A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ironing method) and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James L. Crowley.
Definition
A multiresolution analysis of the Lebesgue space Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L^{2}(\mathbb {R} )} consists of a sequence of nested subspaces
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{0\}\dots \subset V_{1}\subset V_{0}\subset V_{-1}\subset \dots \subset V_{-n}\subset V_{-(n+1)}\subset \dots \subset L^{2}(\mathbb {R} )}
that satisfies certain self-similarity relations in time-space and scale-frequency, as well as completeness and regularity relations.
- Self-similarity in time demands that each subspace Vk is invariant under shifts by integer multiples of 2k. That is, for each Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\in V_{k},\;m\in \mathbb {Z} } the function g defined as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x)=f(x-m2^{k})} also contained in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{k}} .
- Self-similarity in scale demands that all subspaces are time-scaled versions of each other, with scaling respectively dilation factor 2k-l. I.e., for each Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\in V_{k}} there is a with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \forall x\in \mathbb {R} :\;g(x)=f(2^{k-l}x)} .
- In the sequence of subspaces, for k>l the space resolution 2l of the l-th subspace is higher than the resolution 2k of the k-th subspace.
- Regularity demands that the model subspace V0 be generated as the linear hull (algebraically or even topologically closed) of the integer shifts of one or a finite number of generating functions or Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi _{1},\dots ,\phi _{r}} . Those integer shifts should at least form a frame for the subspace Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{0}\subset L^{2}(\mathbb {R} )} , which imposes certain conditions on the decay at infinity. The generating functions are also known as scaling functions or father wavelets. In most cases one demands of those functions to be piecewise continuous with compact support.
- Completeness demands that those nested subspaces fill the whole space, i.e., their union should be dense in , and that they are not too redundant, i.e., their intersection should only contain the zero element.
Important conclusions
In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number of deductions. The proof of existence of this class of functions is due to Ingrid Daubechies.
Assuming the scaling function has compact support, then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{0}\subset V_{-1}} implies that there is a finite sequence of coefficients Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{k}=2\langle \phi (x),\phi (2x-k)\rangle } for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |k|\leq N} , and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{k}=0} for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |k|>N} , such that
Defining another function, known as mother wavelet or just the wavelet
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \psi (x):=\sum _{k=-N}^{N}(-1)^{k}a_{1-k}\phi (2x-k),}
one can show that the space Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle W_{0}\subset V_{-1}} , which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to inside Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{-1}} .[citation needed] Or put differently, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{-1}} is the orthogonal sum (denoted by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \oplus } ) of and . By self-similarity, there are scaled versions Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle W_{k}} of and by completeness one has[citation needed]
thus the set
is a countable complete orthonormal wavelet basis in .
See also
References
- Chui, Charles K. (1992). An Introduction to Wavelets. San Diego: Academic Press. ISBN 0-585-47090-1.
- Akansu, A.N.; Haddad, R.A. (1992). Multiresolution signal decomposition: transforms, subbands, and wavelets. Academic Press. ISBN 978-0-12-047141-6.
- Crowley, J. L., (1982). A Representations for Visual Information, Doctoral Thesis, Carnegie-Mellon University, 1982.
- Burrus, C.S.; Gopinath, R.A.; Guo, H. (1997). Introduction to Wavelets and Wavelet Transforms: A Primer. Prentice-Hall. ISBN 0-13-489600-9.
- Mallat, S.G. (1999). A Wavelet Tour of Signal Processing. Academic Press. ISBN 0-12-466606-X.