Questions tagged [wavelets]
For questions related to wavelets and wavelet theory.
297
questions
2
votes
0answers
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 ...
0
votes
0answers
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\...
1
vote
0answers
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
...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
2
votes
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 ...
0
votes
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(...
0
votes
0answers
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 ...
0
votes
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 ...
1
vote
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 ...
3
votes
2answers
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 ...
1
vote
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 ...
1
vote
0answers
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
$$
f(x) = \sum_{k=-\infty}^{\...
1
vote
0answers
29 views
Mathematical Functions That Generate Waves
I know that $ \sin(x), \cos(x), \csc(x)\text{(cosec(x), I think)}, \sec(x)$ and such functions are capable of generating waves. Also, $\varphi(x)\sin(x)$ or $\varphi(x)cos(x)$ and such functions (...
3
votes
0answers
23 views
Using wavelet decompositions to solve Problems in additive combinatorics
Fourier analysis is used in additive combinatorics as a way to detect structure in sets(Roth’s theorem, Szemeredi’s Theorem, Erdos-Szemeredi conjecture, Green-Tao Theorem etc). In particular(from what ...
2
votes
1answer
49 views
Relationship between autocorrelation function and wavelet coefficient
the autocorrelation function can be represented using the spectral density in Fourier space. Is there a similar relationship between the autocorrelation function and the coefficient in the wavelet ...
2
votes
1answer
33 views
Construction of Compact Boundary-Corrected Scaling Functions
I'm trying to construct the (left) boundary-corrected Daubechies scaling function, following the article
A. Cohen, I. Daubechies, P. Vial. "Wavelets on the Interval and Fast Wavelet Transforms&...
0
votes
0answers
16 views
Help understanding this passages about Fourier transforms
Let $$\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)\:...
0
votes
0answers
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 ...
0
votes
0answers
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 $f \in L^2(\mathbb{R})$ ...
0
votes
0answers
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.
0
votes
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, ...
0
votes
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 ...
0
votes
0answers
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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....
1
vote
0answers
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\...
2
votes
0answers
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, ...
0
votes
0answers
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.
1
vote
0answers
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) ...
1
vote
0answers
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 ...
0
votes
0answers
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 \...
1
vote
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 $$\int_{...
0
votes
1answer
34 views
Wavelet Analysis of non-stationary time series
In this paper: A Practical Guide to Wavelet Analysis, I read that "The wavelet transform can be used to analyze time series that contain nonstationary power at many different frequencies."
I also ...
1
vote
0answers
12 views
Energy of a short time fourier transform involving father wavelets
Lots of definitions here.
My definition for the fourier transform is
$$\widehat{f}(w)=\int_{-\infty}^{\infty}f(t)e^{-iwt}dt$$
$1$) I must show that
$$\int_{-\infty}^{\infty}|Sf(\mu, \varepsilon)|^...
0
votes
0answers
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:
$$x(t)=C_{\psi }^{{-1}}...
2
votes
0answers
26 views
Can we design a basis or frame of discrete wavelets for spherical harmonics on hexagonal grids?
Introduction:
Spherical harmonics are functions which are often very useful in science and engineering when trying to model things which obey some kind of spherically symmetric differential equation, ...
0
votes
0answers
27 views
What is the $\Phi$ and $\Psi$ in Wavelets analysis?
I am studying wavelets and somenthing that catch my attention is the existance of a so caled scaling function $\Phi$. Let's take the Haar wavelet. I do not understand why is the scalling function ...
2
votes
1answer
78 views
Complete family of smooth, orthogonal functions with compact support exists?
are there families of function that are:$\def\R{\mathbb R}$
Smooth, i.e. $C^\infty(\R\to\R)$, and
Complete, i.e. they can point-wise approximate* any piece-wise continuous function $\R\to\R$ almost ...
0
votes
1answer
18 views
Whats the $L_2$ Norm in relation to Wavelets and functions?
I have read that in order for a function to be a wavelet, it needs to fufull the L2 Norm property.
But I don't know what that is and there wasn't an explanation either. I know theres a L2 norm in ...
2
votes
1answer
38 views
Do we have $| \langle f, \psi \rangle_{L^2} | \le C_j \| f \|_{\mathcal{C}^2} \| \psi \|_{1}$?
For $j,k \in \mathbb{Z}$, $m \in \mathbb{Z}$ and $c > 0$ define
$$
\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^...
1
vote
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 ...
0
votes
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 ...
0
votes
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 $t_0$. ...
0
votes
0answers
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?
...
0
votes
0answers
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 ...
0
votes
0answers
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 $\...
0
votes
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 $$\int \psi(t)dt =...
1
vote
1answer
103 views
Twice continuously differentiable wavelets with compact support
Do there exists wavelets such that both mother wavelet and the scaling function have compact support and they are both twice continuously differentiable?
0
votes
1answer
42 views
Wavelets in Fourier domain
Given a wavelet family $\psi_{s, a}$ generated by translations and dilations of a mother wavelet $\psi$
$$
\psi_{s, a}(x)=\frac{1}{s} \psi\left(\frac{x-a}{s}\right)
$$
we can show a wavelet ...
