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1answer
94 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) = ...
2
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2answers
48 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 ...
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1answer
43 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
69 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 ...
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2answers
229 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 ...
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1answer
98 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 ...
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1answer
61 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 ...
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3answers
136 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
78 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
110 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
184 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
573 views

Wavelets: Cone Of Influence

While reading this paper I came across the term Cone of Influence which is described as ...
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1answer
832 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
77 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
38 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) ...
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0answers
128 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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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
132 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 $p_i(x)$: ...
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0answers
108 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
65 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
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1answer
226 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
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1answer
281 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 ...
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0answers
158 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 ...
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1answer
74 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 ...
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1answer
182 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 ...
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0answers
77 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
444 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 ...
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5answers
8k 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
101 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
157 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 ...
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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
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1answer
94 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
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2answers
747 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 ...
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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 ...
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0answers
63 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 $$ ...
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0answers
152 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 $$ ...
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0answers
75 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
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0answers
220 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 ...
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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 ?
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2answers
6k 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
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1answer
136 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 ...
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1answer
463 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 ...
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1answer
212 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 ...
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1answer
126 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
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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 ...
2
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1answer
146 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 ...
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0answers
154 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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1answer
257 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 ...
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1answer
544 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 ...