For questions related to wavelets and wavelet theory

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1answer
73 views

Amplitude Spectrum, Nyquist Frequency, mixed/min/max wavelets

The problem is here. Now I know the definition of mixed/max/min phase wavelets, whether the roots lie within the unit circle or not. Starting from n = 1, let $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
2
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1answer
33 views

A relation for Fourier series

For $f$ and $f'$ in $L^2(0,1)$, define $e_k(x)=e^{2\pi ikx}$, $k \in \mathbb{Z}$. And define the Fourier series: $f=\sum _{k \in \mathbb{Z}}c_ke_k$, where $c_k=\left \langle f,e_k \right ...
4
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0answers
107 views

Matlab's dwt2 in publication quality notation?

I used Matlab's dwt2in my algorithm, what notation have you folks seen that describes the 4 resulting wavelets? How do I express ...
0
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1answer
30 views

How to show that the limit of $\frac{\omega_1}{N^n}+\frac{\omega_2}{N^{n-1}}+…+\frac{\omega_n}{N}$ exist

$$\omega=(\omega_1, \omega_2,...)\in \{0, 1, 2,...,N-1\}^\mathbb N$$ How to show that the ...
0
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1answer
44 views

Bessel Sequence proof check.

I have a similar question to definition of Bessel sequence, where it was solved using Banach-Steinhaus. A sequence $\{f_k\}_{k=1}^{\infty}$ is called a Bessel sequence in a Hilbert space ...
3
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2answers
126 views

In which case should a wavelet transform be applied instead of a Fourier transform?

I was wondering what are the advantages (and possibly the drawbacks) of using a wavelet transform instead of a Fourier transform for the signal processing, are there simple examples to illustrate that ...
3
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0answers
81 views

Regularity of Daubechies wavelet

I am reading the book Wavelets: Theory and applications by A. K. Louis, D. Maass, A. Rieder ...
3
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1answer
84 views

definition of Bessel sequence

A sequence $\{f_{k}\}_{k=1}^{\infty}$ is called a Bessel sequence in a Hilbert space $H$, if there exists $B>0$ such that $$\sum_{k=1}^{\infty}|\langle f,f_{k}\rangle|^{2}\leq B\|f\|^{2}$$ for all ...
2
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1answer
157 views

Need to learn wavelet, suggest steps and resources

I am looking for a good introduction to wavelets and wavelet transforms. that covers the following: Basics Vector Spaces – Properties– Dot Product – Basis – Dimension, Orthogonality and ...
3
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0answers
44 views

Error bounds in representing a vector using a truncated Moore-Penrose biorthogonal basis

I was reading and trying to reproduce the results in the arXiv preprint of Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression by Asaf ...
2
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1answer
120 views

Biorthogonal Projector Construction

I was trying to prove that the biorthogonal projector matrix $P$ given on the Wikipedia page on biorthogonal systems does in fact construct biorthogonal systems from input bases $\mathbf{u}$ and ...
0
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1answer
47 views

Wavelets on $\mathbb{R}^{n}$

I want to study Wavelet theory on $\mathbb{R}^{n}$ and I can't find any book on this topic. Can you recommend me any good book that considers that?
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0answers
144 views

Continous wavelet transform and shannon Entropy.

Note: I have asked the same question on signal processing forum,but didn't get any answer. so it might be more like a math or physics question. Hope you don't consider it as cross-post. I am trying to ...
1
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0answers
53 views

Strange values of approximating coefficients returned by matlab's wavelets decomposition

I'm trying to get wavelet decomposition of arcsin(x) using, say, haar wavelets When using both Matlab's dwt or wavedec functions, I get strange values for approximating coefficients. Since applying ...
1
vote
1answer
108 views

Denoise using wavelet transform

My mathematical class task is to de-noise a function using wavelet transform. I am to select a function $f(x)$ and noise function with zero-mean $n(x)$. I am to add noise like this: $$f_{noise}(x) = ...
1
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2answers
55 views

Find support of a wavelet

I am a complete novice in wavelet analyses, but I was given a task to find a support set of a given wavelet (in 1D, e.g. db4). Please, point me a way of doing this. (Since, how I understood from some ...
1
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1answer
49 views

definition of wavelet in $L_p$ space

Function $\psi$ is called wavelet, if there is a dual $\widetilde{\psi}$ such that a function $f \in L_2(R)$ can be decomposed as $$ f(t)=\sum_{ j \in Z}\sum_{\nu \in Z} \langle f, ...
2
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1answer
87 views

Daubechies wavelet

I have two question about Daubechies wavelet are they really orthogonality? for example daub4 has the support equal $[0-3]$, and for scale function we have must: $$ \int_0^3\phi(x-n)*\phi(x-m)dx ...
2
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3answers
349 views

Morlet's wavelet reconstruction formula

The CWT (continuous wavelet transform) of a signal $x(t)$ is $$X_w(a,b)=\frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t)\psi^{\ast}\left(\frac{t-b}{a}\right)\, dt$$ In order to reconstruct the ...
1
vote
1answer
122 views

Linear algebra explanation of wavelet transform

Is there an easy way to explain wavelets / wavelet transform using only linear algebra? The discrete Fourier transform is a linear operator on $\mathbb C^N$ that simply changes basis to a special ...
1
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1answer
65 views

Integrating a squared Haar wavelet function

I'm studying about wavelets and here is one derivation I couldn't understand: The constant that makes this orthogonal basis orthonormal is $2^{j/2}$. Indeed, from the definition of norm$^2$ in ...
4
votes
3answers
170 views

Orthogonality of Haar wavelet functions

I'm reading about wavelets and I bumped into the follwing: $\text{Haar wavelet is a step function}\; \psi(x), \text{which takes values 1 and -1, when}\; x \;\text{is in the ranges}\; [0, \frac{1}{2}) ...
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3answers
83 views

Reducing an infinite dimensional expansion to a finite dimensional algorithm

The following is a big-picture question about the interplay between infinite dimensional function expansions and finite dimensional algorithms. I feel like I have a good understanding of these ideas ...
0
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1answer
160 views

Weak form for Linear Dynamic Wave Equation of Dirichlet/Neumann's boundaries?

I have a linear problem with double derivate of space and time, which has Dirichlet boundary condition in $(1)_{2}$ and Neumann's boundary condition in $(1)_{3}$: \begin{equation} \frac{\delta^{2} ...
2
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1answer
212 views

compare wavelet and Fourier transform

i would like to compare each other wavelet and Fourier transform on given signal,let us consider following signal ...
3
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0answers
776 views

Wavelets: Cone Of Influence

While reading this paper I came across the term Cone of Influence which is described as ...
0
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1answer
1k views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
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0answers
93 views

Output of wavelet transforms

I am working on a time sensitive computer science and fluid dynamics project that requires me to find applications of wavelet analysis. I know that at its core, a wavelet transform simply takes a ...
4
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1answer
41 views

Show that the endpoints of a compactly supported function satisfying the scaling relation are integers

Suppose $ \phi \in C_0( \mathbb{R}) $ (compact support) satisfy the scaling relation $$ \phi(x) = \sum_{k \in \mathbb{Z}} p_k \phi(2x-k) , $$ with $$ p_k = 2^{1/2} \int_{- \infty}^\infty \phi(x) ...
2
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0answers
140 views

An exercise. A property of the Fourier transform of wavelet

In the book "An Introduction To Wavelet Analysis" by David F. Walnut, there is, Exercise 7.45. Show that if $\psi(x)$ is a wavelet, then $\sum\limits_{j}{\left|\hat{\psi}(2^j\gamma)\right|^2} = 1$ ...
2
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0answers
35 views

Finding correlations between many unknown functions.

Given an arbitrarily large number of black-box functions of one variable, is it possible to produce expressions that approximate their relationships to each other over their shared domain? Is it ...
2
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1answer
137 views

haar wavelet: unable to understand what limits they have used

Please explain what limits of integration they took for solving $p_i(x)$ and $q_i(x)$. like for 1st step of $p_i(x)$: $$\int_\alpha^x 1 \, \mathrm dx = x-\alpha$$ but for 2nd step of $p_i(x)$: ...
2
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0answers
118 views

Operators from $L^{\infty}$ to $L^{\infty}$, below bound of the norm

If $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded on $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le \|T\|_{L^{\infty}\rightarrow ...
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0answers
14 views

Fittig N-D periodic functions

Besides fourier series, are there any stable way to fit N-D nonlinear functions that demonstrate some degree of periodicity, sigmoid neural networks works poorly in this domain.
2
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0answers
73 views

Proving norm inequality with Schwarz's inequality

I'm stuck on the following problem: Suppose that $\{V_j : j \in \mathbb{Z}\}$ is a multiresolution analysis with scaling function $\phi$, and that $\phi$ is continuous and compactly supported. ...
4
votes
1answer
249 views

Mexican Hat wavelet in polar coordinates

I'm interested in wavelet framework for polar coordinates. In the paper of Hou&Qin (2012) was proposed a general method for definition of MH wavelets on a certain manifold. In short, first we ...
3
votes
1answer
313 views

A theorem about Lipschitz regularity and Fourier transform

How to prove that: A function $f$ is uniformly Lipschitz $\alpha$ over $\mathbb R$ if $$\int_{-\infty}^{+\infty}|\hat f(\omega)|(1+|\omega|^\alpha)d\omega<+\infty$$ A function $f$ is uniformly ...
2
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0answers
180 views

Testing whether a finite measure is absolutely continuous with respect to Lebesgue measure using wavelets

I've been working through Fundamentals of Stochastic Filtering (Bain, Crisan) and am a little perplexed by the following (initially) seemingly straightforward exercise and its given solution. We are ...
0
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1answer
76 views

inner product with scaling function

$$ \int_0^1 sin(x)\phi(2^jx-k)dx $$ Are there any software that can compute the above integral where $\phi(x)$ is scaling function of db3 (or dbN) family, j scaling parameter and k translation ...
0
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1answer
212 views

Sine-wave Phase shift

This is more of a doubt rather than question. Just correct me if I am wrong, tks So i have this question, (not exactly maths but my doubt is related to calculation understanding) The question here ...
0
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0answers
79 views

Three bases orthogonal to each other

Is it possible to construct 3 wavelet or harmonic bases which are 'orthogonal' to each other? I'm curious, because if this is the case the product integral of some functions in these bases would ...
3
votes
1answer
563 views

Calderon's admissibility condition for wavelets explained

Calderon's admissibility condition is a central argument in a number of recent wavelet-like constructs, like curvelets, shearlets, to name a few. It states that if $\psi$'s Fourier transform conforms ...
22
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5answers
11k views

Difference between Fourier transform and Wavelets

While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency whereas ...
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0answers
105 views

what's the difference between Cohen-Daubechies-Fauraue 9/7 transformation and Discrete cosine transform?

can someone explain, what's the difference between this 2 tranformations? DCT and CDF Greetings
2
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0answers
185 views

Scaling function and dilation equation in MRA

In the book I am reading, the author explains how we can use the dilation equation to obtain the scaling function through the process of iteration. In particular, we use the dilation equation, which ...
3
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1answer
1k views

How to intuitively interpret Gabor lambda param?

I have troubles understanding in an intuitive way (not by writing complicated math formulas) what is the meaning of the lambda parameter in the Gabor functions. (I have basic math understanding, grad ...
1
vote
1answer
97 views

Boundary condition error, correlation of a function with a wavelet.

I'm trying to compute some wavelet transforms and I seem to be running into some boundary condition errors. I have a randomly generated signal pictured at the top of the figure, and the real part of ...
3
votes
2answers
849 views

Need a formula for a quadratic spline

I'm trying to reproduce some results from a paper and I need an explicit formula for a specific quadratic spline to do so. The problem is, I've only got a plot of it. The quadratic spline is from ...
6
votes
0answers
250 views

expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ I am wondering if one can expand ...
2
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0answers
67 views

Lower bound for the eigenvalue

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ ...