For questions related to wavelets and wavelet theory

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2
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0answers
78 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
260 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
338 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 ...
7
votes
2answers
266 views

Absolute Continuity of Finite Borel Measure Characterized by Orthonormal Basis

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
votes
1answer
77 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
votes
1answer
229 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
votes
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
685 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 ...
24
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5answers
13k 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 ...
1
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0answers
106 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
votes
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 ...
3
votes
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
100 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
885 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
251 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
votes
0answers
70 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
votes
0answers
168 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 $$ ...
1
vote
0answers
83 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 $$ ...
2
votes
0answers
223 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 ...
0
votes
2answers
3k views

DFT and DWT difference?

what is the basic difference between the Discrete Fourier Transform and the Wavelet Transform ? and why does JPEG2000 preferred DWT over DCT or DFT ?
12
votes
2answers
8k views

What is a “vanishing moment”?

In this paper, Sweldens says about desireable properties of wavelets: To analyze and represent such signals we need wavelets that are local in space and frequency. Typically this is achieved by ...
2
votes
1answer
169 views

How do I divide Laurent polynomials?

I have an example from a paper (listed below) that I cannot figure out. I can divide normal polynomials, but the alternative ways to divide Laurent polynomials is beyond me at the moment. The paper ...
1
vote
1answer
598 views

Extrapolating signals using wavelets

I am an absolute beginner to wavelets, and I've read a few articles on how wavelets are used for predicting future points of a dataset, notably Wavelet prediction for Oil Prices and 1D Signal ...
0
votes
1answer
264 views

Why is the dot product of 2 wavelet domain functions a real value?

I'm working on some code here, and here is what I have done. It is based on the work by Ng et al. An example of what this looks like is here. Background: Here, a "lighting cubemap" is a bunch of ...
1
vote
1answer
144 views

What other wavelets (besides the Haar system) form a basis of $L^2(0,1)$?

The Haar system of wavelets forms a basis of $L^2[0,1]$. What other wavelets are there that also form bases of $L^2[0,1]$ (or $L^2[0,a]$ in general)?. Thanks
1
vote
0answers
114 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 ...
2
votes
1answer
168 views

Best way to find magnitude and phase of a specific frequency in an empirical time series…

I've a discrete, univariate time series, and I'm interested in to investigate a specific frequency component. Assume I'm interested in a frequency with a cycle-time of $f$ samples - and I need to get ...
2
votes
0answers
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 ...
1
vote
1answer
277 views

What is the “fourth” Haar basis function of the 4×4 Haar basis?

I know the three Haar basis functions //mother wavelet: [1 1 -1 -1] //dilation: [1 -1 0 0] //translation: [0 0 1 -1] However, the Haar basis consists of four ...
5
votes
1answer
612 views

Wavelet Theory — where do I start?

I am in the process of implementing a Fixed-Point Fast Fourier Transform. The Fixed-Point FFT requires mathematical background in the area of wavelets and lifting schemes. What are good ...
4
votes
1answer
166 views

Question in solving $\phi(t)=\phi(2t)+\phi(2t-1)$, $\phi\ne0$

Actually one can resort to the two-scale equation in multiresolution analysis. Perform Fourier transformation on both side of $\phi(t)=\phi(2t)+\phi(2t-1)$, it turns out that ...
0
votes
1answer
100 views

How to calculate dual frames under constraints?

Denote orthonormal basis in $\mathbb{R}^2$: $(\epsilon_1,\epsilon_2)=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ and ...
1
vote
2answers
586 views

How to implement the Daubechies wavelet?

http://en.wikipedia.org/wiki/Daubechies_wavelet#Transform.2C_D4 I find it is difficult to understand the pseudo-code on this Wiki page. ...
1
vote
0answers
554 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 ...
7
votes
3answers
781 views

Decomposing a discrete signal into a sum of rectangle functions

Hello math@stackexchange community ! I have a simple question that seems to have a non trivial answer. Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar ...
6
votes
3answers
613 views

An introduction to wavelets, and the wavelet transform

I am looking for a good introduction to the wavelet transform, particularly in the context of image processing. I am very comfortable with the Fourier transforms, and I've got a good background in ...
5
votes
2answers
595 views

Which time-frequency coefficients does the Wavelet transform compute?

(I asked this on Stack Overflow a while ago and didn't get a satisfying answer, so I'm trying again here.) The Fast Fourier Transform takes O(N log N) operations, while the Fast Wavelet Transform ...