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IEEE International Conference on Acoustics Speech and Signal Processing

Beyond time-frequency analysis: energy densities in one and many dimensions

Richard G. Baraniuk

Published Dec 17, 2002• 26 citations
Publisher

Abstract

Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and measures the "A" content of signals. The construction is quite general and is also easily extended to the multi-operator case, which generalizes previously derived joint densities such as the time-frequency and time-scale distributions.>

DOI
10.1109/icassp.1994.390016

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Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and measures the "A" content of signals. The construction is quite general and is also easily extended to the multi-operator case, which generalizes previously derived joint densities such as the time-frequency and time-scale distributions.>
Given a unitary operator A representing a physical quantity of interest, we employ concepts from group representation theory to define two natural signal energy densities for A. The first is invariant to A and proves useful when the effect of A is to be ignored; the second is covariant to A and measures the "A" content of signals. We also consider joint densities for multiple operators and, in the process, provide an alternative interpretation of Cohen's (see Englewood Cliffs, NJ: Prentice-Hall, 1995) general construction for joint distributions of arbitrary variables.
Fundamental to the theory of joint signal representations is the idea of associating a variable, such as time or frequency, with an operator, a concept borrowed from quantum mechanics. Each variable can be associated with a Hermitian operator, or equivalently and consistently, as we show, with a parameterized unitary operator. It is well known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal, and is hence useful when such changes in the signal are to be ignored. However, for detection or estimation of such changes, a signal representation covariant to them is needed. Using well-known results in functional analysis, we show that there always exists a translationally covariant representation; that is, an application of the operator produces a corresponding translation in the representation. This is a generalization of a recent result in which a transform covariant to dilations is presented. Using Stone's theorem, the "covariant" transform naturally leads to the definition of another, unique, dual parameterized unitary operator. This notion of duality, which we make precise, has important implications for joint distributions of arbitrary variables and their interpretation. In particular, joint distributions of dual variables are structurally equivalent to Cohen's class of time-frequency representations, and our development shows that, for two variables, the Hermitian and unitary operator correspondences can be used consistently and interchangeably if and only if the variables are dual.
This thesis presents a signal representation in terms of operators. The signal is assumed to be an element of a vector space and subject to transformations of operators. The operators form continuous groups, so-called Lie groups. The representation can be used for signals in general, in particular if spatial relations are undefinied and it does not require a basis of the signal space to be useful. Special attention is given to orthogonal operator groups which are generated by anti-Hermitian operators by means of the exponential mapping. It is shown that the eigensystem of the group generator is strongly related to properties of the corresponding operator group. For one-parameter orthogonal operator groups, a phase concept is introduced. This phase can for instance be used to distinguish between spatially even and odd signals and, therefore, corresponds to the usual phase for multi-dimensional signals. Given one operator group that represents the variation of the signal and one operator group that represents the variation of a corresponding feature descriptor, an equivariant mapping maps the signal to the descriptor such that the two operator groups correspond. Suficient conditions are derived for a general mapping to be equivariant with respect to a pair of operator groups. These conditions are expressed in terms of the generators of the two operator groups. As a special case, second order homo-geneous mappings are considered, and examples of how second order mappings can be used to obtain different types of feature descriptors are presented, in particular for operator groups that are homomorphic to rotations in two and three dimensions, respectively. A generalization of directed quadrature lters is made. All feature extraction algorithms that are presented are discussed in terms of phase invariance. Simple procedures that estimate group generators which correspond to one-parameter groups are derived and tested on an example. The resulting generator is evaluated by using its eigensystem in implementations of two feature extraction algorithms. It is shown that the resulting feature descriptor has good accuracy with respect to the corresponding feature value, even in the presence of signal noise.