Multi-scale analysis

The multi-scale analysis ( MRA , English: m ulti r esolution a nalysis ) or -approximation ( MSA , English: m ulti s cale a pproximation ) of the function space is a functional analytical basic construction of the wavelet theory, which describes the approximation properties of the discrete wavelet transformation . In particular, it explains the possibility and functionality of the fast wavelet transform algorithm . ${\displaystyle L^{2}(\mathbb {R} )}$

definition

A multiscale analysis of space L² (R) consists of a series of nested subspaces

${\displaystyle \{0\}\subset \dots \subset V_{2}\subset V_{1}\subset V_{0}\subset V_{-1}\subset V_{-2}\subset \dots V_{-n}\subset \dots \subset L^{2}(\mathbb {R} )}$,

which fulfills both self-similarity conditions in time / space and scale / frequency as well as completeness and regularity conditions.

• Self-similarity in time requires that every subspace is invariant with shifts by integral multiples of . This means that for every function there is a function with . ${\displaystyle V_{k}}$ ${\displaystyle 2^{k}}$ ${\displaystyle f\in V_{k},\;m\in \mathbb {Z} }$ ${\displaystyle g\in V_{k}}$ ${\displaystyle f(x)=g(x+m2^{k})}$
• Self-similarity between different scales requires that all subspaces are time- scaled copies of one another, with the scaling or stretching factor being. This means that for every function there is a function with . For example , if the beam has a restricted beam , the beam is compressed by a factor of . In other words, the resolution (in the sense of points on a screen) of the l th subspace is higher than the resolution of the k th subspace. ${\displaystyle V_{k}\subset V_{l},\;k>l,}$ ${\displaystyle 2^{k-l}}$ ${\displaystyle f\in V_{k}}$ ${\displaystyle g\in V_{l}}$ ${\displaystyle g(x)=f(2^{k-l}x)}$ ${\textstyle f}$ ${\textstyle g}$ ${\displaystyle 2^{k-l}}$

• Regularity requires that the model subspace the linear hull (or even algebraically topologically closed) of the integer shifts one or a finite number of generating functions or is. These integer shifts should at least form a Riesz basis, but better a Hilbert basis of the subspace , from which a rapid decrease in the infinite of the generating functions follows. The latter is trivially fulfilled for functions with a compact carrier. The generating functions are called scaling functions or father wavelets . Often they are constructed as (piece-wise) continuous functions with a compact carrier . ${\displaystyle V_{0}}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi _{1},\dots ,\varphi _{r}}$ ${\displaystyle V_{0}\subset L^{2}(\mathbb {R} )}$

• Completeness requires that these nested subspaces fill the entire space, that is, their union should be tight in ; also that they are not redundant, that is, their average may only contain the zero element. ${\displaystyle \textstyle \bigcup _{k\in \mathbb {Z} }V_{k}}$ ${\displaystyle L^{2}(\mathbb {R} )}$ ${\displaystyle \textstyle \bigcap _{k\in \mathbb {Z} }V_{k}}$

Scaling function

In practically the most important case, that there is only one scaling function with compact carrier in the MRA and that this generates a Hilbert basis in the subspace , this fulfills a two-scale equation (in English literature: refinement equation ) ${\displaystyle \varphi }$ ${\displaystyle V_{0}}$

${\displaystyle \varphi (x)=\sum _{n=-N}^{N}a_{n}\cdot \varphi (2x-n)}$.

The sequence of numbers occurring there is called the scaling sequence or mask and must be a discrete low-pass filter , which in this case means that ${\displaystyle a=\{\dots ,0,a_{-N},\dots ,a_{N},0,\dots \}}$

${\displaystyle \sum _{n=-N}^{N}a_{n}=2}$ and ${\displaystyle \sum _{n=-N}^{N}(-1)^{n}a_{n}=0}$

is fulfilled, or that the Fourier series

${\displaystyle {\hat {a}}(\omega ):={\frac {1}{2}}\sum _{k=-N}^{N}a_{k}e^{i\omega k}}$

at the zero point the value 1 and at the location of a zero point has . ${\displaystyle \pi }$ ${\displaystyle a(\pi )=0}$

It is a basic task of wavelet design to determine conditions under which the desired properties of , such as continuity , differentiability etc. follow. Should be orthogonal, i.e. H. be perpendicular to all integer shifts of itself, so must ${\displaystyle a}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi }$

${\displaystyle \sum _{n=-N}^{N}a_{n}^{2}=2}$ and ${\displaystyle \sum _{n=-N}^{N}a_{n}a_{n+2m}=0}$

for apply, using the Fourier series the condition is . ${\displaystyle 0\neq m\in \mathbb {Z} }$ ${\displaystyle |{\hat {a}}(\omega )|^{2}+|{\hat {a}}(\omega +\pi )|^{2}\equiv 1}$

Usually these sequences are given as coefficient sequences of a Laurent polynomial , that is . The normalization is thus written as , the low-pass property as or for a , the orthogonality condition as . ${\displaystyle \textstyle a(Z)=\sum _{n=-N}^{N}a_{n}Z^{n}}$ ${\displaystyle {\hat {a}}(\omega )=a(e^{i\omega })}$ ${\displaystyle a(1)=2}$ ${\displaystyle a(-1)=0}$ ${\displaystyle a(Z)=(1+Z)^{A}p(Z)}$ ${\displaystyle 0<A\in \mathbb {N} }$ ${\displaystyle a(Z)a(Z^{-1})+a(-Z)a(-Z^{-1})=4}$

Examples

• The Haar wavelet has a scaling mask ${\displaystyle a(Z)=1+Z}$
• The wavelet with order of the Daubechies family has the scaling mask ${\displaystyle A=2}$

${\displaystyle a(Z)={\frac {1}{4}}(1+Z)^{2}((1+Z)+{\sqrt {3}}(1-Z))}$

Nested subspaces

Let be an orthogonal scaling function. Then an affine function system and a sequence of scaling subspaces can be defined. So then applies and is an orthonormal basis of . ${\displaystyle \varphi }$ ${\displaystyle \varphi _{j,k}(x)=2^{-j/2}\varphi (2^{-j}x-k)}$ ${\displaystyle V_{j}=\operatorname {span} (\varphi _{j,k}:k\in \mathbb {Z} )}$ ${\displaystyle V_{j+1}\subset V_{j}}$ ${\displaystyle \{\varphi _{j,k}:k\in \mathbb {Z} \}}$ ${\displaystyle V_{j}}$

The wavelet sequence can now be defined with any odd number , where . The wavelet is thus defined as ${\displaystyle K\in \mathbb {Z} }$ ${\displaystyle b=\{\dots ,b_{-1},b_{0},b_{1},\dots \}}$ ${\displaystyle b_{n}:=(-1)^{n}a_{K-n}}$

${\displaystyle \psi (x):=\sum _{n=K-N}^{K+N}b_{n}\cdot \varphi (2x-n)}$

and the wavelet subspaces as . This results in an orthogonal decomposition of the scaling spaces known as herringbone ${\displaystyle W_{j}=\operatorname {span} \left(\psi _{j,k}(x)=2^{-j/2}\psi (2^{-j}x-k):\;k\in \mathbb {Z} \right)}$

${\displaystyle V_{0}=W_{1}\oplus V_{1}=W_{1}\oplus W_{2}\oplus V_{2}=\dots }$and generally at . ${\displaystyle V_{J}=W_{J+1}\oplus \dots \oplus W_{M}\oplus V_{M}}$ ${\displaystyle J<M}$

The basic analytical requirement of an MRA is that the wavelet subspaces fully utilize the, that is, should be a dense subspace of . ${\displaystyle L^{2}(\mathbb {R} )}$ ${\displaystyle \textstyle \bigoplus _{n=-\infty }^{\infty }W_{n}}$ ${\displaystyle L^{2}(\mathbb {R} )}$

literature

• Alfred Louis , Peter Maaß , Andreas Rieder : Wavelets: Theory and Applications . 2nd Edition. Teubner, 1998, ISBN 3-519-12094-1 .