NONORTHOGONAL EXPANSIONS OF SIGNALS AND SOME OF THEIR APPLICATIONS C. Cenker H. G. Feichtinger K. Grochenig Abstract We want to present a quick overview of new types of series expansions which have been \discovered" in the last ve years. They nd more and more applications in signal analysis, image processing, numerical analysis and other areas. This wide range of applicability comes from the \unusual" exibility of these series expansions. Furthermore good algorithms allowing practical calculations based on these expansions are available. That maybe why an unusual high percentage of funding is going into wavelet theory these days in the USA and in France. 1 Introduction If we expand a signal (image, function, distribution) f by P i2i i g i we want to get a (discrete) representation of f that \extracts" informations from f which are not obvious by just looking at f. For instance, given the time behavior f(t), the frequency spectrum ^f(w) is hard to see, and, vice versa, in the frequency decomposition ^f(w) the amplitudes and phases of f are not visible. It is one of the objectives of signal analysis to nd intermediate representations in a simultaneous time and frequency picture (phase and space), e. g., the Wigner distribution. As discussed in [7], the recently discovered series expansions are an excellent tool for time frequency description of signals. Before describing the various non{classical expansions let us shortly review possible requirements to be imposed on series representations of a signal of the form P i2i i g i. (A) Locality: The basic functions g i should be well concentrated, e. g. of compact support or exponential decay, in a way that allows to infer from the size of a coecient i how much of f is concentrated near the support of the corresponding g i, and decay of the coecient{sequence should reect decay of f (at innity).
(B) Smoothness: It should be possible to recognize by the coecients whether and where the functions under consideration are smooth. This means once again, that the functions g i should have good localization properties. Like in classical theory one should have only a few large coecients and faster convergence of the series for smooth functions. Having to work with only few coecients is also important for numerical calculations, because the computational eort is reduced and roundo errors are thus minimized. Furthermore we want smooth building blocks, as this guarantees e. g., that good approximation properties imply good smoothness properties of the function itself. Conversely it is clear that a bad building block (e. g. a rectangular function) forces a large number of terms, even/especially for smooth functions (because any nite sum of these rectangular blocks is very likely to have some sharp corners somewhere, which have to be lled by additional terms). (C) Linearity: The sequence of coecients = ( i ) i2i should depend linearly on the signal f, since non-linearity makes calculations complicated. (D) Constructivity: We look for constructive methods to generate such expansions as easy as possible. (E) Flexibility: We want to choose the basic functions g i from a class of (nice) signals in order to be able to adapt the expansions to a given problem (or to an operator, or to a proposed interpretation). The classical orthogonal expansions with respect to special functions (such as Hermite functions) usually lack this property. (F) Redundancy & Stability: In some cases the coecients i are unique and the functions g i form a basis in the mathematical sense. This leads to very concise and elegant representations of signals. However, stability is often more important, and modest redundancy is prefered to achieve it. Thus in order to guarantee (A){(F) one should not insist on uniqueness! It turns out that until now the best way to meet these requirements is to choose frames fg i j i 2 Ig with coherent states g i as basic functions, that is, they are all simple transforms of a single function g, e. g. shifts in time and/or frequency and/or dilations of g. Denition: A frame in a Hilbert space IH is a system of functions fe i j i 2 Ig such that Akfk X i2i j < e i ; f > j 2! 1 2 Bkfk; 0 < A B < 1 : (1) This formula is equivalent to the following assertion: The operator D : f 7! P i2i < e i; f > e i is (boundedly) invertible on IH with inverse
D 1. If we set g i = D 1 e i, fg i j i 2 Ig is then called the dual frame to fe i j i 2 Ig, we obtain two types of expansions of f: f = D(D 1 f) = X i2i < e i ; D 1 f > e i = X i2i < g i ; f > e i (2) and f = D 1 (Df) = X i2i < e i ; f > D 1 e i = X i2i < e i ; f > g i : (3) Equation (2) gives the desired expansion with respect to the frame fe i j i 2 Ig, whereas (3) states that f can be completely reconstructed from the frame coecients < e i ; f >. Moreover, since D 1 can be written as a Neumann series, i. e. D 1 = P 1 n=0(i D) n, the reconstruction can be done by iteration of (I D) (essentially). In order to simplify the presentation we give the details only for the one{dimensional case, but all results stated will hold true with only slight modications in higher dimensions (with applications in image analysis, for example). On IR the basic functions g i are considered to be of the form g mn (t) = e imt g(t n) or g jk (t) = j 2 g( j t k) (4) for a xed g. The rst system can be understood as a collection of building blocks which are obtained from one single function by time{frequency shifts. So we expect that its coecients reect the time{frequency content of a given signal at time n and frequency m (roughly). The second system gives a family of peaked functions, each of them having the shape of a wave (with positive and negative values). The eect of the dilations is, that there are very peaked versions of the function (for these the translation parameter is correspondingly smaller) and also at versions. From this it should be clear that the functions g mn and g jk have the same time and frequency localization as the original g (In the case of scaling the frequency localization is adapted to the frequency band under consideration!). Thus we are aiming at Gabor{type expansions or at wavelet expansions f(t) = X m;n mn e imt g(t n) (5) f(t) = X j;k jk j 2 g( j t k) : (6) The existence and construction of coherent frames (both, Gabor and wavelet type) has been extensively investigated in [7, 14, 15, 16, 23]. Furthermore it has to be mentioned, that all orthonormal bases are frames because of (1). Gabor expansions already have a long history [21], but because of stability problems [26], which have been settled in a
rigorous mathematical theory only in the last decade, they were not much used. Wavelet theory is quite young, starting with Y. Meyer's invention (cf. [34, 35, 29]), and has become a booming eld (cf. [24, 7, 8, 30, 10],... ) with many applications. In the following sections we describe the main types of how to fet the coecients of our series expansions. 2 Orthonormal Bases It can be shown that there exist special functions g 2 L 2 (IR) such that the family fg jk j j; k 2 Zg ; with g jk (t) = 2 j 2 g(2 j t k); (7) forms a complete orthonormal basis (wavelet basis) of L 2 (IR) (cf. [29]). Thus any f in L 2 (IR) has an orthonormal expansion f = X j;k < g jk ; f > g jk ; (8) which meets at least the requirements (A){(D) and (F) of above. It is obvious, that the function g must be a very carefully chosen function (actually, the existence of such functions was quite doubtful before Y. Meyer's important discovery, cf. [34, 35]). Nowadays it is clear, that there are many ways to construct wavelet bases. The function g can be chosen to be a Schwartz function, bandlimited (i. e. with compactly supported Fourier transform, cf. [29]), a spline function (see [36, 28, 30, 31]) or a compactly supported function with arbitrary but nite degree of smoothness (cf. [8]), which appears to be very useful from the point of practical calculations (see also [6]). The importance of these wavelet bases goes far beyond the Hilbert space L 2 (IR). Actually, they are unconditional bases for a variety of classical function spaces (cf. [29, 22]), including the Besov spaces B s p;q (IRm ) and the Bessel potential spaces L p s (IRm ). Actually, the membership of f to one of these spaces can be characterized by the size j < g jk ; f > j of the coecients. The situation is much worse with respect to a system of the form fe 2int g(t k) ; n; k 2 Zg (9) if such a system should be an orthonormal basis for L 2 (IR), e. g. in the case where g = c (0;1] or = = 1. It can be shown that such sytems miss either requirement (A) or (B), i. e. they cannot be well concentrated (in time) and smooth at the same time (cf. the arguments due to the Theorem of Balian{Coifman{Semmes given in [4, 7, 6]).
This incompatibility of the requirements of orthonormality and simultaneous time{frequency concentration (which will be relevant if the coecients should have an interpretation as the contribution of a certain frequency at a given time) makes it necessary to abandon the rst requirement and to look for a more exible concept than that of an orthonormal expansion, trying to keep as many good (wanted) properties as possible. It turns out that the next best tools are systems that work like orthonormal systems. 3 Tight Frames A tight frame for a Hilbert space IH is a system of functions fe i j i 2 Ig such that: f = X i2i < e i ; f > e i and kfk = X i2i j < e i ; f > j 2! 1 2 8f 2 IH : (10) This denition is equivalent to the assumption A = B in (1). In contrast to an orthonormal basis the system fe i j i 2 Ig need not even be linear independent and is thus overcomplete in general. The coecients of the e i 's in the representation of f are not unique anymore, but still the choice of the coecients to be < e i ; f > is convenient and suitable ([9]). In the wavelet case the rst tight frames of the form g jk (t) = 2 j 2 g(2 j t k) were given by Frazier and Jawerth (see [18, 19]) in order to obtain atomic decompositions of Besov{ Triebel{Lizorkin spaces. Again, a good time and frequency localization (i. e. good decay and smoothness properties) of g allows the characterization of smoothness of a signal f in terms of the decay properties of the coecients < g jk ; f >. Likewise, there are basic functions g 2 L 2 (IR) of arbitrary smoothness, actually C 1 in this case, such that the functions g mn (t) = e imt g(t n) with, depending on g form a tight frame (which should be called a tight Gabor frame) for L 2 (IR) and also for Bessel{potential spaces. A systematic study of tight frames with g having compact support for both the Gabor and the wavelet case has been carried out in [9]. From there we know that tight frames are relative easy to construct and exist in abundance (in contrast to the rare phenomenon of a basis). Later on research on tight frames has been abandoned in favor of wavelet bases. Except for the uniqueness of the representation these tight frames satisfy all the properties we have asked for and moreover, since they yield easy expansions of signals with low computational costs, it may be expected that they will be of increasing importance in numerical and signal analysis.
4 Frames Sometimes orthonormal bases or tight frames are still too rigid concepts. Suppose that g is the ideal response of a loudspeaker (or the measurement of an ideal heartbeat). In order to compare the actual response f to the ideal one we would like to decompose f as a series of the form (5) or (6). The coecients in such an expansion then eciently and quantitatively measure the dierence between f and g and allow to detect hidden irregularities. Thus we have to expand f with respect to transformations (shifts, dilates,... ) of a given function in contrast to sections 2 and 3, where the basic g has to be constructed. In order to manage these problems we use the concept of (general) frames (cf. (1),(3),(2)). Let us just state two very special results, which show the existence of frames of Gabor and wavelet type respectively, using \nice" basic functions. Gabor-type frames: If g 2 L 2 (IR) is a \nice" function, e. g. satisfying Z IR jg(x)j2 + j^g(x)j 2 (1 + jxj) 2+ dx < 1 for some > 0 ; then there exist 0 ; 0 > 0 such that for any 0 < < 0 and 0 < < 0 the set n o g mn (t) = e imt g(t n); m; n 2 Z constitutes a frame for L 2 (IR). According to (3) this implies that the coecients < g mn ; f >= Z IR e imt g(t n) f(t) dt uniquely determine f. But these values are nothing but a sampled version of the Short Time Fourier Transform with window g! Thus the general theory of coherent frames has immediate and useful consequences in signal analysis (cf. also [2, 3]). Frames of wavelet type: In a similar way if g 2 L 2 (IR) is not too wild, e. g. belongs to some Besov space B 1=2 1;1 there exist 0 > 0 and 0 > 1 such that for any 0 < 0 and 1 < < 0 the family n o g jk (t) = j 2 g( j t k); j; k 2 Z is a frame for L 2 (IR). Again by (3) this is equivalent to the fact that it is sucient to know certain sampling values of the so called wavelet transform V g (f)(x; s) = Z IR s 1 2 g(s 1 (t x)) f(t) dt : (11)
5 Banach Frames, Atomic Decompositions Most approaches to the problem of coherent frames use direct methods involving the Fourier transform, Poisson's formula etc., and mainly Hilbert space methods. Our approach in [14, 15, 16, 6] starts with the observation that the transformations by which the systems fg jk g and fg mn g are obtained from their mother atom g are related to certain group representations. Convolutions over (non-)abelian groups (involving suitable function spaces on these groups) and group representation theory are then the essential tools for the construction of atomic decompositions [14, 15] and frames [23]. What do we mean by atomic decompositions? Let f be a function in some Banach space IB. Then X f = < g i ; f > e i ; (12) i2i where the frame functions g i have to be in the dual space IB 0 and the atoms e i are in IB. Furthermore atoms and frames are dual to each other, that is, a set of atoms fe i j i 2 Ig for IB is a frame for IB 0 and vice versa (cf. [23, 6]). This approach admits for an unexpected large class of basic functions g and also allows to deal with irregular sampling of the short time Fourier transform and the wavelet transform ([23], cf. [2, 3]). Finally, as in the case of orthonormal systems and tight frames smoothness and decay properties of the signal can be characterized by growth properties of the frame coecients (cf. [23], where the concept of frames is extended to Banach spaces; cf. also [6], where a short introduction to the frame and wavelet theory of L 2 (IR) is given, and reasons and ways of the expansion of this theory to other function spaces are shown). 6 Some Applications We conclude by mentioning a few of the most interesting applications. The one{dimensional wavelet transform has been found to be a useful tool for music analysis and synthesis (cf. [27]). The scaling property can be used to slow down or speed up audio signals without distortion. Especially for transient signals the wavelet transform seems to perform better than the method of \harmonic" Fourier analysis. Gabor{type and wavelet expansions will also be a tool for describing the auditory system, as the receptors are not orthonormal ones but each of them covers a small frequency range. Hence such expansions may be used not only for en- and decoding music but also for speech recognition.
There are now models of vision which are based on Gabor{type expansions (cf. [37, 11, 12]). In this context it is important to have more freedom to choose the basic function g as it is assumed in the classical theory [14, 15, 16]. Since wavelet expansions contain the idea of scaling, they are a powerful tool for the dedection of irregularities and singularities in a signal function f (cf. [20]). The local maxima of the wavelet transform e. g. correspond with the edges of the original signal [33]. As an important application of wavelet theory to image processing one has to mention the work of S. Mallat [32] on multiscale resolution of images which provides very ecient algorithms and has caused much interest in the eld. His ideas, establishing the connection to pyramidal image representations, are intimately connected with the construction of wavelet bases over IR 2. Two dimensional Gabor functions are proposed to be used in texture analysis. In order to recognize textures images are encoded into multiple narrow spatial frequency and orientation channels. Thus such functions are useful, as they have easy tuneable orientation and radial frequency bandwidths, furthermore you get optimal joint resolution in space and frequency [5]. The coecients in Gabor{type or wavelet expansions yield a very compact representation of a signal f. Numerical experiments show that very good data compression rates can be obtained in this way (cf. [11, 12, 33]). Summarizing we can say that these new methods of (coherent) series expansions from the perfect, but quite rigid wavelet bases to the extremely exible frames (involving more eort in the reconstruction) are more and more recognized as an important branch of applied mathematics. References [1] BASTIAANS, M. J.: Gabor's expansion of a signal into Gaussian elementary signals. Proc. IEEE Vol. 68, (1980), 538{539. [2] BASTIAANS, M. J.: A sampling theorem for the complex spectrogram and Gabor's expansion of a signal in Gaussian elementary signals. Opt. Eng. 20 (1981), 594{598. [3] BASTIAANS, M. J.: Local{frequency description of optical signals and systems. Techn. Report Eindhoven Univ. Techn., April 1988. [4] BENEDETTO, J./HEIL, C./WALNUT, D.: Remarks on the proof of the Balian{Low theorem. Preprint, MITRE 1990.
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