) = 1$ for $0\le x\le 0.5$ and $\...
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wavelet energy at each scale j

By the wavelet representation, we know that, for $f \in L_2[0,1]$, we can write $f = \sum_{j=0}^{\infty} \sum_{k=0}^{2^j-1} \theta_{jk} \psi_{jk}(x)$, where $\theta_{jk} = \langle f, \psi_{jk}\rangle$....
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Inverse Fourier transform of equation to define a wavelet

First time asking the internet a question... I am trying to learn wavelet analyses using the filter bank method described in this article (https://www.sciencedirect.com/science/article/pii/...
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How to Calculate a Wave Travelling Along the Circumference of a Circle?

I'm wondering how you might calculate a wave that traverses the circumference of a circle. This image is a terribly drawn attempt at showing what I mean, Imagine that the black wavy pattern is a ...
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Intersection of wavelet spaces is trivial

Let $\Phi\in C_c(\mathbb R)$ be a continuous and compactly supported function, satisfying $\int\Phi(x-m)\Phi(x-n)=\delta_{m,n}$. Set $V_j=\mathrm{span}\{\Phi(2^jx-k)\}_{k\in\mathbb Z}$ and consider $\...
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How space of wavelets is created from {0}?

I have learned about wavelets and there is a space $$... V_{-1} \subset V_0 \subset V_1 $$ and they create as a closure of sum of $V_j$ all space $L^2(\Bbb{R})$. There are some other points, but one ...
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Information-based function approximation. Series of bits instead of series of real/complex coefficients

There are multiple series and functional bases (e.g. Taylor, Fourier) that allow approximating a function. In some sense, such approximations associate a function with an infinite sequence of real or ...
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Existence of Schauder wavelet basis in $L^2([0,1], \mathbb{R})$

I know there are several wavelet (Schauder) bases in $L^2(\mathbb{R})$. Is there a wavelet basis when we consider functions defined on a compact domain, for example $L^2([0,1], \mathbb{R})$ ? If yes, ...
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Singular distributions

I am doing a report on wavelets (the book is A. Cohen, Numerical Analysis of Wavelets Methods), but I am not really into Hilbert spaces, so I have some problems with notation. The book use quite often ...
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How to plot the scaling space Vj using the Haar Wavelet Transform?

How do I plot the scaling space Vj of the Haar Wavelet Transform if I am given scaling and wavelet coefficients? I have some definitions involved in the Haar wavelet transform, but when I am asked to ...
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How to calculate Gaussian wavelets

What is the formula for the $n$th order derivative Gaussian wavelet? A Gaussian wavelet is given by $$\psi(t) = C \exp(-\frac{(t-\mu)^2}{2\sigma})$$ with $\mu$ and $\sigma$ the mean and standard ...
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Mother wavelet and Lorentz invariance

Can we choose a mother wavelet that is a Lorentz variant in practical applications of wavelet transform in Minkowski space?
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Orthonormal basis for $W_j$

Given $\{V_j: j \in \mathbb{Z}\}$ is multi-resolution analysis in $L^2(\mathbb{R})$. Suppose $W_0$ is an orthogonal complement of $V_0$ relative to $V_1$. So that we can write, $V_1 = V_0 \bigoplus ...
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Wave Packet Decay Estimates

In Tao's notes on time frequency analysis, the following theorem is stated (Theorem 5.5 of Part 1 of https://www.math.ucla.edu/~tao/254a.1.01w/): Fix $\xi_0 \in \mathbf{R}$, and positive values $\...
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Interchanging summation and integration on a wavelets proof

I have some problems understanding a couple of proofs about wavelets, all theese proofs have the same computation i do not understand (i can not see why it is true). Firts, i give you some context. ...
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When can a function be written as superposition of Gaussians?

I read the following definition called Bump Algebra $B$: Let $g(t)=e^{-t^2}$ be a normal Gaussian function. $$B := \left\{ f \in C(\mathbb{R}) : f = \sum_i a_i \, g \left( \frac{t-t_i}{s_i} \right) \...
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Show that Haar Wavelet is admissible?

In Ten Lectures on Wavelets Chapter 3.3, we have the following claim: if $\psi_{m,n}$ constitute a tight frame and also form an orthogonal basis of $L^2(\mathbb{R})$ , we have equation (3.3.8) $$\...
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Alternative ways to show that a set of elements span a space?

Could we rigorously prove the following? In a Hilbert Space $\mathcal{H}$, if $f, \phi_j \in \mathcal{H}$ for all $j \in J$, and $$\left<f, \phi_j\right> = 0 \text{ for all } j \in J \implies ...
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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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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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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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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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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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1 vote
1 answer
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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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3 votes
2 answers
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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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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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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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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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2 votes
1 answer
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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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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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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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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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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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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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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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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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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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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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