# 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,}$${\ displaystyle 2 ^ {kl}}$${\ displaystyle f \ in V_ {k}}$${\ displaystyle g \ in V_ {l}}$${\ displaystyle g (x) = f (2 ^ {kl} x)}$${\ textstyle f}$ ${\ textstyle g}$${\ displaystyle 2 ^ {kl}}$
• 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 ${\ displaystyle a (Z) a (Z ^ {- 1}) + a (-Z) a (-Z ^ {- 1}) = 4}$

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}$${\ displaystyle \ varphi _ {j, k} (x) = 2 ^ {- j / 2} \ varphi (2 ^ {- j} xk)}$${\ 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_ {Kn}}$

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

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

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

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})}$