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Wavelet (Rev #3, changes)

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Contents

Idea

This page is about wavelets that constitute a further development of central ideas of the Fourier transform. Wavelets are important in numerical mathematics, statistical analysis and simulations, especially in the Earth sciences.

As is the case with the Fourier transform, there is both a continuous theory of functions of the real numbers and a discrete theory of finite and infinite series.

Motivation for Physicists, Continuous Transform

Heisenberg’s uncertainty relation says that one cannot know both the location and the impulse of an elementary quantum particle: If we know the location exactly (the wave function is a delta function in position space), we don’t know anything about the impulse (the wave function is multiple of e ixpe^{i x p} in impulse space). Vice versa, if we know the impulse exactly, we don’t know anything about the location.

The Fourier transform of a function in position space is in a generalized sense a coefficient expansion in a basis of functions e iωte^{i \omega t} that are optimally localized in impulse space and maximal not localized in position space.

Wavelets on the other hand are orthonormal bases or, more general, frames of the Hilbert space of wave functions such that if we learn about the coefficients of a wave function with respect to this basis, we learn simultaneously about the location and the impulse of the particle described by the wave function. That is, wavelets are constructed in a way that they are both localized in position and in impulse space. This makes them often more useful in practical problems than the Fourier transform.

Motivation from sparseness

“Dual representations” such as a Fourier transform or wavelet transform are generally used to map a complete “function” (e.g., set of (time,value) pairs over a time interval) to an alternate representation, often with little knowledge of how the function is formed. Mostly, although not always, the advantage of using a dual representation is that it converts a dense (i.e., not sparse) representation into a representation which is either

  • exactly sparse, or

  • effectively sparse, in the sense that many small coefficients can be zeroed with negligable deviation of the overall signal from the original.

This sparsity is useful both for:

  1. Dramatically reducing the amount of computation in actual problems, eg, allowing sparse matrix solvers to be used.

  2. Restricting the number of non-zero wavelet coefficients can be used as a mechanism to prevent overfitting.

Whilst Fourier representations are sparse for certain kinds of problem, problems, many of the problems occurring in practical problems application are sparse when using a wavelet representation. This is often related to the simultaneous localisation in time and space refered to above.

Motivation from Music

Imagine listening to a symphony. The representation of the signal in time is e.g. the air pressure at your ears. The Fourier transform of the signal is the representation in frequency space that specifies the amount that each frequency contributes to the whole symphony. Both representations are obviously not very useful to humans :-)

The representation of the signal via notes is a wavelet representation, every note represents a contribution of a signal that is both localized in time and in frequency.

Details

We will first define several concepts of the continuous wavelet transform before we concentrate on the discrete transform and its applications.

The Continuous Transform

The following definition of a wavelet may appear rather technical right know, but we will explain the motivation behind it further down.

Definition

Wavelet

A function ψL 2()\psi \in L^2(\mathbb{R}) and unit norm ψ=1\|\psi\| = 1 is called a wavelet if it fullfills the following consistency condition

C ψ=2π |ψ(ω)^| 2|ω|dω=:C ψ< C_{\psi} = 2 \pi \int_{- \infty}^{\infty} \frac{|\widehat{\psi(\omega)}|^2}{|\omega|} d\omega =: C_{\psi} \lt \infty

where ψ^\widehat{\psi} denotes the Fourier transform.

We will often write

ψ a,b(t):=1(|a|)ψ(tba) \psi_{a, b}(t) := \frac{1}{\sqrt(|a|)} \psi(\frac{t-b}{a})

Now that we know what a wavelet is, we are ready to define the wavelet transform:

Definition

Continuous Wavelet Transform

Let ψL 2()\psi \in L^2(\mathbb{R}) be a wavelet. The continuous wavelet transform Wf of a function fL 2()\f \in L^2(\mathbb{R}) is the following function:

Wf: 0x Wf: \mathbb{R}_{\neq 0} x \mathbb{R} \to \mathbb{C}
Wf(a,b)=f,ψ a,b=1(|a|) f(t)ψ(tba)¯dt Wf(a, b) = \langle f, \psi_{a, b} \rangle = \frac{1}{\sqrt(|a|)} \int_{- \infty}^{\infty} \; f(t) \; \overline{\psi(\frac{t -b}{a})} \; dt

The numbers Wf(a,b)Wf(a, b) are called the wavelet coefficients of f.

A wavelet coefficient of a function is supposed to tell us about the mean of this function with respect to a certain time and frequency interval (“interval” in an approximate sense). In order to be a mean value, the wavelet itself should put equal weight to the positive and the negative part of the sample function. The next proposition tells us that the consistency condition implies this, if we add an additional assumption.

Proposition

Wavelet Mean is Zero Let ψ\psi be a wavelet. If in addition tψL 1t \psi \in L^1{\mathbb{R}}, then we have

ψ(t)dt=0 \infty_{- \infty}^{\infty} \psi(t) \; dt = 0

Just like the Fourier transform is invertible, there is an inversion formula for the wavelet transform too. It was discovered by A.P. Calderón in 1964 and rediscovered by A.Grossmann and J.Morlet in 1984:

Definition Proposition

Inversion Formula (Calderón , Grossmann-Morlet )

Let ψ\psi be a wavelet and in addition ψL 1()\psi \in L^1(\mathbb{R}). Then for every function fL 2()\f \in L^2(\mathbb{R}) we have the following inversion formula of the continuous wavelet transform:

f=lim ϵ01C ψ |a|ϵ Wf(a,b)ψ a,b()dbdaa 2 f = \lim_{\epsilon \to 0} \; \frac{1}{C_{\psi}} \; \int_{|a| \ge \epsilon} \int_{- \infty}^{\infty} \; Wf(a, b) \; \psi_{a, b} (\cdot) \frac{db \; da}{a^2}

in L 2L^2{\mathbb{R}}. To be more precise, the right hand side defines for every ϵ>0\epsilon \gt 0 a function L 2()\in L^2(\mathbb{R}) which converges to f in L 2()L^2(\mathbb{R}) as ϵ0\epsilon \to 0.

This “full fledged” inversion formula may look a little bit complicated, and its proof certainly is. If we add some “nicety” assumptions, we get an inversion formula that is considerably easier to prove, and guarantees point-wise convergence instead of convergence in L 2L^2, which is more interesting for many applications:

Proposition

Simplified Inversion Formula

Let ψ\psi be a wavelet. Let fL 2()f \in L^2(\mathbb{R}) be continuous and both fL 1()f \in L^1(\mathbb{R}) and f^L 1()\hat{f} \in L^1(\mathbb{R}). Then for every tt \in \mathbb{R} we have:

f(t)=1C ψ Wf(a,b)ψ a,b()dbdaa 2 f(t) = \; \frac{1}{C_{\psi}} \; \int_{- \infty}^{\infty} \; \int_{- \infty}^{\infty} \; Wf(a, b) \; \psi_{a, b} (\cdot) \frac{db \; da}{a^2}

Note that the convergence is pointwise for every tt.

References

There are a lot of textbooks about wavelets so any list of references is biased.

Numerical Mathematics and Signal Processing

  • Stéphane Mallat: A wavelet tour of signal processing. The sparse way. 3rd ed. (ZMATH)

  • Yves Meyer: Wavelets and operators. (ZMATH)

  • Stéphane Jaffard, Yves Meyer and Robert D. Ryan: Wavelets. Tools for science and technology. (ZMATH)

  • Karsten Urban: Wavelets in Numerical Simulation (ZMATH)

Applications in Physics

Wavelets also play a role in physics:

  • Syed Twareque Ali, Jean-Pierre Antoine and Jean-Pierre Gazeau: Coherent states, wavelets and their generalizations. (ZMATH)

  • G. Battle: Wavelets and renormalization. (ZMATH)