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Wavelet (Rev #2, 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, many of the problems occurring in practical problems are sparse when using a wavelet representation. This is often related to the simultaneous localisation in time and space refered to above.

Details

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

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.

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

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.

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)

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)