Questions tagged [wavelets]
For questions related to wavelets and wavelet theory.
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Time varying parameter estimation
I am working with a linear time-varying system described by the equation:
dx/dt=f(t)U(t)-alpha(t)*x(t).
Here, U(t) and x(t) represent known input and output signals, respectively. I lack information ...
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I know that $ \int_0^\infty \frac{|f(x)|^2}{x} < \infty $, which condition should I have to prove $ \int_0^\infty \frac{|f(x)|^2}{x^2} < \infty $?
I'm working with the wavelet transform, and I'm facing a problem proving that
$$ \int_0^\infty \frac{|f(x)|^2}{x^2}\mathrm{d}x $$
converges.
The only thing I know is that $f(x)$ is the Fourier ...
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Compactly supported orthonormal basis of $L^2(\mathbf R)$ with certain properties
Take a positive integer $\alpha$. I am looking for an orthonormal basis $(\phi_n)$ for $L^2(\mathbf R)$ with roughly speaking the following properties:
each $\phi_n$ is compactly supported and $C^\...
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Orthonormal Basis in $L^2[0,5/8]$
It is well-known that the Haar wavelet system forms an orthonormal basis for $L^2[0,1]$. I am interested in forming a similar orthonormal basis for $L^2[0,\frac58]$. We write $[0, \frac58] = [0,\...
- 44
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3D wavelet transform in the form of a matrix?
I was wondering if anyone may know of any method for the construction of a 3D wavelet transform in matrix form? I've been able to build matrices to perform 1D & 2D transforms. Yet, am finding very ...
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Is a sequence in a Hilbert space that forms biorthogonal with a Schauder basis also a Schauder basis?
Suppose $H$ is a Hilbert space and has a Schauder basis $\{e_{j}\}_{j=1}^{\infty} $. $\{\hat{e_{j}}\}_{j=1}^{\infty}$ is a sequence biorthogonal with $\{e_{j}\}_{j=1}^{\infty}$; that is, $<e_{j},\...
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What is the functional space of (general) Wavelet transform of squared-integrable functions?
Let $\psi(x) = \exp(-x^2)$. The (general) Wavelet transform $W: L^2(\mathbb R) \rightarrow X$ is defined by
$$Wf(a, b) = b^{-1/2}\int_{x\in \mathbb R} f(x) \psi\left( \frac{x-a}{b}\right)dx$$
for $(a,...
- 1,344
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Quadratic function and the Gaussian integral
I'm trying to find the CWT (Continuous Wavelet Transform) for the function $$f(t) = \alpha t^2 + \beta t + c$$ and the wavelet $$\psi(t) = (1-\frac{t^2}{\sigma^2})\exp(-\frac{t^2}{2\sigma^2})$$Note ...
- 4,559
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Haar Representation of Brownian Motion Understanding Proof of Continuity
I have begun reading about the Haar function representation of Brownian motion and am trying to understand the proof of why the representation defines a Brownian motion. Specifically, I am looking at ...
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Two different formulations of Haar basis in $L^2(\mathbb R)$
I found two different formulations of Haar basis in $L^2(\mathbb R)$. For example, in Heil, Christopher. A basis theory primer: expanded edition. Springer Science & Business Media, 2010. we found:
...
- 7,807
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How to orthogonalize the Fourier-Iagolnitzer transform in the same direction of a wave train?
The Fourier-Bros-Iagolnitzer transform is given by the functional
$$
\mathcal {F}\{f(t)\}=(2\pi )^{{-n/2}}\int _{{{{\mathbf R}}^{n}}}f(x)e^{{-a|x-y|^2/2}}e^{ix\cdot t},dx.$$
I applied this on a set of ...
- 2,596
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Duals In The Sense Of Riesz Representation Theorem
Can somebody help me understand this quote from Wikipedia:
In mathematics, a dual wavelet is the dual to a wavelet. In general, the wavelet series generated by a square-integrable function will have ...
- 3,522
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What is the exact minimal condition for the emergence of stationary intensity patterns in a chaotic wave field?
Consider the fact that a superposition of two wave functions with different frequencies $\omega_A$ and $\omega_B$ ($\omega_A \neq \omega_B$),
$$\begin{align}
\Psi(\vec x, t) &= \Psi_A(\vec x, t) + ...
user1156115
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Diffusion Process/Operator Starting on a Subgraph
I'm reading through the paper on Diffusion Wavelets and wanted to ask if there is a coherent notion for a diffusion operator on a non-negative weighted undirected simple graph $G=(V,E)$ that is some ...
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How to calculate the covariance between two signals using the wavelet transform?
I am currently in my last year of Master in Bioengineering, and am studying the application of the wavelet transform for various uses.
For instance, wavelet analysis can be used to calculate the ...
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Maximal Overlap Wavelet Transform Energy Conservation Properties
When I perform the haar maximal overlap wavelet transform of a signal I get a series of coefficients and one approximation of the signal itself.
...
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Is $\left\{ \frac{1}{x-a\pm iw} \right\}_{a\in\mathbb R, w>0}$ dense in $L^2(\mathbb R)$?
For $f\in L^2(\mathbb R)$, one may want to expand $f$ in terms of other functions. Fourier transform is a prominent example.
I want to consider another type of `basis functions'. Specifically, for $f\...
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multiresolution analysis scaling identity
I'm reading the First Course in Wavelets with Fourier Analysis by Boggess & Narcowich. In chapter 5 section 1.2, it defines
$$\newcommand{\scal}[1]{\langle{#1}\rangle}p_k = \sqrt2 \scal{ \phi, \...
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Cascade algorithm for wavelet and scaling functions
It has been described to me that the wavelet and scaling functions can be computed using the Cascade Algorithm applied to the low and high pass filter coefficients.
Where can I find some proofs of how ...
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Using low and high pass filter coefficients with cascade algorithm to observe the wavelet and scaling functions
If want to analyze seismic trace data using low and high pass filter coefficients. I want to see what the wavelet and scaling functions look like, I would then have to use the Cascade Algorithm.
I do ...
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Computing db2 to db20 coefficients
I want to implement db2 to db20. Is there a specific formula for the wavelet function coefficients, which determine its shape and properties? How about the associated scaling function?
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low pass and high pass filter defined as a pair
In Strang's Wavelets and Filter Banks, Problem 1.3.6 asks,
If $H_0$ is the response of a lowpass filter, what is the response $H_1$ of a corresponding highpass filter? If $h(0), \dots, h(N)$ are ...
- 41
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Which methods are best to analyse wave-packets?
I would like to find several methods to analyse wave-packets. My primary aim is to find a way to decompose wave-packets into "sub-wave packets" that, when subjected to some operation give ...
- 2,596
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Which methods apply best in the analysis of non-linear phenomena?
I am going to set up my Bachelors project in analysis of non-linear wave phenomena, and I am planning to study plots and graphs of nonlinear spectra.
In this context, I am quite familiar with the ...
- 2,596
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Haar wavelet decomposition, show that: $\psi_{j,k}=\sum_{n\in\mathbb{N}}b_{n-2k}\phi_{j-1,k}$
We have a $L^2(\mathbb{R})$ multiresolution $(V_j)_j$ through the family of scaling functions $\phi_j$ and the orthogonal wavelets $\psi_j$ such that
$$\phi(x)=\sum_{n\in\mathbb{N}}a_n\phi(2x-n)$$
and
...
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What is a waveform dictionary?
If I google "waveform dictionary" or "waveform dictionaries" I find some papers that use these mathematical objects e.g. The Analysis of Foreign Exchange Data Using Waveform ...
- 7,807
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Extracting frequency derivative from STFT
I have a signal that oscillates on short timescales and changes frequency of that oscillation on larger scales. If I do a Short Time Fourier Transform(STFT) or a constant Q transform, I get a feature ...
- 31
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When are LU factors of sparse matrices surely sparse?
The other week I revisited an old classic factorization from my bachelors studies, the LU-factorization.
The LU factorization of a (square) matrix M finds lower and upper triangular matrices (L and U ...
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Relationship between primal and dual wavelet filter coefficients
I've been working with wavelets for a project, more specifically implementing the Fast Wavelet Transform, which requires the use of the dual high pass and low pass filters (for the forward FWT). In ...
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Looking to Learn more about compressed sensing. Looking for guidances (on algorithms).
I am looking to learn more into compressed sensing algorithms (especially parallelizable ones). I am familiar with some formulations of the problem (minimize L2 norm of recovered data against original ...
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Applying Riesz Representation Theorem to show existence of dual basis
Given a square-integrable function $\psi\in L^2(\mathbb{R})$, define the series $\{\psi_{jk}\}$ by
$$\psi_{jk}(x) = 2^{j/2}\psi(2^jx-k)$$
for integers $j,k\in \mathbb{Z}$.
Such a function is called an ...
- 2,511
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Inner product of a function with a dual function
I have been reading a textbook on wavelets. I faced the following statement,
Let us relax the condition that $A_s$ and $W_s$ be orthogonal to each other and assume that the wavelet $\psi_{k,s}\in W_s$ ...
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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.
user886636
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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 ...
- 341
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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/...
- 11
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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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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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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\...
- 125
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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 ...
