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Multiresolution analysis

Contents

Idea

Multiresolution analysis is an approach to construct wavelets.

Let’s say person A chooses a function (f:[0,1][0,1]f: [0, 1] \to [0, 1], ff sufficiently nice) and would like to tell person B about it. According to the philosophy of multiresolution analysis, A could, for starters, tell B the overall average of the function: 0 1f(x)dx\int_0^1 f(x) \; d x. In a second step, A tells B how much the function deviates from this average on the interval [0,12][0, \frac{1}{2}] and on the interval [12,1][ \frac{1}{2}, 1]. Both players could continue this game until B can reconstruct the average of the function on any prescribed subinterval, so that she knows all values of the integral of f over every element of the Borel algebra of [0,1][0, 1].

This is a multiresolution analysis of ff in the sense that one starts with information of averages of ff on a big scale, and adds information on smaller scales to it, until ff is completely specified. (It is a multiresolution in the precise sense of the word if one uses the Haar wavelet on [0,1][0, 1].

Details

Definition

A multiresolution analysis is usually defined via 5 conditions, although we will see that these conditions are not completely independent. But for simplicity we will stick to the tradition.

Multiresolution analysis

A two-sided sequence of closed subspaces V jV_j of L 2()L^2(\mathbb{R}) is a multiresolution (analysis), if

(1) V j+1V jV_{j+1} \subset V_j, that is the higher the index, the “coarser” is the approximation of a given function in it.

(2) for every fV jf \in V_j we have f(t2)V j+1f(\frac{t}{2}) \in V_{j+1}, that is if we “spread” ff by a factor of 2, then the “next coarser” subspace contains it.

(3) lim jV j={0}\lim_{j \to \infty} V_j = \{ 0 \}

(4) lim jV j¯=L 2() \overline{ \lim_{j \to -\infty} V_j} = L^2(\mathbb{R})

(5) there is a function ϕ(t)\phi(t) in V 0V_0 such that the set consisting of ϕ(tk),k\phi(t - k), k \in \mathbb{Z} is an orthonormal basis of V 0V_0. This function is called a scaling function of the multiresolution.

The conditions are not independent:

Dependency of Conditions

Conditions (1), (2) and (5) imply (3).

For a proof see e.g. theorem 1.6 in

  • Eugenio Hernández and Guido Weiss: A first course on wavelets. (ZMATH, first edition 1996)

Wavelet Construction

In this paragraph we will sketch how one can construct a wavelet starting from a multiresolution analysis.

Let us first note that every V j+1V_{j+1} is contained in V jV_{j} and therefore has an orthogonal complement W j+1W_{j+1}, such that:

V j=V j+1W j+1 V_j = V_{j+1} \oplus W_{j+1}

Continuing this by induction letting the index on the right side go to \infty, we see that

V j= k=j+1 W k V_j = \cup_{k = j+1}^{\infty} W_{k}

Now, when we let jj \to -\infty, we see that

L 2()= W k L^2(\mathbb{R}) = \cup_{- \infty}^{\infty} W_{k}

To find an orthonormal wavelet, therefore, all we need to do is to find a function ψ\psi such that the set {ψ(tk),k}\{ \psi(t - k), k \in \mathbb{Z} \} is an orthonormal basis for W 0W_0. If we find one, then we will get a basis of every W jW_{-j} from {2 j2ψ(2 jtk),k}\{ 2^{\frac{j}{2}} \psi (2^j t - k), k \in \mathbb{Z} \}, which shows that ψ\psi is a wavelet.

We will simply state the construction: From the assumptions we get that the following expansion has to exist:

12ϕ(12t)= k= a kϕ(tk) \frac{1}{2} \phi(\frac{1}{2} t) = \sum_{k = - \infty}^{\infty} a_k \phi(t -k)

We can therefore a function that is usually called the low pass filter associated to a given multiresolution analysis:

m 0(χ):= k= a kexp(ikχ) m_0(\chi) := \sum_{k = - \infty}^{\infty} a_k \exp(i k \chi)
Wavelet Construction

Suppose ϕ\phi is a scaling function for a given multiresolution analysis, and m 0m_0 is the associated low-pass filter; then a function ψW 0\psi \in W_0 is an orthonormal wavelet if and only if

ψ^(2χ)=exp(iχ)η(2χ)m 0(χ+π)¯ϕ^(χ) \hat{\psi} (2 \chi) = \exp{(i \chi)} \; \eta(2 \chi) \; \overline{m_0 (\chi + \pi)} \; \hat{\phi}(\chi)

with a function η\eta that is measurable, 2π2 \pi-periodic with η=1\| \eta \| = 1 almost everywhere. (The hat denotes the Fourier transform.)

References

See wavelet.