The Azimuth Project
Multiresolution analysis (changes)

Showing changes from revision #4 to #5: Added | Removed | Changed



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].



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.)


See wavelet.

sex shop sex shop sex shop sex shop sex shop lingerie sex shop atacado calcinhas uniformes profissionais uniformes dicas de sexo