Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

Transcription

1 SIAM REVIEW Vol. 43,No. 4, c 2001 Society for Industrial and Alied Mathematics Nonuniform Samling and Reconstruction in Shift-Invariant Saces Akram Aldroubi Karlheinz Gröchenig Abstract. This article discusses modern techniques for nonuniform samling and reconstruction of functions in shift-invariant saces. It is a survey as well as a research aer and rovides a unified framework for uniform and nonuniform samling and reconstruction in shiftinvariant saces by bringing together wavelet theory, frame theory, reroducing kernel Hilbert saces, aroximation theory, amalgam saces, and samling. Insired by alications taken from communication, astronomy, and medicine, the following asects will be emhasized: (a) The samling roblem is well defined within the setting of shift-invariant saces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform samling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the resence of noise. (d) To model the natural decay conditions of real signals and images, the samling theory is develoed in weighted L -saces. Key words. nonuniform samling, irregular samling, samling, reconstruction, wavelets, shift-invariant saces, frame, reroducing kernel Hilbert sace, weighted L -saces, amalgam saces AMS subject classifications. 41A15,42C15, 46A35, 46E15, 46N99, 47B37 PII. S Introduction. Modern digital data rocessing of functions (or signals or images) always uses a discretized version of the original signal f that is obtained by samling f on a discrete set. The question then arises whether and how f can be recovered from its samles. Therefore, the objective of research on the samling roblem is twofold. The first goal is to quantify the conditions under which it is ossible to recover articular classes of functions from different sets of discrete samles. The second goal is to use these analytical results to develo exlicit reconstruction schemes for the analysis and rocessing of digital data. Secifically, the samling roblem consists of two main arts: (a) Given a class of functions V on R d, find conditions on samling sets X = {x j R d : j J}, where J is a countable index set, under which a function f V can be reconstructed uniquely and stably from its samles {f(x j ): x j X}. Received by the editors January 24, 2001; acceted for ublication (in revised form) Aril 2, 2001; ublished electronically October 31, The U.S. Government retains a nonexclusive, royalty-free license to ublish or reroduce the ublished form of this contribution, or allow others to do so, for U.S. Government uroses. Coyright is owned by SIAM to the extent not limited by these rights. htt:// Deartment of Mathematics, Vanderbilt University, Nashville, TN (aldroubi@math. vanderbilt.edu). This author s research was suorted in art by NSF grant DMS Deartment of Mathematics U-3009, University of Connecticut, Storrs, CT (groch@ math.uconn.edu). 585

2 586 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG Fig. 1.1 The samling roblem. To: A function f defined on R has been samled on a uniform grid. Bottom: The same function f has been samled on a nonuniformly saced set. The samling locations x j are marked by the symbol, and the samled values f(x j ) by a circle o. (b) Find efficient and fast numerical algorithms that recover any function f V from its samles on X. In some alications, it is justified to assume that the samling set X = {x j : j J} is uniform, i.e., that X forms a regular n-dimensional Cartesian grid; see Figures 1.1 and 1.2. For examle, a digital image is often acquired by samling light intensities on a uniform grid. Data acquisition requirements and the ability to rocess and reconstruct the data simly and efficiently often justify this tye of uniform data collection. However, in many realistic situations the data are known only on a nonuniformly saced samling set. This nonuniformity is a fact of life and revents the use of the standard methods from Fourier analysis. The following examles are tyical and indicate that nonuniform samling roblems are ervasive in science and engineering. Communication theory: When data from a uniformly samled signal (function) are lost, the result is generally a sequence of nonuniform samles. This scenario is usually referred to as a missing data roblem. Often, missing samles are due to the artial destruction of storage devices, e.g., scratches on a CD. As an illustration, in Figure 1.3 we simulate a missing data roblem by randomly removing samles from a slice of a three-dimensional magnetic resonance (MR) digital image. Astronomical measurements: The measurement of star luminosity gives rise to extremely nonuniformly samled time series. Daylight eriods and adverse nighttime weather conditions revent regular data collection (see, e.g., [111] and the references therein). Medical imaging: Comuterized tomograhy (CT) and magnetic resonance imaging (MRI) frequently use the nonuniform olar and siral samling sets (see Figure 1.2 and [21, 90]).

3 NONUNIFORM SAMPLING AND RECONSTRUCTION Cartesian uniform samling grid 5 Polar samling grid Siral samling grid 5 Nonuniform samling grid Fig. 1.2 Samling grids. To left: Because of its simlicity the uniform Cartesian samling grid is used in signal and image rocessing whenever ossible. To right: A olar samling grid used in comuterized tomograhy (see [90]). In this case, the two-dimensional Fourier transform ˆf is samled with the goal of reconstructing f. Bottom left: Siral samling used for fast MRI by direct signal reconstruction from sectral data on sirals [21]. Bottom right: A tyical nonuniform samling set as encountered in sectroscoy, astronomy, geohysics, and other signal and image rocessing alications. Original digital image Digital image with missing data Fig. 1.3 The missing data roblem. Left: Original digital MRI image with samles. Right: MRI image with 50% randomly missing samles. Other alications using nonuniform samling sets occur in geohysics [92], sectroscoy [101], general signal/image rocessing [13, 22, 103, 106], and biomedical imaging [20, 59, 90, 101] (see Figures 1.2 and 1.4). More information about modern techniques for nonuniform samling and alications can be found in [16].

4 588 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG Fig. 1.4 Samling and boundary reconstruction from ultrasonic images. Left: Detected edge oints of the left ventricle of a heart from a two-dimensional ultrasound image constitute a nonuniform samling of the left ventricle s contour. Right: Boundary of the left ventricle reconstructed from the detected edge samle oints (see [59]) Samling in Paley Wiener Saces: Bandlimited Functions. Since infinitely many functions can have the same samled values on X = {x j } R d, the samling roblem becomes meaningful only after imosing some a riori conditions on f. The standard assumtion is that the function f on R d belongs to the sace of bandlimited functions B Ω ; i.e., the Fourier transform ˆf(ξ) = R d f(x)e 2πi ξ,x dx of f is such that ˆf(ξ) = 0 for all ξ / Ω=[ ω, ω] d for some ω < (see, e.g., [15, 44, 47, 55, 62, 72, 78, 88, 51, 112] and the review aers [27, 61, 65]). The reason for this assumtion is a classical result of Whittaker [114] in comlex analysis which states that, for dimension d = 1, a function f L 2 (R) B [ 1/2,1/2] can be recovered exactly from its samles {f(k) : k Z} by the interolation formula (1.1) f(x) = k Z f(k) sinc(x k), sin πx where sinc(x) = πx. This series gave rise to the uniform samling theory of Shannon [96], which is fundamental in engineering and digital signal rocessing because it gives a framework for converting analog signals into sequences of numbers. These sequences can then be rocessed digitally and converted back to analog signals via (1.1). Taking the Fourier transform of (1.1) and using the fact that the Fourier transform of the sinc function is the characteristic function χ [ 1/2,1/2] shows that for any ξ [ 1/2, 1/2] ˆf(ξ) = k f(k)e 2πikξ = k ˆf,e i2πk L 2 ( 1/2,1/2) e i2πkξ. Thus, reconstruction by means of the formula (1.1) is equivalent to the fact that the set {e i2πkξ,k Z} forms an orthonormal basis of L 2 ( 1/2, 1/2) called the harmonic Fourier basis. This equivalence between the harmonic Fourier basis and the reconstruction of a uniformly samled bandlimited function has been extended to treat some secial cases of nonuniformly samled data. In articular, the results by Paley and Wiener [87], Kadec [71], and others on the nonharmonic Fourier bases {e i2πx kξ, k Z} can be translated into results about nonuniform samling and reconstruction of bandlimited functions [15, 62, 89, 94]. For examle, Kadec s theorem

5 NONUNIFORM SAMPLING AND RECONSTRUCTION 589 [71] states that if X = {x k R : x k k L<1/4} for all k Z, then the set {e i2πxkξ,k Z} is a Riesz basis of L 2 ( 1/2, 1/2); i.e., {e i2πxkξ,k Z} is the image of an orthonormal basis of L 2 ( 1/2, 1/2) under a bounded and invertible oerator from L 2 ( 1/2, 1/2) onto L 2 ( 1/2, 1/2). Using Fourier transform methods, this result imlies that any bandlimited function f L 2 B [ 1/2,1/2] can be comletely recovered from its samles f(x k ),k Z, as long as the samling set is of the form X = {x k R : x k k < 1/4} k Z. The samling set X = {x k R : x k k < 1/4} k Z in Kadec s theorem is just a erturbation of Z. For more general samling sets, the work of Beurling [23, 24], Landau [74], and others [18, 58] rovides a dee understanding of the one-dimensional theory of nonuniform samling of bandlimited functions. Secifically, for the exact and stable reconstruction of a bandlimited function f from its samles {f(x j ):x j X}, it is sufficient that the Beurling density (1.2) D(X) = lim inf #X (y +[0,r]) r y R r satisfies D(X) > 1. Conversely, if f is uniquely and stably determined by its samles on X R, then D(X) 1 [74]. The marginal case D(X) = 1 is very comlicated and is treated in [79, 89, 94]. It should be emhasized that these results deal with stable reconstructions. This means that an inequality of the form f C 1/ f(x j ) x j X holds for all bandlimited functions f L B Ω. A samling set for which the reconstruction is stable in this sense is called a (stable) set of samling. This terminology is used to contrast a set of samling with the weaker notion of a set of uniqueness. X is a set of uniqueness for B Ω if f X = 0 imlies that f = 0. Whereas a set of samling for B [ 1/2,1/2] has a density D 1, there are sets of uniqueness with arbitrarily small density. See [73, 25] for examles and characterizations of sets of uniqueness. While the theorems of Paley and Wiener and Kadec about Riesz bases consisting of comlex exonentials e i2πxkξ are equivalent to statements about samling sets that are erturbations of Z, the results about arbitrary sets of samling are connected to the more general notion of frames introduced by Duffin and Schaeffer [40]. The concet of frames generalizes the notion of orthogonal bases and Riesz bases in Hilbert saces and of unconditional bases in some Banach saces [2, 5, 6, 12, 14, 15, 20, 28, 29, 46, 66, 97] Samling in Shift-Invariant Saces. The series (1.1) shows that the sace of bandlimited functions B [ 1/2,1/2] is identical with the sace { } (1.3) V 2 (sinc) = c k sinc(x k) : (c k ) l 2. k Z Since the sinc function has infinite suort and slow decay, the sace of bandlimited functions is often unsuitable for numerical imlementations. For instance, the ointwise evaluation f f(x 0 )= k Z c k sinc(x 0 k)

6 590 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG is a nonlocal oeration, because, as a consequence of the long-range behavior of sinc, many coefficients c k will contribute to the value f(x 0 ). In fact, all bandlimited functions have infinite suort since they are analytic. Moreover, functions that are measured in alications tend to have frequency comonents that decay for higher frequencies, but these functions are not bandlimited in the strict sense. Thus, it has been advantageous to use non-bandlimited models that retain some of the simlicity and structure of bandlimited models but are more amenable to numerical imlementation and are more flexible for aroximating real data [13, 63, 64, 86, 103, 104]. One such examle are the shift-invariant saces which form the focus of this aer. A shift-invariant sace is a sace of functions on R d of the form r V (φ 1,...,φ r )= c i jφ i (x j). i=1 j Z d Such saces have been used in finite elements and aroximation theory [34, 35, 67, 68, 69, 98] and for the construction of multiresolution aroximations and wavelets [32, 33, 39, 53, 60, 70, 82, 83, 95, 98, 99, 100]. They have been extensively studied in recent years (see, for instance, [6, 19, 52, 67, 68, 69]). Samling in shift-invariant saces that are not bandlimited is a suitable and realistic model for many alications, e.g., for taking into account real acquisition and reconstruction devices, for modeling signals with smoother sectrum than is the case with bandlimited functions, or for numerical imlementation [9, 13, 22, 26, 85, 86, 103, 104, 107, 110, 115, 116]. These requirements can often be met by choosing aroriate functions φ i. This may mean that the functions φ i have a shae corresonding to a articular imulse resonse of a device, or that they are comactly suorted, or that they have a Fourier transform ˆφ i (ξ) that decays smoothly to zero as ξ Uniform Samling in Shift-Invariant Saces. Early results on samling in shift-invariant saces concentrated on the roblem of uniform samling [7, 9, 10, 11, 37, 64, 105, 108, 107, 113, 116] or interlaced uniform samling [110]. The roblem of uniform samling in shift-invariant saces shares some similarities with Shannon s samling theorem in that it requires only the Poisson summation formula and a few facts about Riesz bases [7, 9]. The connection between interolation in sline saces, filtering of signals, and Shannon s samling theory was established in [11, 109]. These results imly that Shannon s samling theory can be viewed as a limiting case of olynomial sline interolation when the order of the sline tends to infinity [11, 109]. Furthermore, Shannon s samling theory is a secial case of interolation in shiftinvariant saces [7, 9, 113, 116] and a limiting case for the interolation in certain families of shift-invariant saces V (φ n ) that are obtained by a generator φ n = φ φ consisting of the n-fold convolution of a single generator φ [9]. In alications, signals do not in general belong to a rescribed shift-invariant sace. Thus, when using the bandlimited theory, the common ractice in engineering is to force the function f to become bandlimited before samling. Mathematically, this corresonds to multilication of the Fourier transform ˆf of f by a characteristic function χ Ω. The new function f a with Fourier transform ˆf a = ˆfχ Ω is then samled and stored digitally for later rocessing or reconstruction. The multilication by χ Ω before samling is called refiltering with an ideal filter and is used to reduce the errors in reconstructions called aliasing errors. It has been shown that the three stes of the traditional uniform samling rocedure, namely refiltering, samling, and ostfiltering for reconstruction, are equivalent to finding the best L 2 -aroximation of

7 NONUNIFORM SAMPLING AND RECONSTRUCTION 591 a function in L 2 B Ω [9, 105]. This rocedure generalizes to samling in general shiftinvariant saces [7, 9, 10, 85, 105, 108]. In fact, the reconstruction from the samles of a function should be considered as an aroximation in the shift-invariant sace generated by the imulse resonse of the samling device. This allows a reconstruction that otimally fits the available samles and can be done using fast algorithms [106, 107] Nonuniform Samling in Shift-Invariant Saces. The roblem of nonuniform samling in general shift-invariant saces is more recent [4, 5, 30, 66, 75, 76, 77, 102, 119]. The earliest results [31, 77] concentrate on erturbation of regular samling in shift-invariant saces and are therefore similar in sirit to Kadec s result for bandlimited functions. For the L 2 case in dimension d = 1, and under some restrictions on the shift-invariant saces, several theorems on nonuniform samling can be found in [76, 102]. Moreover, a lower bound on the maximal distance between two samling oints needed for reconstructing a function from its samles was given for the case of olynomial slines and other secial cases of shift-invariant saces in [76]. For the general multivariate case in L, the theory was develoed in [4], and for the case of olynomial sline shift-invariant saces, the maximal allowable ga between samles was obtained in [5]. For general shift-invariant saces, a Beurling density D 1is necessary for stable reconstruction [5]. As in the case of bandlimited functions, the theory of frames is central in nonuniform samling of shift-invariant saces, and there is an equivalence between a certain tye of frame and the roblem of samling in shift-invariant saces [5, 66, 75]. The aim of the remainder of this aer is to rovide a unified framework for uniform and nonuniform samling in shift-invariant saces. This is accomlished by bringing together wavelet theory, frame theory, reroducing kernel Hilbert saces, aroximation theory, amalgam saces, and samling. This combination simlifies some arts of the literature on samling. We also hoe that this unified theory will rovide the ground for more interactions between mathematicians, engineers, and other scientists who are using the theory of samling and reconstruction in secific alications. The aer is intended as a survey, but it contains several new results. In articular, all the well-known results are develoed in weighted L -saces. Extensions of frame theory and reroducing kernel Hilbert saces to Banach saces are discussed, and the connections between reroducing kernels in weighted L -saces, Banach frames, and samling are described. In the sirit of a review, we focus on the discussion of the samling roblem and results, and we ostone the technical details and roofs to the end of each section or to section 8. The reader more interested in the alications and techniques can omit the roofs in a first reading. The aer is organized as follows. Section 2 introduces the relevant saces for samling theory and resents some of their roerties. Weighted L -saces and sequence saces are defined in section 2.1. Wiener amalgam saces are discussed in section 2.2, where we also derive some convolution relations in the style of Young s inequalities. The weighted L -shift-invariant saces are introduced in section 2.3, and some of their main roerties are established. The samling roblem in weighted shift-invariant saces is stated in section 3. In sections 4.1 and 4.2 some asects of reroducing kernel Hilbert saces and frame theory are reviewed. The discussion includes an extension of frame theory and reroducing kernel Hilbert saces to Banach saces. The connections between reroducing kernels in weighted L -saces, Banach frames, and samling are discussed in section 4.3. Frame algorithms for the recon-

8 592 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG struction of a function from its samles are discussed in section 5. Section 6 discusses iterative reconstructions. In alications, a function f does not belong to a articular rescribed sace V, in general. Moreover, even if the assumtion that a function f belongs to a articular sace V is valid, the samles of f are not exact due to digital inaccuracy, or the samles are corruted by noise when they are obtained by a real measuring device. For this reason, section 7 discusses the results of the various reconstruction algorithms when the samles are corruted by noise, which is an imortant issue in ractical alications. The roofs of the lemmas and theorems of sections 6 and 7 are given in section Function Saces. This section rovides the basic framework for treating nonuniform samling in weighted shift-invariant saces. The shift-invariant saces under consideration are of the form (2.1) V (φ) = c k φ( k), k Z d where c =(c k ) k Z is taken from some sequence sace and φ is the so-called generator of V (φ). Before it is ossible to give a recise definition of shift-invariant saces, we need to study the convergence roerties of the series k Z c kφ( k). In the context d of the samling roblem the functions in V (φ) must also be continuous. In addition, we want to control the growth or decay at infinity of the functions in V (φ). Thus the generator φ and the associated sequence sace cannot be chosen arbitrarily. To settle these questions, we first discuss weighted L -saces with secific classes of weight functions (section 2.1), and we then develo the main roerties of amalgam saces (section 2.2). Only then will we give a rigorous definition of a shift-invariant sace and derive its main roerties in section 2.3. Shift-invariant saces figure rominently in other areas of alied mathematics, notably in wavelet theory and aroximation theory [33, 34]. Our resentation will be adated to the requirements of samling theory Weighted L -Saces. To model decay or growth of functions, we use weighted L -saces [41]. A function f belongs to L (R d ) with weight function if f belongs to L (R d ). Equied with the norm f L = f L, the sace L is a Banach sace. If the weight function grows raidly as x, then the functions in L decay roughly at a corresonding rate. Conversely, if the weight function decays raidly, then the functions in L may grow as x. In general, a weight function is just a nonnegative function. We will use two secial tyes of weight functions. The weight functions denoted by ω are always assumed to be continuous, symmetric, i.e., ω(x) = ω( x), ositive, and submultilicative: (2.2) 0 <ω(x + y) ω(x)ω(y) x, y R d. This submultilicativity condition imlies that 1 ω(0) ω(x) for all x R d. For a technical reason, we imose the growth condition n=1 log ω(nk) n 2 < k Z d. Although most of the results do not require this extra condition on ω, we use it in Lemma For simlicity we refer to ω as a submultilicative weight. A rototyical

9 NONUNIFORM SAMPLING AND RECONSTRUCTION 593 examle is the Sobolev weight ω(x) =(1+ x ) α, with α 0. When ω = 1, we obtain the usual L -saces. In addition, a weight function is called moderate with resect to the submultilicative weight ω, or simly ω-moderate, if it is continuous, symmetric, and ositive and satisfies (x + y) Cω(x)(y) for all x, y R d. For instance, the weights (x) =(1+ x ) β are moderate with resect to ω(x) =(1+ x ) α if and only if β α. If is ω-moderate, then (y) =(x + y x) Cω( x)(x + y), and it follows that 1 (x + y) Cω(x) 1 (y). Thus, the weight 1 is also ω-moderate. If is ω-moderate, then a simle comutation shows that f( y) L Cω(y) f L, and in articular, f( y) L ω ω(y) f L ω. Conversely, if L is translation-invariant, then ω(x) = su f L 1 f(. x) L is submultilicative and is ω-moderate. To see this, we note that ω(x) = su f(. x) L f L 1 is the oerator norm of the translation oerator f f( x). Since oerator norms are submultilicative, it follows that ω(x + y) ω(x)ω(y). Moreover, f(t x) (t) dt = f(t) (t + x) dt ω(x) f(t) (t) dt. R d R d R d Thus, (x + y) ω(x)(y), and therefore the weighted L -saces with a moderate weight are exactly the translation-invariant saces. We also consider the weighted sequence saces l (Z d ) with weight : a sequence {(c k ): k Z d } belongs to l if ((c) k )=(c k k ) belongs to l with norm c l = c l, where ( k ) is the restriction of to Z d Wiener Amalgam Saces. For the samling roblem we also need to control the local behavior of functions so that the samling oeration f (f(x j )) is at least well defined. This is done conveniently with the hel of the Wiener amalgam saces W (L ). These consist of functions that are locally in L and globally in L. To be recise, a measurable function f belongs to W (L ), 1 <, if it satisfies (2.3) f W (L ) = k Z d ess su{ f(x + k) (k) ; x [0, 1] d } <. If =, a measurable function f belongs to W (L ) if it satisfies (2.4) f W (L ) = su {ess su{ f(x + k) (k); x [0, 1] d }} <. k Z d Note that W (L ) coincides with L. Endowed with this norm, W (L ) becomes a Banach sace [43, 45]. Moreover, it is translation invariant; i.e., if f W (L ), then f( y) W (L ) and f( y) W (L ) Cω(y) f W (L ).

10 594 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG The subsace of continuous functions W 0 (L )=W (C, L ) W (L ) is a closed subsace of W (L ) and thus also a Banach sace [43, 45]. We have the following inclusions between the various saces. Theorem 2.1. Let be ω-moderate and 1 q. Then the following inclusions hold: (i) W 0 (L ) W 0 (L q ) and W (L ) W (L q ) L q. (ii) W 0 (L ω) W 0 (L ), W (L ω) W (L ), and L ω L. The following convolution relations in the style of Young s theorem [118] are useful. Theorem 2.2. Let be ω-moderate. (i) If f L and g L 1 ω, then f g L and f g L C f L g L 1 ω. (ii) If f L and g W (L 1 ω), then f g W (L ) and f g W (L ) C f L g W (L 1 ω ). (iii) If c l and d l 1 ω, then c d l and c d l C c l d l 1 ω. Remark 2.1. Amalgam saces and their generalizations have been investigated by Feichtinger, and the results of Theorem 2.1 can be found in [42, 43, 45, 44]. The results and methods develoed by Feichtinger can also be used to deduce Theorem 2.2. However, for the sake of comleteness, in section 2.4 we resent direct roofs of Theorems 2.1 and 2.2 that do not rely on the dee results of amalgam saces Shift-Invariant Saces. This section discusses shift-invariant saces and their basic roerties. Although some of the following observations are known in wavelet and aroximation theory, they have received little attention in connection with samling. Given a so-called generator φ, we consider shift-invariant saces of the form (2.5) V (φ) = c k φ( k) : c l. k Z d If = 1, we simly write V (φ). The weight function controls the decay or growth rate of the functions in V (φ). To some extent, the arameter also controls the growth of the functions in V (φ), but more imortantly, controls the norm we wish to use for measuring the size of our functions. For some alications in image rocessing, the choice = 1 is aroriate [36]; = 2 corresonds to the energy norm, and = is used as a measure in some quality control alications. Moreover, the smoothness of a function and its aroriate value of, 1 <, for a given class of signals or images can be estimated using wavelet decomosition techniques [36]. The determination of and the signal smoothness are used for otimal comression and coding of signals and images. For the saces V (φ) to be well defined, some additional conditions on the generator φ must be imosed. For = 1 and = 2, the standard condition in wavelet theory is often stated in the Fourier domain as (2.6) 0 <m â φ (ξ) = ˆφ(ξ + j) 2 M< for almost every ξ, j Z d

11 NONUNIFORM SAMPLING AND RECONSTRUCTION 595 for some constants m>0 and M>0 [80, 81]. This condition imlies that V 2 (φ) isa closed subsace of L 2 and that { φ( k) : k Z d} is a Riesz basis of V 2 (φ), i.e., the image of an orthonormal basis under an invertible linear transformation [33]. The theory of Riesz bases asserts the existence of a dual basis. Secifically, for any Riesz basis for V 2 (φ) of the form { φ( k) : k Z d}, there exists a unique function φ V 2 (φ) such that { φ( k) :k Z d } is also a Riesz basis for V 2 (φ) and such that φ satisfies the biorthogonality relation φ( ),φ( k) = δ(k), where δ(0) = 1 and δ(k) = 0 for k 0. Since the dual generator φ belongs to V 2 (φ), it can be exressed in the form (2.7) φ( ) = k Z d b k φ( k). The coefficients b k are determined exlicitly by the Fourier series b k e 2πikξ = 1 2 φ(ξ + k) ; k Z d k Z d i.e., (b k ) is the inverse Fourier transform of ( k Z φ(ξ + k) 2 ) 1 (see, for examle, d [8, 9]). Since a φ (ξ) 1 1/m by (2.6), the sequence (b k ) exists and belongs to l 2 (Z d ). In order to handle general shift-invariant saces V (φ) instead of V 2 (φ), we need more information about the dual generator. The following result is one of the central results in this aer and is essential for the treatment of general shift-invariant saces. Theorem 2.3. Assume that (1) φ W (L 1 ω) and that (2) { φ( k) : k Z d} is a Riesz basis for V 2 (φ). Then the dual generator φ is in W (L 1 ω). As a corollary, we obtain the following theorem. Theorem 2.4. Assume that φ W (L 1 ω) and is ω-moderate. (i) The sace V (φ) is a subsace (not necessarily closed) of L and W (L ) for any with 1. (ii) If { φ( k) : k Z d} is a Riesz basis of V 2 (φ), then there exist constants m > 0,M > 0 such that (2.8) m c l c k φ( k) M c l c l (Z d ) k Z d L { is satisfied for all } 1 and all ω-moderate weights. Consequently, φ( k) : k Z d is an unconditional basis for V (φ) for 1 <, and V (φ) is a closed subsace of L and W (L ) for 1. The theorem says that the inclusion in Theorem 2.4(i) and the norm equivalence (2.8) hold simultaneously for all and all ω-moderate weights, rovided that they hold for the Hilbert sace V 2 (φ). But in V 2 (φ), the Riesz basis roerty (2.8) is much easier to check. In fact, it is equivalent to inequalities (2.6). Inequalities (2.8) imly that l and V (φ) are isomorhic Banach saces and that the set { φ( k) : k Z d} is an unconditional basis of V (φ). In aroximation theory we say that φ has stable integer translates and is a stable generator [67, 68, 69]. When = 1 the conclusion (2.8) of Theorem 2.4 is well known and can be found in [67, 68, 69].

12 596 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG As a corollary of Theorem 2.4, we obtain the following inclusions among shiftinvariant saces. Corollary 2.5. Assume that φ W (L 1 ω) and that is ω-moderate. Then V 1 ω (φ) V ω (φ) V q ω (φ) for 1 q and Vω q (φ) V q (φ) for 1 q Proof of Theorems. We begin with the following roerties of weight functions. Lemma 2.6. Let K be a comact subset of R d and let be an ω-moderate weight. Then there exists a constant C 1 > 0 such that and C1 1 (j) (x + j) C 1(j) j Z d, x K. Proof. Using the submultilicative roerty, we have (x + j) Cω(x)(j) (j) =(x + j x) C(x + j)ω( x). We may take C 1 = C max x K ω(x), since ω is continuous and symmetric and K is comact. As a consequence of Lemma 2.6 and the definition of W (L ) we obtain a slightly different characterization of the amalgam saces. Corollary 2.7. The following are equivalent. (i) f W (L ). (ii) f k Z c kχ d [0,1] d( k) a.e. for some c l, for instance, c k = ess su x [0,1] d f(x + k). In the corollary above, we used the standard notation χ [0,1] d to denote the characteristic function of [0, 1] d. Proof (of Theorem 2.1). Write b l = ess su x [0,1] d f(x + l)(l). Then b l = f W (L ) for 1. Therefore, the inclusions W 0(L ) W 0 (L q ) and W (L ) W (L q ) in (i) follow immediately from the inclusion l l q when 1 q. The inclusion W 0 (L ) W (L ) is obvious. Next, using Lemma 2.6, we deduce that f(x)(x) dx = f(x + j)(x + j) dx R d [0,1] d j Z (2.9) d C 1 [0,1] d j Z d f(x + j)(j) dx C 1 f W (L ) holds for 1 <. Consequently, the inclusion W (L ) L holds. Similarly, for = we have (2.10) f L = su j Z d {ess su{ f(x + j)(x + j) : x [0, 1] d }} C 1 su j Z d {ess su{ f(x + j)(j) : x [0, 1] d }} = C 1 f W (L ).

13 NONUNIFORM SAMPLING AND RECONSTRUCTION 597 The inclusion L ω L follows immediately from the inequality (x) C(0)ω(x) for all x R d. Likewise, the inclusion W (L ω) W (L ) follows from l ω l. Proof (of Theorem 2.2). To rove (i), let f L, g L 1 ω, and 1. Then using the fact that (x) =(x y + y) C(x y)ω(y), we have (f g)(x) (x) = g(y)f(x y)dy R (x) d (2.11) C g(y) ω(y) f(x y) (x y)dy R d C( g ω f )(x). From the ointwise estimate above and Young s inequality for the convolution of an L 1 function with an L function, it follows that (f g) L C f L gω L 1. Thus f g L and f g L C f L g L 1. ω To rove (ii), consider first the case g = χ [0,1] d for 1 <. Write b k = ess su x [0,1] d f χ [0,1] d(x + k). Then, using Hölder s inequality, we obtain b k ess su x [0,1] f(x + k y) dy [0,1] d d f(k y) dy. [0,1] d [0,1] d Using Lemma 2.6 with K =[0, 1] d [0, 1] d =[ 1, 1] d, it follows that b l f(k y) (k) dy (2.12) [0,1] d [0,1] d k Z d C 1 f(k y) (k y) dy = C [0,1] d 1 f L. k Z d Thus we have f χ[0,1] C f d W (L ) L. For general g W (L 1 ω) we use the reresentation of Corollary 2.7, which imlies that g k Z c kχ d [0,1] d( k) and c l 1 ω = g W (L 1 ω ). We estimate f g f g k Z d c k ( f χ[0,1] d) ( k), and consequently The last inequality imlies f g W (L ) k Z d c k f χ [0,1] d( k) W (L ) C 2 k Z d c k ω(k) f L. f g W (L ) C 2 f L g W (L 1 ω ). The case = is roved in a similar fashion. The roof of (iii) is similar to the roof of (i). To finish the roofs of this section, we need the following three lemmas.

14 598 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG Lemma 2.8. If φ W (L 1 ω) then the autocorrelation sequence (2.13) a k = φ(x)φ(x k)dx R d belongs to l 1 ω, and we have a l 1 ω C φ 2 W (L 1 ω ). Proof. Write b k = ess su x [0,1] d φ(x + k) and b k = b k = ess su x [0,1] d φ(x k). Then φ W (L 1 ω ) = b l = b 1 ω l 1 and ω a k φ(x) φ(x k) dx R d φ(x + j) φ(x + j k) dx b j b j k [0,1] d j Z d j Z d =(b b )(k). Theorem 2.2(iii) imlies that a l 1 C b 2 ω l = C φ 2 1 ω W (L 1 ω ) Lemma 2.9. If φ W (L 1 ω) and c l, then the function f = k Z c kφ(x k) d belongs to W (L ) and f W (L ) C c l φ W (L 1 ω ). Proof. Write b k = ess su x [0,1] d φ(x + k), d k = ess su x [0,1] d f(x + k). Then φ W (L 1 ω ) = b l and f 1 ω W (L ) = d l, and we have d k = ess su x [0,1] d c j φ(x + k j) j Z c j b k j =( c b)(k). d j Z d Theorem 2.2(iii) then imlies that d l C c l b l 1 ; in other words, f ω W (L ) C c l φ W (L 1 ω ). Lemma If f L and g W (L 1 ω), then the sequence d defined by d k = f(x)g(x k)dx belongs to l R d and we have d l C f L g W (L 1 ), 1. Remark 2.2. The fact that the autocorrelation sequence in Lemma 2.8 belongs to l 1 ω is a direct consequence of Lemma Proof. Since g W (L 1 ω) L 1/ by Theorem 2.1 and f L, the terms d k are well defined. For 1 < we have d k (k) = f(x)g(x k)(k)dx R d f(x + j) g(x + j k)(k) dx [0,1] d j Z d f(x + j) g(x + j k)(k) dx. [0,1] d j Z d

15 NONUNIFORM SAMPLING AND RECONSTRUCTION 599 We sum over k and aly Theorem 2.2(iii) to the sequences {f(x + j) :j Z d } and {g(x j) :j Z d } for fixed x R d, and we obtain d l f(x + j) g(x + j k)(k) dx [0,1] d k Z d j Z d C f(x + k)(k) g(x k)ω(k) dx [0,1] d k Z d k Z d C g W (L 1 ω ) f L. The case = is roved in a similar fashion. For the roof of Theorem 2.3 we need the following weighted version of Wiener s lemma on absolutely convergent Fourier series. Lemma Assume that the submultilicative weight ω satisfies the so-called Beurling Domar condition (mentioned in section 2.1) (2.14) n=1 log ω(nk) n 2 < k Z d. If f(ξ) = k Z a ke 2πikξ is an absolutely convergent Fourier series with coefficient d sequence a =(a k ) k Z d l 1 ω(z d ) and if f(ξ) 0for all ξ R d, then 1 f also has an 1 absolutely convergent Fourier series f(ξ) = k Z b ke 2πikξ with coefficient sequence d b =(b k ) k Z d l 1 ω(z d ). Remark 2.3. The unweighted version is a classical lemma of Wiener. The weighted version is imlicit in [38] and stated in [91]. We are now ready to rove Theorem 2.3. Proof (of Theorem 2.3). We have already seen that the dual generator φ V 2 (φ) has the exansion φ = k Z d b k φ( k), where the coefficients b k are the Fourier coefficients of â 1 (ξ) =( k Z d ˆφ(ξ+k) 2 ) 1. We wish to aly Lemma 2.11 to â. Since {φ( k) :k Z d } is a Riesz basis for V 2 (φ), we have â(ξ) 0 for all ξ R d by (2.6). Furthermore, using the Poisson summation formula, â has the Fourier series â(ξ) = k Z d ˆφ(ξ + k) 2 = k Z d φ, φ( k) e 2πikξ. Consequently, by Lemma 2.8, the Fourier coefficients of â are in l 1 ω(z d ). Thus the hyotheses of Wiener s lemma are satisfied, and we conclude that the Fourier coefficients of â 1 are also in l 1 ω(z d ). Now Lemma 2.9 imlies that φ W (L 1 ω). Proof (of Theorem 2.4). Part (i) and the right-hand inequality in (2.8) follow directly from Lemma 2.9. To rove the remaining statements, we consider the oerator T φ defined by (2.15) T φ c = k Z d c k φ( k), c l,

16 600 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG and the oerator T φ defined by (2.16) (T φf) k = f(x) φ(x k)dx. R d Lemma 2.9 imlies that T φ is a bounded ma from l to L with range V (φ). Furthermore, Lemma 2.10 imlies that T φ is a bounded ma from L to l. Let f = k Z c kφ( k) V d (φ) = Range(T φ ). Since { φ( k) :k Z d } is biorthogonal to {φ( k) :k Z d }, we find that c k = f, φ( k) =(T φf) k,or c =T φf. Consequently, (2.17) c l T φ o f L, 1 and we may choose m = T φ o as the lower bound in (2.8). The other statements of the theorem follow immediately from (2.8). Proof (of Corollary 2.5). Since (k) = (k + 0) C(0)ω(k), we immediately have the inclusions l q ω(z d ) l q (Z d ). Since l 1 ω(z d ) l ω(z d ) l q ω(z d ) l q for 1 q, the inequality (2.8) then imlies the inclusions Vω 1 (φ) Vω (φ) Vω q (φ) V q (φ) for 1 q. 3. The Samling Problem in Weighted Shift-Invariant Saces V (φ). For a reasonable formulation of the samling roblem in V (φ) the oint evaluations f f(x) must be well defined. Furthermore, a small variation in the samling oint should roduce only a small variation in the samling value. As a minimal requirement, we need the functions in V (φ) to be continuous. This is guaranteed by the following statement. Theorem 3.1. Assume that φ W 0 (L 1 ω), that φ satisfies (2.6), and that is ω-moderate. (i) V (φ) W 0 (L ) for all, 1. (ii) If f V (φ), then we have the norm equivalences f L c l f W (L ). (iii) If X = {x j : j J} is such that inf j,l x j x l > 0, then ( ) 1/ (3.1) f(x k ) (x k ) C f L f V (φ). x k X In articular, if φ is continuous and has comact suort, then the conclusions (i) (iii) hold. A set X = {x j : j J} satisfying inf j,l x j x l > 0 is called searated. Inequality (3.1) has two interretations. It imlies that the samling oerator S X : f f X is a bounded oerator from V (φ) into the corresonding sequence sace l (X) = (c j): c j (x j ) 1/ <.

17 NONUNIFORM SAMPLING AND RECONSTRUCTION 601 Equivalently, the weighted samling oerator S X : f f X is a bounded oerator from V (φ) intol. To recover a function f V (φ) from its samles, we need a converse of inequality (3.1). Following Landau [74], we say that X is a set of samling for V (φ) if (3.2) c f L f(x j ) (x j ) x j X 1/ C f L, where c and C are ositive constants indeendent of f. The left-hand inequality imlies that if f(x j ) = 0 for all x j X, then f =0. Thus X is a set of uniqueness. Moreover, the samling oerator S X can be inverted on its range and S 1 X is a bounded oerator from Range(S X ) l (X) tov (φ). Thus (3.2) says that a small change of a samled value f(x j ) causes only a small change of f. This imlies that the samling is stable or, equivalently, that the reconstruction of f from its samles is continuous. As ointed out in section 1.1, every set of samling is a set of uniqueness, but the converse is not true. For ractical considerations and numerical imlementations, only sets of samling are of interest, because only these can lead to robust algorithms. A solution to the samling roblem consists of two arts: (a) Given a generator φ, we need to find conditions on X, usually in the form of a density, such that the norm equivalence (3.2) holds. Then, at least in rincile, f V (φ) is uniquely and stably determined by f X. (b) We need to design reconstruction rocedures that are useful and efficient in ractical alications. The objective is to find efficient and fast numerical algorithms that recover f from its samles f X, when (3.2) is satisfied. Remark 3.1. (i) The hyothesis that X be searated is for convenience only and is not essential. For arbitrary samling sets, we can use adative weights to comensate for the local variations of the samling density [48, 49]. Let V j = {x R d : x x j x x k for all k j} be the Voronoi region at x j, and let γ j = λ(v j ) be the size of V j. Then X is a set of samling for V (φ) if c f L f(x j ) γ j (x j ) x j X 1/ C f L. In numerical alications the adative weights γ j are used as a chea device for reconditioning and for imroving the ratio C /c, the condition number of the set of samling [49, 101]. (ii) The assumtion that the samles {f(x j ): j J} can be measured exactly is not realistic. A better assumtion is that the samled data is of the form (3.3) g xj = f(x)ψ xj (x)dx, R d where {ψ xj : x j X} is a set of functionals that act on the function f to roduce the data {g xj : x j X}. The functionals {ψ xj : x j X} may reflect the characteristics of the samling devices. For this case, the well-osedness

18 602 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG condition (3.2) must be relaced by (3.4) c f L x j X g xj (f) 1/ C f L, where g xj are defined by (3.3) and where c and C are ositive constants indeendent of f [1] Proof of Theorem 3.1. Proof. To rove (i), let f = k Z c kφ( k) V d (φ). Then Lemma 2.9 imlies that (3.5) f W (L ) C c l φ W (L 1 ω ). To verify the continuity of f in the case 1 <, we observe that W (L ) W (L ) L and thus (3.6) f L C f W (L ). Let f n = ( ) k n c kφ( k) be a artial sum of f. Then (3.5) and (3.6) imly that f f n L C φ W (L 1 ω ) 1/ c k (k). k >n Therefore, the sequence of continuous functions f n converges uniformly to the continuous function f. Since is ositive and continuous, f must be continuous as well. To treat the case = we choose a sequence φ n of continuous functions with comact suort such that φ φ n W (L 1 ω ) 0asn. Set f n (x) = k Z c kφ n (x k). Since the sum is locally finite, each f n is continuous. Using (3.5) we estimate f f n L C c l φ φ n W (L 1 ω ) 0. It follows that the sequence f n converges uniformly to f.thusf is continuous as well. Regarding the roof of (ii), the norm equivalence f L c l was roved earlier in Theorem 2.4. Theorem 2.1 imlies that f L C f W (L ). Finally, if f = k c kφ( k) V (φ), then we obtain f W (L ) C c l φ W (L 1 ω ) C 1 f L by Lemma 2.9 and (2.8). This roves that f L and f W (L ) are equivalent norms on V (φ). For the roof (iii), if inf j,l x j x l = δ>0, then there are at most N = N(δ) samling oints in every cube k +[0, 1] d. Thus, using Lemma 2.6, we obtain f(x j ) (x j ) N su f(x + k) (x + k) x j k+[0,1] d x [0,1] d CN su f(x + k) (k). x [0,1] d

19 NONUNIFORM SAMPLING AND RECONSTRUCTION 603 Taking the sum over k Z d and alying the norm equivalence roved in (ii), we obtain f(x j ) (x j ) CN su f(x + k) (k) k Z d x [0,1] d x j X = NC 1 f W (L ) C 2 f L for all f V (φ). 4. Reroducing Kernel Hilbert Saces, Frames, and Nonuniform Samling. As mentioned in the introduction, results of Paley and Wiener and Kadec relate Riesz bases consisting of comlex exonentials to samling sets that are erturbations of Z. More generally, the aroriate concet for arbitrary sets of samling in shiftinvariant saces is the concet of frames discussed in section 4.2. Frame theory generalizes and encomasses the theory of Riesz bases and enables us to translate the samling roblem into a roblem of functional analysis. The connection between frames and sets of samling is established by means of reroducing kernel Hilbert saces (RKHSs), discussed in the next section. Frames are introduced in section 4.2, and the relation between RKHSs, frames, and sets of samling is develoed in section RKHSs. Theorem 3.1(iii) holds for arbitrary searated samling sets, so in articular Theorem 3.1(iii) shows that all oint evaluations f f(x) are continuous linear functionals on V (φ) for all x R d. Since V (φ) L and the dual sace of L is L 1/, where 1/ +1/ = 1, there exists a function K x L 1/ such that f(x) = f,k x = f(t)k x (t) dt R d for all f V (φ). In addition, it will be shown that K x V 1/ (φ). In the case of a Hilbert sace H of continuous functions on R d, such as V 2 (φ), the following terminology is used. A Hilbert sace is an RKHS [117] if, for any x R d, the ointwise evaluation f f(x) is a bounded linear functional on H. The unique functions K x Hsatisfying f(x) = f,k x are called the reroducing kernels of H. With this terminology we have the following consequence of Theorem 3.1. Theorem 4.1. Let be ω-moderate. If φ W 0 (L 1 ω), then the evaluations f f(x) are continuous functionals, and there exist functions K x Vω 1 (φ) such that f(x) = f,k x. The kernel functions are given exlicitly by (4.1) K x (y) = k Z d φ(x k) φ(y k). In articular, V 2 (φ) is an RKHS. The above theorem is a reformulation of Theorem 3.1. We only need to rove the formula for the reroducing kernel. Note that K x in (4.1) is well defined: since φ W 0 (L 1 ω), Theorem 2.3 combined with Theorem 3.1(iii) imlies that the sequence { φ(x k) : k Z d } belongs to l 1 ω. Thus, by the definition of Vω 1 (φ), we have K x Vω 1 (φ), and so K x V (φ) for any with 1 and any ω-moderate weight. Furthermore, K x is clearly the reroducing kernel, because if f(x) = k c kφ(x k),

20 604 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG then f,k x = j,k c j φ(x k) φ( j), φ( k) = k c k φ(x k) =f(x) Frames. In order to reconstruct a function f V (φ) from its samles f(x j ), it is sufficient to solve the (infinite) system of equations (4.2) k Z d c k φ(x j k) =f(x j ) for the coefficients (c k ). If we introduce the infinite matrix U with entries (4.3) U jk = φ(x j k) indexed by X Z d, then the relation between the coefficient sequence c and the samles is given by Uc = f X. Theorem 3.1(ii) and (iii) imly that f X l (X). Thus U mas l (Z d )into l (X). Since f(x) = f,k x, the samling inequality (3.2) imlies that the set of reroducing kernels {K xj,x j X} sans V 1/. This observation leads to the following abstract concets. A Hilbert frame (or simly a frame) {e j : j J} of a Hilbert sace H is a collection of vectors in H indexed by a countable set J such that (4.4) A f 2 H j f,e j 2 B f 2 H for two constants A, B > 0 indeendent of f H[40]. More generally, a Banach frame for a Banach sace B is a collection of functionals {e j : j J} B with the following roerties [54]. (a) There exists an associated sequence sace B d on the index set J, such that A f B ( f,e j ) Bd B f B for two constants A, B > 0 indeendent of f B. (b) There exists a so-called reconstruction oerator R from B d into B, such that R(( f,e j ) )=f Relations between RKHSs, Frames, and Nonuniform Samling. The following theorem translates the different terminologies that arise in the context of samling theory [2, 40, 74].

21 NONUNIFORM SAMPLING AND RECONSTRUCTION 605 Theorem 4.2. The following are equivalent: (i) X = {x j : j J} is a set of samling for V (φ). (ii) For the matrix U in (4.3), there exist a, b > 0 such that a c l Uc l (X) b c l c l. (iii). There exist ositive constants a>0 and b>0 such that a f L f(x j ) (x j ) x j X 1/ b f L f V (φ). (iv) For =2, the set of reroducing kernels {K xj : x j X} is a (Hilbert) frame for V 2 (φ). Remark 4.1. (i) The relation between RKHSs and uniform samling of bandlimited functions was first reorted by Yao [117] and used to derive interolating series similar to (1.1). For the case of shift-invariant saces, this connection was established in [9]. Samling for functions in RKHSs was studied in [84]. For the general case of nonuniform samling in shift-invariant saces, the connection was established in [5]. (ii) The relation between Hilbert frames and samling of bandlimited functions is well known [14, 48]. Samling in shift-invariant saces is more recent, and the relation between frames and samling in shift-invariant saces (with =2 and =1) can be found in [5, 30, 75, 77, 102]. (iii) The relation between Hilbert frames and the weighted average samling mentioned in Remark 3.1 can be found in [1]. This relation is obtained via kernels that generalize the RKHS. 5. Frame Algorithms for L -Saces. Theorem 4.2 states that a searated set X = {x j : j J} is a set of samling for V 2 (φ) if and only if the set of reroducing kernels {K xj : x j X} is a frame for V 2 (φ). It is well known from frame theory that there exists a dual frame { K xj : x j X} V 2 (φ) that allows us to reconstruct the function f V 2 (φ) exlicitly as (5.1) f(x) = f,k xj K xj (x) = f(x j ) K xj (x). However, a dual frame { K xj : x j X} is difficult to find in general, and this method for recovering a function f V 2 (φ) from its samles {f(x j ): x j X} is often not ractical. Instead, the frame oerator (5.2) T f(x) = f,k xj K xj (x) = f(x j )K xj (x) can be inverted via an iterative that we now describe. The oerator I 2 A+B Tis contractive, i.e., the oerator norm on L 2 (R d ) satisfies the estimate I 2 A + B T B A o A + B < 1,

22 606 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG where A, B are frame bounds for {K xj : x j X}. Thus, 2 the Neumann series A + B ( T 1 = I 2 ) n 2 A + B T. n=0 A+B T can be inverted by This analysis gives the iterative frame reconstruction algorithm, which is made u of an initialization ste f 1 = f(x j )K xj and iteration (5.3) f n = 2 A + B f 1 + ( I 2 ) A + B T f n 1. As n, the iterative frame algorithm (5.3) converges to f =T 1 f 1 =T 1 T f = f. Remark 5.1. (i) The comutation of T requires the comutation of the reroducing frame functions {K xj : x j X}, which is a difficult task. Moreover, for each samling set X we need to comute a new set of reroducing frame functions {K xj : x j X}. (ii) Even if the frame functions {K xj : x j X} are known, the erformance of the frame algorithm deends sensitively on estimates for the frame bounds. Since accurate and exlicit frame bounds, let alone otimal ones, are hardly ever known for nonuniform samling roblems, the frame algorithm converges very slowly in general. For efficient numerical comutations involving frames, the rimitive iteration (5.3) should therefore be relaced almost always by acceleration methods, such as Chebyshev or conjugate gradient acceleration. In articular, conjugate gradient methods converge at the otimal rate, even without any knowledge of the frame bounds [49, 56]. (iii) The convergence of the frame algorithm is guaranteed only in L 2, even if the function belongs to other saces L. It is a remarkable fact that in Hilbert sace the norm equivalence (4.4) alone guarantees that the frame oerator is invertible. In Banach saces the situation is much more comlicated and the existence of a reconstruction rocedure must be ostulated in the definition of a Banach frame. In the secial case of samling in shift-invariant saces, the frame oerator T is invertible on all V (φ) whenever T is invertible on V 2 (φ) and φ ossesses a suitable olynomial decay [57]. 6. Iterative Reconstruction Algorithms. Since the iterative frame algorithm is often slow to converge and its convergence is not even guaranteed beyond V 2 (φ), alternative reconstruction rocedures have been designed [4, 76]. These rocedures are also iterative and based on a Neumann series. For the sake of exosition, the roofs of the results of this section and the next section are ostoned to section 8. The first ste is to aroximate the function f from its samles {f(x j ): x j X} using an interolation or a quasi-interolation Q X f. For examle, Q X f could be a iecewise linear interolation of the samles f X or even an aroximation by ste functions, the so-called samle-and-hold interolant.

23 NONUNIFORM SAMPLING AND RECONSTRUCTION 607 The aroximation Q X f is then rojected in the sace V (φ) to obtain the first aroximation f 1 =PQ X f V (φ). The error e = f f 1 between the functions f and f 1 belongs to the sace V (φ). Moreover, the values of e on the samling set X can be calculated from {f(x j ): x j X} and (P Q X f)(x j ). Then we reeat the interolation-rojection rocedure on e and obtain a correction e 1. The udated estimate is now f 2 = f 1 + e 1. By reeating this rocedure, we obtain a sequence f n = f 1 + e 1 + e 2 + e e n 1 that converges to the function f. In order to rove convergence results for this tye of algorithm, we need the samling set to be dense enough. The aroriate definition for the samling density of X is again due to Beurling. Definition 6.1. A set X = {x j : j J} is γ 0 -dense in R d if (6.1) R d = j B γ (x j ) γ>γ 0. This definition imlies that the distance of any samling oint to its nearest neighbor is at most 2γ 0. Thus, strictly seaking, γ 0 is the inverse of a density; i.e., if γ 0 increases, the number of oints er unit cube decreases. In fact, if a set X is γ 0 -dense, then its Beurling density defined by (1.2) satisfies D(X) γ0 1. This last relation states that γ 0 -density imoses more constraints on a samling set X than the Beurling density D(X). To create suitable quasi-interolants, we roceed as follows. Let {β j } be a artition of unity such that (1) 0 β j 1 for all j J; (2) su β j B γ (x j ); and (3) β j =1. A artition of unity that satisfies these conditions is sometimes called a bounded artition of unity. Then the oerator Q X defined by Q X f = f(x j)β j is a quasi-interolant of the samled values f X. In this situation we have the following qualitative statement. Theorem 6.1. Let φ in W 0 (L 1 ω) and let P be a bounded rojection from L onto V (φ). Then there exists a density γ>0(γ = γ(,, P)) such that any f V (φ) can be recovered from its samles {f(x j ): x j X} on any γ-dense set X = {x j : j J} by the iterative algorithm (6.2) { f1 =PQ X f, f n+1 =PQ X (f f n )+f n. Then iterates f n converge to f uniformly and in the W (L )- and L -norms. convergence is geometric, that is, f f n L C f f n W (L ) C f f 1 L α n, The for some α = α(γ) < 1. The algorithm based on this iteration is illustrated in Figure 6.1. Figure 6.2 shows the reconstruction of a function f by means of this algorithm, and Figure 6.3 shows the reconstruction of an MRI image with missing data. Remark 6.1. For =1, Theorem 6.1 was roved in [4]. For a secial case of the weighted average samling mentioned in Remark 3.1, a modified iterative algorithm and a theorem similar to Theorem 6.1 can be found in [1].

24 608 AKRAM ALDROUBI AND KARLHEINZ GRÖCHENIG f ( X ) f n S X f n ( X ) ( f f n )( X ) Q X Q X ( f f n ) P PQ X ( f f n ) f n 1 Fig. 6.1 The iterative reconstruction algorithm of Theorem Samled signal x Final error Convergence Reconstructed signal Fig. 6.2 Reconstruction of a function f with f using the iteration algorithm (6.2) of Theorem 6.1. To left: Function f belonging to the shift-invariant sace generated by the Gaussian function e x2 /2σ 2, σ 0.81, and its samle values {f(x j ): x j X} marked by + (density γ 0.8). To right: Error f f n L 2 against the number of iterations. Bottom left: Final error f f n after 10 iterations. Bottom right: Reconstructed function f 10 (continuous line) and original samles {f(x j ): x j X}. Universal Projections in Weighted Shift-Invariant Saces. Theorem 6.1 requires bounded rojections from L onto V (φ). In contrast to the situation in Hilbert sace, the existence of bounded rojections in Banach saces is a difficult roblem. In the context of nonuniform samling in shift-invariant saces, we would like the rojections to satisfy additional requirements. In articular, we would like rojectors that can be imlemented with fast algorithms. Further, it would be useful to find a universal rojection, i.e., a rojection that works simultaneously for all L, 1, and all weights. In shift-invariant saces such universal rojections do indeed exist.

25 NONUNIFORM SAMPLING AND RECONSTRUCTION 609 Fig. 6.3 Missing data reconstruction. To left: Original digital MRI image with samles. To right: MRI image with 50% randomly missing samles. Bottom left: Reconstruction using the iterative reconstruction algorithm (6.2) of Theorem 6.1. The corresonding shiftinvariant sace is generated by φ(x, y) =β 3 (x) β 3 (y), where β 3 = χ [0,1] χ [0,1] χ [0,1] χ [0,1] is the B-sline function of degree 3. Theorem 6.2. Assume φ W 0 (L 1 ω). Then the oerator P:f k Z d f, φ( k) φ( k) is a bounded rojection from L onto V (φ) for all, 1, and all ω-moderate weights. Remark 6.2. The oerator P can be imlemented using convolutions and samling. Thus the universal rojector P can be imlemented with fast filtering algorithms [3]. 7. Reconstruction in Presence of Noise. In ractical alications the given data are rarely the exact samles of a function f V (φ). We assume more generally that f belongs to W 0 (L ); then the samling oerator f {f(x j ):x j X} still makes sense and yields a sequence in l (X). Alternatively, we may assume that f V (φ), but that the samled sequence is a noisy version of {f(x j ): x j X}, e.g., that the samling sequence has the form {f x j = f(x j )+η j } l (X). If a reconstruction algorithm is alied to noisy data, then the question arises whether the algorithm still converges, and if it does, to which limit it converges. To see what is involved, we first consider samling in the Hilbert sace V 2 (φ). Assume that X = {x j : j J} is a set of samling for V 2 (φ). Then the set of reroducing kernels {K xj : x j X} forms a frame for V 2 (φ), and so f V 2 (φ) can