For questions related to wavelets and wavelet theory

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What mathematical background is preliminary for reading and understanding books/papers on wavelets?

Please excuse my english. I have had the following math courses for mechatronics engineering education: Calculus (single and multivariable) Linear algebra (introductory) Differential equations (ode'...
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26 views

Why is this true about MRA-wavelets?

In the article Characterization of wavelets and MRA wavelets on local fields of positive characteristic, in page 11, As seen in the picture, The author refers to 16(Multiresolution analysis on local ...
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How theorem 3.1 is a consequence of lemmas 3.2-3.4?

In the article "Tight wavelet frames on local fields", the author states that "Theorem 3.1 is an easy consequence of lemmas 3.2-3.4". How?
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20 views

The range of the coefficients of 2 level Wavelet LeGall 5/3 2D transform

As in the title, I am confused about the range of the coefficients of Wavelet LeGall 5/3 (has to be exact this filter) 2D transform (only for a 8*8 block) if the value of the input matrix are within ...
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1answer
510 views

Wavelet or FFT for Transient signal analysis?

For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frenquency domain. On the other hand I ...
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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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3answers
210 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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1answer
53 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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38 views

How to prove that $f(t)=t\sin (\frac{1}{t})$ is 1-Hölder continuous for fixed $t_0 \in [-1,1]$?

How to prove that $f(t)=t\sin (\frac{1}{t})$ is 1-Hölder continuous (Lipschitz continuous) for fixed $t_0 \in [-1,1]$? $f(t)$ is defined as $0$ at $t_0 =0$. That is, for $t_0 \in [-1,1]$, there ...
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How is the study of wavelets not just a special case of Fourier analysis?

As far as I can tell, "wavelets" is just a neologism for certain "non-smooth" families of functions which constitute orthonormal bases/families for $L^2[0,1]$. How is wavelet analysis anything new ...
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22 views

Wavelet reference for PDE

Can anyone recommend a very readable introduction to wavelets for use in theoretical PDE/harmonic analysis? I'm frustrated with the account in Lemarie-Rieusset's Navier-Stokes book since he provides ...
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1answer
19 views

Is there any approach to computer vision that doesn't make use of geometry?

I've long been interested in applying my background in functional analysis (especially wavelets) and other related areas to actually create something with "real world" value (not that I don't enjoy ...
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13 views

What different between huffman and EBCOT?

I have a question about entropy coding and Image compression. Huffman coding and EBCOT coding are entropy coding. correct or not? I knew about huffman coding but I don't know EBCOT, How it working?...
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65 views

where can I find the proof of this theorem of wavelet frames?

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
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58 views

A basis for $\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\}$, and how compute coordinates

Let $$\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\},$$ where we take $$\eta_\alpha(t)= \left\{ \frac{\alpha}{t} \right\} -\alpha \left\{ \frac{1}{t} \right\},$$ and $ \left\{ x ...
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2answers
47 views

What does $\psi \in \mathscr{L}^2(\mathbb{R}^2)^7$ mean?

I'm aware that $\mathscr{L}^2(S)$ usually means the $L^p$-norm over some set in the context of metric spaces. So I guess that the first part "$\mathscr{L}^2(\mathbb{R}^2)$" means a set of $L^2$-norms ...
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Daubechies wavelet for structural dynamic analysis

I am working on the structural dynamic response with Daubechies wavelet. The Equation of Motion of dynamic system with single degree of freedom can be described as $$ m\ddot{x}(t) +c\dot{x}(t)+k{x}(t)...
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1answer
74 views

Function Approximation by Wavelet method

If $ f(x)\in{L^{2}(R)}$ , then $ f(x) $ can be projected into the subspace $ V_{j}$ as, $$ P_{j}f(x) = \sum\limits_{k\in{Z}}c_{j,k}\phi_{j,k}(x) $$ where $ k\in{Z}$. The projection equation is ...
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22 views

Why is the scaling and shifting variable for wavelet transform is chosen to be multiple of 2?

The continuous wavelet transform is $f \mapsto \gamma(s,\tau)=\int f(t)\psi(\frac{t-\tau}{s})dt$ where we choose the variables s and $ \tau $ to be discrete integers from -$\infty$ to $+\infty$. We ...
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8 views

Can you find shearlets in CNNs?

I read that you can find wavelets in the first layers of a CNN. Can you also find shearlets, and if not, why? Edit (further explanations): I read that, when you train a CNN the filters in the first ...
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3answers
75 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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751 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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31 views

Non-uniform convolution with discrete wavelet transform

I understand that if you have a circular N-dimensional convolution matrix, it can be diagonalized with the Fourier transform of the convolution operator. This makes it easy to calculate the density of ...
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1answer
23 views

Admissibility condition for wavelets

The admissibility condition for a wavelet $\psi$ is: $\int \frac{|\hat\psi(x)|^2}{|x|} dx < \infty$, with $\hat\psi$ the Fourier transform of $\psi$. A necessary and sufficient condition should ...
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40 views

How can I analyse signal with discrete wavelet transform?

With CWT it's clear enough. We have function of two variables which are scale and translation. But what about DWT? Here is Matlab code: ...
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Smoothness in Haar basis

The rate of decay of fourier co-efficients of a function $f$ determines the order of differentiability of $f$. Is there an equivalent result for the case when Fourier basis is replaced by Haar wavelet ...
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1answer
40 views

What is wavelet tranform in simple words?

I have read wiki and other sources and have still problem understanding the wavelet transform. What is the basic idea in simple words? Does the Fourier uncertainty hold for wavelet transform?
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80 views

How the second and third equalities can be achieved?

I am reading this paper. In the Proof of Lemma 3.3, How the second(*) equality can be achieved? How can i use Parseval's identity in third(**) equality?
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2answers
35 views

Find an function that oscillates between a given upper and lower envelope

Suppose I'm given two real, continuous functions $f(x)$ and $g(x)$ such that $f(x)\ge g(x)$ for all real $x$. I'd like to determine an oscillating function $h(x)$ that has $f(x)$ as its upper-envelope ...
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How to obtain an analytic approximation of this wavelet-looking function?

I would like the approximating function to be infinitely differentiable. I have tried fitting (1) a polynomial and (2) a combination of sin, cos, and exp functions to approximate the function shown ...
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14 views

Orthogonal basis for scaling function

I computed a scaling function of a wavelet [phi, ~, ~] = wavefun('dB4',6); . Because I want to calculate the coefficients to annhilate polynomials, I need a dual ...
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Choose a Daubechies wavelet to approximate a polynomial function

Assume I have a polynomial function $s(x) = \sum\limits_{i=0}^d a_i x^i$, and $\boldsymbol{s}_{d}$ is a discretized sample from $s(x)$, i.e. $\boldsymbol{s}_{d}=(s(0), s(\Delta x), \ldots, s((n-1)\...
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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 ...
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930 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 (...
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Multi dimensional multiresolution analysis, designing biorthogonal wavelets.

With inspiration from this question, I'm wondering if biorthogonal bases for arbitrary dimensions are possible to construct with the same mechanism. I am thinking a subsampling of a factor of $N$ in ...
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Biorthogonal (discrete wavelet) noise bases?

I am slightly interested in discrete wavelet transforms (DWT), but so far I have mostly used already-derived and existing well-known wavelets, such as Daubechies, Cohen-Daubechies-Faveau, Symlets and ...
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Using wavelets to capture the $L^2$ norm of $f''$

The goal I have in mind is to express the $L^2$ norm of the second derivative of a quite regular function as a sum of some coefficients. My idea was that such coefficients involved some system of ...
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14 views

Interpretation of diagonal detail in Haar Wavelet Transforms

I am a statistics grad student, and I have just begun exploring the topic of wavelet regression (specifically, Haar wavelets for discrete functions). I understand the generalization from a one ...
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32 views

Wavelet zero mean condition

I'm trying to understand more about wavelets so I went and read the wikipedia article and some other papers on the topic. I have learned that wavelet functions are compact supported and belong to the ...
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74 views

Scale resolution and frequency resolution in continuous wavelet transform

I'm reading the well known wavelets tutorial by Robi Polikar here. In part 3, about figure 3.7 and 3.8, it says "lower scales (higher frequencies) have better scale resolution (narrower in scale, ...
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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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What are the different type of Daubechies Wavelet transform?

Like Daub4 are there others named as Daub2, daub3 or we only have daub4 , daub8, daub16? What is the order of a transform(represented usually by N)? Does this order have any resemblance with the ...
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2answers
105 views

A polynomial that satisfies $x^pf(1-x) + (1-x)^pf(x) = 1$

The context of this question is the construction of the Daubechies wavelet. $f$ is a polynomial of degree $p-1$ which satisfies the equation: $$ x^pf(1-x) + (1-x)^pf(x) = 1 \tag{1} $$ Since $$ f(x)...
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Invariance of stationary wavelet transform

Suppose we are given 64 points $x_1,\ldots ,x_{64}$ and divide them into two groups $x_1,\ldots, x_{32}$ and $x_{33},\ldots , x_{64}$. Then we apply stationary wavelet transform to both these groups ...
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223 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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171 views

Signal differentiation by wavelet transform

I have a noisy signal x(t) that I want to differentiate in time, in order to obtain x'(t). Though numerical finite difference approximantion does not work well on noisy signal, I would like to ...
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39 views

Wavelet through the lens of Harmonic analysis

I need some references for starting to study about wavelet. I have enough information about abstract Harmonic analysis. Thanks!
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Build polynomial orthogonal to set of other pre-defined polynomials

Basically. the question is simple. Is there any algorithm so I can build a polynomial, orthogonal for the set of pre-defined polynomials? I need the algorithm(like Gram-Schmidt) that would be ...
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29 views

How to orthogonalize piecewise polynomial with respect to set of piecewise polynomials?

Currently I'm working with multiwavelet basis(need it for solving the stochastic ODE system as described here) and can't understand how to build it. Basically, I have a set of functions $p_{i} = x^{i}$...
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38 views

Absolute convergence related to Fourier analysis

If the Fourier transform of a function $f \in L^2$, whose frequency $\xi$ satisfies $|\xi| \leq \pi$, has compact support, it is famous that \begin{align*} f(x) = \sum_n f(n)\frac{\sin\pi(x-n)}{\pi(...