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 .
![L ^ {2} ({\ mathbb R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/8722fb232f689925a4baa0e4ba478e43ee346672)
definition
A multiscale analysis of space L² (R) consists of a series of nested subspaces
-
,
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 .
![V_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f43bfe96795a33589c12e1500b843f6268d35f2f)
![2 ^ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d82641ae2702b0db07dd11830af27b9ee0cd196)
![f \ in V_ {k}, \; m \ in \ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/814af0e779b631226db2c556b53c3f18e40481da)
![g \ in V_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc9bd4b91e49f30d364c4140f4f5f2334a515ff)
![f (x) = g (x + m2 ^ {k})](https://wikimedia.org/api/rest_v1/media/math/render/svg/0da54d0e76e13756d3df8cd80fda2c2438b3d8df)
-
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.
![V_ {k} \ subset V_ {l}, \; k> l,](https://wikimedia.org/api/rest_v1/media/math/render/svg/838f6c34aa7220eedef0873706b2628489e6bbee)
![{\ displaystyle 2 ^ {kl}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b40a2a8e03b04831674fc08817a6007b0b8d5164)
![f \ in V_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8987aa5a6243b12a22faed7f93027b1d684bcb)
![g \ in V_ {l}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80c1bac662c1d0e9bb9b89ffc605da829f740d08)
![g (x) = f (2 ^ {{kl}} x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/21c8f2bdadc0822e42ee20ddc5abc97840407e3a)
![{\ textstyle g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38dc9ad184fe5486391b456b9e68767ff77f3719)
![{\ displaystyle 2 ^ {kl}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b40a2a8e03b04831674fc08817a6007b0b8d5164)
-
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 .
![V_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae15ff9b845587dc4e1816f59c3fed0e71a132f)
![\ varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
![\ varphi _ {1}, \ dots, \ varphi _ {r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc9e0b34b39f3ef866896377bc2e72c8f296896)
![V_ {0} \ subset L ^ {2} (\ mathbb {R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/593a2c2805d4e1eb04c26140ca30eaacd2e4f73f)
-
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.
![\ textstyle \ bigcup _ {{k \ in \ mathbb {Z}}} V_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d0f794fa6a46a659e5a57132e83526a3ae2b2a2)
![L ^ {2} (\ mathbb {R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/8722fb232f689925a4baa0e4ba478e43ee346672)
![\ textstyle \ bigcap _ {{k \ in \ mathbb {Z}}} V_ {k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2abc2a9a9017f474a282bb2fb1c923314b113cc)
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 )
![\ varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
![V_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae15ff9b845587dc4e1816f59c3fed0e71a132f)
-
.
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
![a = \ {\ dots, 0, a _ {{- N}}, \ dots, a_ {N}, 0, \ dots \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63040469c9bf3e2cebe9581acdd9ba158022771b)
-
and
is fulfilled, or that the Fourier series
![{\ hat a} (\ omega): = {\ frac 12} \ sum _ {{k = -N}} ^ {N} a_ {k} e ^ {{i \ omega k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f328a5b1d7def6eaa4432cc85617fd56c7465c9)
at the zero point the value 1 and at the location of a zero point has .
![\pi](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
![{\ displaystyle a (\ pi) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae1f26824fa8a7b1ab03fd75501afac5dc7c7cca)
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
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![\ varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
![\ varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
-
and
for apply, using the Fourier series the condition is
.
![0 \ neq m \ in {\ mathbb Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d481a63fc285e6f9b0d9749c0a286ad8fdf301ff)
![| {\ hat a} (\ omega) | ^ {2} + | {\ hat a} (\ omega + \ pi) | ^ {2} \ equiv 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6de6be2b8a62b5c67a48334f7bf01fac425d5a2)
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 .
![\ textstyle a (Z) = \ sum _ {{n = -N}} ^ {N} a_ {n} Z ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/147a4103fc74801b66e41e4337e757d1120bb4bc)
![{\ hat a} (\ omega) = a (e ^ {{i \ omega}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/3988132c6f3103c020e13013a4f1d0048e6da8ce)
![a (1) = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/77d65df0c82994ce517ee373e48d47d35082354f)
![a (-1) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e73262b1aacb6efa09069e67b69ab911d72baa9)
![a (Z) = (1 + Z) ^ {A} p (Z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/126792bd8209590deed78afa08e8349c2138166b)
![0 <A \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca2f36834f09824b47b94d92989e2b0b7add6f2)
![a (Z) a (Z ^ {{- 1}}) + a (-Z) a (-Z ^ {{- 1}}) = 4](https://wikimedia.org/api/rest_v1/media/math/render/svg/b236e61a03cfd40aaf145683bb1f5576482c7afb)
Examples
- The Haar wavelet has a scaling mask
- The wavelet with order of the Daubechies family has the scaling mask
![A = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c93f57e3368569f0c80dc007e7602177572ddc4)
![a (Z) = {\ frac 14} (1 + Z) ^ {2} ((1 + Z) + {\ sqrt 3} (1-Z))](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac7b9d7483d737ae0e8c7b2c65a4ae8028d8f3b)
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 .
![\ varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
![\ varphi _ {{j, k}} (x) = 2 ^ {{- j / 2}} \ varphi (2 ^ {{- j}} xk)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ecc78238359e104cffa9a2584e55d4dafb28800)
![V_ {j} = \ operatorname {span} (\ varphi _ {{j, k}}: k \ in {\ mathbb Z})](https://wikimedia.org/api/rest_v1/media/math/render/svg/75d9801b60cd0e7290657db54fe0c4771dac78e3)
![V _ {{j + 1}} \ subset V_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6306cb579dcb464ce414755ce4c6ee8a084a21e2)
![\ {\ varphi _ {{j, k}}: k \ in {\ mathbb Z} \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71a571a5b1829688effc2c9be119481368934f3d)
![V_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c9bb9af363d8550bde7bfaee674d3fb2bba343)
The wavelet sequence can now be
defined with any odd number , where . The wavelet is thus defined as
![K \ in {\ mathbb Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecedbab629781d236a6ed9456a6f5fde705a9943)
![b = \ {\ dots, b _ {{- 1}}, b_ {0}, b_ {1}, \ dots \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17d79ae7274c5fc6818c1988f58a67c932b4e8ad)
![b_ {n}: = (- 1) ^ {n} a _ {{Kn}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0237a77a3196bd88931782e3af17be161152843d)
![\ psi (x): = \ sum _ {{n = KN}} ^ {{K + N}} b_ {n} \ cdot \ varphi (2x-n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/402eb0d07840a72ac568895456195397605d440d)
and the wavelet subspaces as . This results in an orthogonal decomposition of the scaling spaces known as herringbone
![W_ {j} = \ operatorname {span} \ left (\ psi _ {{j, k}} (x) = 2 ^ {{- j / 2}} \ psi (2 ^ {{- j}} xk) : \; k \ in {\ mathbb Z} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6c4e2a3fa9440530d2ca8f32d48d1c9fafa3d9)
and generally
at .
![V_ {J} = W _ {{J + 1}} \ oplus \ dots \ oplus W_ {M} \ oplus V_ {M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0172cb928bba81ea0ce168f65f39f9be4162948e)
![J <M](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8666cb2b3bad3cc3bb781edbdc600904fcaec8)
The basic analytical requirement of an MRA is that the wavelet subspaces fully utilize the, that is, should be a dense subspace of .
![L ^ {2} ({\ mathbb R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/8722fb232f689925a4baa0e4ba478e43ee346672)
![\ textstyle \ bigoplus _ {{n = - \ infty}} ^ {\ infty} W_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1385afcacddd60c87fb4e86f566ded196c9b988a)
![L ^ {2} ({\ mathbb R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/8722fb232f689925a4baa0e4ba478e43ee346672)
literature