Comparison of Discrete and Continuous Wavelet Transforms

Disclaimer: This glossary has the structure of four columns. A number of terms are listed line by line, and each line is followed by explanation. Some “terms” have up to four separate (yet commonly accepted) names. Mathematically, functions may map between any two sets, say, from X to Y ; but if X is a probability space (typically called Ω),

Work supported in part by the U.S. National Science Foundation. The full version with figures can be found at http://www.siue.edu/ ∼msong/Research/ency.pdf.

it comes with a σ-algebra B of measurable sets, and probability measure P . Elements E in B are called events, and P(E) the probability of E. Corresponding measurable functions with values in a vector space are called random variables, a terminology which suggests a stochastic viewpoint. The function values of a random variable may represent the outcomes of an experiment, for example “throwing of a die.” Yet, function theory is widely used in engineering where functions are typically thought of as signal. In this case, X may be the real line for time, or R d . Engineers visualize functions as signals. A particular signal may have a stochastic component, and this feature simply introduces an extra stochastic variable into the “signal,” for example noise.

Turning to physics, in our present application, the physical functions will be typically be in some L 2 -space, and L 2 -functions with unit norm represent quantum mechanical “states.”

Mathematically, a sequence is a function defined on the integers Z or on subsets of Z, for example the natural numbers N. Hence, if time is discrete, this to the engineer represents a time series, such as a speech signal, or any measurement which depends on time. But we will also allow functions on lattices such as Z d .

In the case d = 2, we may be considering the grayscale numbers which represent exposure in a digital camera. In this case, the function (grayscale) is defined on a subset of Z 2 , and is then simply a matrix.

A random walk on Z d is an assignment of a sequential and random motion as a function of time. The randomness presupposes assigned probabilities. But we will use the term “random walk” also in connection with random walks on combinatorial trees. nested subspaces refinement multiresolution scales of visual resolutions While finite or infinite families of nested subspaces are ubiquitous in mathematics, and have been popular in Hilbert space theory for generations (at least since the 1930s), this idea was revived in a different guise in 1986 by Stéphane Mallat, then an engineering graduate student. In its adaptation to wavelets, the idea is now referred to as the multiresolution method.

What made the idea especially popular in the wavelet community was that it offered a skeleton on which various discrete algorithms in applied mathematics could be attached and turned into wavelet constructions in harmonic analysis. In fact what we now call multiresolutions have come to signify a crucial link between the world of discrete wavelet algorithms, which are popular in computational mathematics and in engineering (signal/image processing, data mining, etc.) on the one side, and on the other side continuous wavelet bases in function spaces, especially in L 2 (R d ). Further, the multiresolution idea closely mimics how fractals are analyzed with the use of finite function systems.

But in mathematics, or more precisely in operator theory, the underlying idea dates back to work of John von Neumann, Norbert Wiener, and Herman Wold, where nested and closed subspaces in Hilbert space were used extensively in an axiomatic approach to stationary processes, especially for time series. Wold proved that any (stationary) time series can be decomposed into two different parts: The first (deterministic) part can be exactly described by a linear combination of its own past, while the second part is the opposite extreme; it is unitary, in the language of von Neumann. von Neumann’s version of the same theorem is a pillar in operator theory. It states that every isometry in a Hilbert space H is the unique sum of a shift isometry and a unitary operator, i.e., the initial Hilbert space H splits canonically as an orthogonal sum of two subspaces H s and H u in H, one which carries the shift operator, and the other H u the unitary part. The shift isometry is defined from a nested scale of closed spaces V n , such that the intersection of these spaces is H u . Specifically,

However, Stéphane Mallat was motivated instead by the notion of scales of resolutions in the sense of optics. This in turn is based on a certain “a

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