Multiresolution analysis
Encyclopedia
A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transform
s (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 equation
s (the ironing method) and the pyramid method
s of image processing
as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James Crowley.
consists of a sequence
of nested subspaces
that satisfies certain self-similarity relations in time/space and scale/frequency, as well as completeness
and regularity relations.
.
There is, because of
, a finite sequence of coefficients 
, for 
and 
for 
, such that
Defining another function, known as mother wavelet or just the wavelet
one can see that the space
, which is defined as the linear hull of the mother wavelets integer shifts, is the orthogonal complement to 
inside 
. Or put differently, 
is the orthogonal sum of 
and 
. By self-similarity, there are scaled versions 
of 
and by completeness one has
thus the set
is a countable complete orthonormal wavelet basis in
.
Discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled...
s (DWT) and the justification for the algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
of the fast wavelet transform
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...
(FWT). It was introduced in this context in 1988/89 by Stephane Mallat
Stéphane Mallat
Stéphane G. Mallat made some fundamental contributions to the development of wavelet theory in the late 1980s and early 1990s...
and Yves Meyer
Yves Meyer
Yves F. Meyer is a French mathematician and scientist and a foremost expert on wavelets.- Biography :Meyer was a professor at the Paris Dauphine University, at École Polytechnique and currently holds a position as Professor Emeritus at the École Normale Supérieure de Cachan.- Awards and...
and has predecessors in the microlocal analysis
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...
in the theory of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s (the ironing method) and the pyramid method
Pyramid (image processing)
Pyramid or pyramid representation is a type of multi-scale signal representation developed by the computer vision, image processing and signal processing communities, in which a signal or an image is subject to repeated smoothing and subsampling...
s of image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...
as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James Crowley.
Definition
A multiresolution analysis of the spaceLp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
consists of a sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of nested subspaces
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
that satisfies certain self-similarity relations in time/space and scale/frequency, as well as completeness
Completeness
In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.-Logical completeness:In logic, semantic completeness is the converse of soundness for formal systems...
and regularity relations.
- Self-similarity in time demands that each subspace Vk is invariant under shifts by integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
multipleMultiple (mathematics)In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n , which is called the multiplier or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer...
s of 2-k. That is, for each
there is a 
with 
. - Self-similarity in scale demands that all subspaces
are time-scaled versions of each other, with scaling respectively dilationDilationDilation refers to an enlargement or expansion in bulk or extent, the opposite of contraction. It derives from the Latin dilatare, "to spread wide".In physiology:* Pupillary dilation, dilation of the pupil of the eye...
factor 2l-k. I.e., for each
there is a 
with 
. If f has limited supportSupport (mathematics)In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
, then as the support of g gets smaller, the resolution of the l-th subspace is higher than the resolution of the k-th subspace.
- Regularity demands that the model subspace V0 be generated as the linear hull (algebraicallyAlgebraic closureIn mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
or even topologically closed) of the integer shifts of one or a finite number of generating functions
or 
. Those integer shifts should at least form a frame for the subspace 
, 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 denseDense setIn topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
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 DaubechiesIngrid Daubechies
Ingrid Daubechies is a Belgian physicist and mathematician. She was between 2004 and 2011 the William R. Kenan Jr. Professor in the mathematics and applied mathematics departments at Princeton University. In January 2011 she moved to Duke University as a Professor in mathematics. She is the first...
.
There is, because of
, a finite sequence of coefficients , for and for , such thatDefining another function, known as mother wavelet or just the wavelet
one can see that the space
, which is defined as the linear hull of the mother wavelets integer shifts, is the orthogonal complement to inside . Or put differently, is the orthogonal sum of and . By self-similarity, there are scaled versions of and by completeness one hasthus the set
is a countable complete orthonormal wavelet basis in
.