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A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant
discrete wavelet transform In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal ...
s (DWT) and the justification for the
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
of the
fast wavelet transform The fast wavelet transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended ...
(FWT). It was introduced in this context in 1988/89 by
Stephane Mallat Stephane may refer to: * Stéphane, a French given name * Stephane (Ancient Greece), a vestment in ancient Greece * Stephane (Paphlagonia) Stephane ( grc, Στεφάνη) was a small port town on the coast of ancient Paphlagonia, according to Arri ...
and
Yves Meyer Yves F. Meyer (; born 19 July 1939) is a French mathematician. He is among the progenitors of wavelet theory, having proposed the Meyer wavelet. Meyer was awarded the Abel Prize in 2017. Biography Born in Paris to a Jewish family, Yves Meyer s ...
and has predecessors in the
microlocal analysis In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes genera ...
in the theory of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s (the ''ironing method'') and the pyramid methods of
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson an
James L. Crowley


Definition

A multiresolution analysis of the Lebesgue space L^2(\mathbb) consists of a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of nested subspaces ::\\dots\subset V_1\subset V_0\subset V_\subset\dots\subset V_\subset V_\subset\dots\subset L^2(\R) that satisfies certain
self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
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 An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
multiples of ''2k''. That is, for each f\in V_k,\; m\in\Z the function ''g'' defined as g(x)=f(x-m2^) also contained in V_k. * ''Self-similarity'' in ''scale'' demands that all subspaces V_k\subset V_l,\; k>l, are time-scaled versions of each other, with
scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
respectively
dilation Dilation (or dilatation) may refer to: Physiology or medicine * Cervical dilation, the widening of the cervix in childbirth, miscarriage etc. * Coronary dilation, or coronary reflex * Dilation and curettage, the opening of the cervix and surgic ...
factor 2''k-l''. I.e., for each f\in V_k there is a g\in V_l with \forall x\in\R:\;g(x)=f(2^x). * In the sequence of subspaces, for ''k''>''l'' the space resolution 2''l'' of the ''l''-th subspace is higher than the resolution 2''k'' of the ''k''-th subspace. * ''Regularity'' demands that the model subspace ''V0'' be generated as the
linear hull In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterize ...
( algebraically or even
topologically closed In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of ...
) of the integer shifts of one or a finite number of generating functions \phi or \phi_1,\dots,\phi_r. Those integer shifts should at least form a frame for the subspace V_0\subset L^2(\R) , which imposes certain conditions on the decay at
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
. The generating functions are also known as scaling functions or
father wavelets A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
. In most cases one demands of those functions to be
piecewise continuous In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
with
compact support In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
. * ''Completeness'' demands that those nested subspaces fill the whole space, i.e., their union should be
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in L^2(\R) , and that they are not too redundant, i.e., their intersection should only contain the
zero element In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive identi ...
.


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 Baroness Ingrid Daubechies ( ; ; born 17 August 1954) is a Belgian physicist and mathematician. She is best known for her work with wavelets in image compression. Daubechies is recognized for her study of the mathematical methods that enhance ...
. Assuming the scaling function has compact support, then V_0\subset V_ implies that there is a finite sequence of coefficients a_k=2 \langle\phi(x),\phi(2x-k)\rangle for , k, \leq N, and a_k=0 for , k, >N, such that :\phi(x)=\sum_^N a_k\phi(2x-k). Defining another function, known as mother wavelet or just the wavelet :\psi(x):=\sum_^N (-1)^k a_\phi(2x-k), one can show that the space W_0\subset V_, which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to V_0 inside V_. Or put differently, V_ is the orthogonal sum (denoted by \oplus) of W_0 and V_0. By self-similarity, there are scaled versions W_k of W_0 and by completeness one has :L^2(\R)=\mbox\bigoplus_W_k, thus the set :\ is a countable complete orthonormal wavelet basis in L^2(\R).


See also

*
Multiscale modeling Multiscale modeling or multiscale mathematics is the field of solving problems which have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic ac ...
*
Scale space Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theor ...
*
Time–frequency analysis In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains ''simultaneously,'' using various time–frequency representations. Rather than viewing a 1-dimensional signal (a ...
*
Wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...


References

* * * Crowley, J. L., (1982)
A Representations for Visual Information
Doctoral Thesis, Carnegie-Mellon University, 1982. * * {{cite book, first=S.G., last=Mallat, url=http://www.cmap.polytechnique.fr/~mallat/book.html, title=A Wavelet Tour of Signal Processing, publisher=Academic Press, year=1999, isbn=0-12-466606-X Time–frequency analysis Wavelets