For questions related to wavelets and wavelet theory

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7
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50 views

What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
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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 ...
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113 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 ...
3
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0answers
83 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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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 ...
3
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856 views

Wavelets: Cone Of Influence

While reading this paper I came across the term Cone of Influence which is described as ...
2
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15 views

Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
2
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36 views

Why the $L^1(R)$ space does not have a unconditional basis

Why the $L^1(R)$ space does not have a unconditional basis. It is a known fact that $L^p(R)$ ($1<p<+\infty$) has unconditional basis. A simple example is the dyadic wavelet or Haar system. I ...
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145 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 $\sum\limits_{j}{\left|\hat{\psi}(2^j\gamma)\right|^2} = 1$ ...
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36 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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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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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. ...
2
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185 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 ...
2
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0answers
186 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 ...
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 $$ ...
2
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166 views

Hermite functions and integral

Let $$ h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}, $$ where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, be Hermite function. Consider $$ ...
2
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222 views

$L_2$-norm representation of the function

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$(see for reference ...
2
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168 views

Singular Value Decomposition

I want to decompose an image $A$ using the Discrete Wavelet Transform and then find the singular values, $S$, such that $A=USV$. I will then do the same to another image such that $B=USV$. I will ...
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9 views

For wavelets, is a multiresolution analysis an example of a filtration? If so, is it a filtration on the algebra of L^2?

For wavelets, is a multiresolution analysis an example of a filtration? If so, is it a filtration on the algebra of $L^2$? Even elementary clarifications of basic concepts on the filtration side, in ...
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0answers
47 views

Compare between Short Time Fourier Transform and Wavelets

Fourier transform is localised in only frequency domain but Short time Fourier transform(STFT) is localised both in time and frequency domain same as in wavelets. I want to know How are STFT and ...
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18 views

How to express a signal in terms of Riesz bases?

Fast discrete wavelet transform allows us to express any discrete signal in terms of wavelet bases by convolution with filter coefficients. How can one express a digital signal in terms of ...
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20 views

What are the wavelet coefficients of a time series that is linear interpolated?

I want to know the relationship between the wavelet coefficients of a time series before and after linear interpolated. Suppose we have a time series $x(0),x(1),x(2),\cdots$. When this time ...
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27 views

Continuous second derivative over the support of a Daubechies4 wavelet

I can not entirely follow the proof from section 3.1.1 from the book "A primer on Wavelets" by Walker. After the first part (listed below), I can grasp the rest so if you could help I would greatly ...
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0answers
34 views

Significance of orthonormal basis in wavelet analysis

I've recently been looking into wavelet analysis and I have the question: What is the importance of wavelets having an orthogonal basis, say as opposed to a bi-orthogonal basis or otherwise? I'm ...
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0answers
11 views

Terminology with wavelets

I have seen in textbooks that the wavelet transform is stated as two different types of filters. When texts are defining the wavelet transform they call it a band pass filter. However when they talk ...
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24 views

Can wavelets be used for texture discrimination?

I've recently been studying wavelet analysis with a view to differentiating certain areas of texture images where the texture differs from the background pattern (which is quite random); for example a ...
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0answers
41 views

The Heisenberg uncertainty principle in the time-frequency plane

The Heisenberg uncertainty principle says that it is impossible to have a signal with finite support on the time axis which is at the same time band limited. Is the following reasoning correct: When ...
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0answers
313 views

How to calculate wavelet energy?

Part of my assignment about signal processing says the following: Compute the Discrete Wavelet Transform for the input signals Group the wavelet coefficients in trees growing across scales ...
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0answers
32 views

Can anyone provide a sketch as to what a wavelet transform of the dirac delta function look like?

I was trying to motivate the idea of a time frequency localization using wavelet transform to a peer today and I thought the impulse function would be a good example. In my mind I was thinking of a ...
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0answers
40 views

Frequency response of wavelets and scaling functions

I am getting started with wavelets! And I am having trouble going from scaling function to the frequency response of the scaling function. The scaling function and wavelet is defined on some axis ...
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156 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 ...
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0answers
54 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 ...
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0answers
100 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 ...
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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.
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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
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82 views

$n$-th derivative of the prolate spheroidal function

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 $$ ...
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0answers
109 views

Is there a wavelet frame for $L^2[0,\infty)$?

What systems of wavelets provide a frame for $L^2[0,\infty)$. For example, the Haar system of wavelets provides a basis for $L^2[0,1]$, and the harmonic wavelets provide a basis for all of ...
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0answers
549 views

Mathematica: How to convert scales to frequencies?

According to the transform $$w(u,s)=\frac{1}{\sqrt{s}}\int _{-\infty }^{\infty }x(t) \psi ^*\left(\frac{t-u}{s}\right)dt,$$ the frequency should be $f=\omega/(2\pi)=1/(2\pi s)$ (is it right?), where ...
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0answers
15 views

Wavelet transform and taking out of frequencies

We use a scaled wavelet and move it across the signal taking out frequencies so that they need not to be processed with a differently scaled wavelet. How does this show up in the math behind wavelet ...
0
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0answers
14 views

Directly compute image gradient from discrete wavelet or curvelet coefficients?

We know that (1) the discrete wavelet or curvelet transformation can be used in data compression, and (2) the Sobel operator can be used to compute the gradient of a 2D array of data. In the case of ...
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0answers
21 views

Fourier transform of complex wavelet for negative frequencies?

One of the requirements that complex wavelets must satisfy is their Fourier transform must be real and vanish for negative frequencies ($f<0$). However, looking at the Fourier transform of Morlet ...
0
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0answers
19 views

Lipschitz property and wavelets

I'm trying to understand Nason et al(2000)'s proof of Proposition 3.3 in this paper: http://www.maths.bris.ac.uk/~guy/Research/papers/WavProcEWS1.pdf The expectation of the uncorrected periodogram is ...
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0answers
8 views

wavelet transformation for discrete vector

I am trying to find the first 3 levels of the Haar wavelet transformation of the vector $sin(\frac{2\pi x}{16})$, where $x=0,1,\cdots,15$. So there will be $2+4+8=14$ numbers, right? How should I ...
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0answers
8 views

using wavelet transform for espisific frequencies

I want to analyze a signal in the time-frequency domain, at the frequency : 10,15 and 20 Hz, so should the input "scales" for matlab function 'cwt' be [.1 .15 .2], since the scale is the inverse of ...
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0answers
15 views

EZW parent-child relation

I’m trying to learn the EZW principle. I’m having trouble understanding the parent-child relationship. In my case, I want to use it on a 1 dimensional signal. So, let’s say for example a signal of 4 ...
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29 views

the meaning of frequency resolution

I have found in the literature the word "frequency resolution" in the comparison between DFT and wavelet transform WT , and they said that DFT provides the same frequency resolution , but WT allow ...
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25 views

two dimensional wavelets construction

I am trying to construct 2D wavelets for function approximation. Please suggest some references for constructing and implementing 2D Legendre Wavelets and other variants of it. thanks
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0answers
31 views

Wavelets - clarification of vanishing moments post

.I have read the post What is a "vanishing moment"? regarding vanishing moments. Here, it is stated in an answer how "In a sense, these conditions mean that the wavelet is "unbiased." It ...
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0answers
33 views

Short-Time-Fourier-Transform: why overlapping the window?

For STFT, we impose window of certain size onto the original signal, then we perform fft on each window. The uncertanty about frequency and time is determined by the width of the window, however, I ...
0
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0answers
21 views

Continuous wavelet over piecewise functions

Is there a wavelet function that spans the space of piecewise: linear functions(as Haar is a basis for piecewise constant functions) polynomial functions I am almost sure there is one for ...