On Stability of Sampling-Reconstruction Models
aa r X i v : . [ m a t h . G M ] M a y ON STABILITY OF SAMPLING-RECONSTRUCTIONMODELS
ERNESTO ACOSTA-REYES, AKRAM ALDROUBI, AND ILYA KRISHTAL
Abstract.
A useful sampling-reconstruction model should be sta-ble with respect to different kind of small perturbations, regardlesswhether they result from jitter, measurement errors, or simply froma small change in the model assumptions. In this paper we provethis result for a large class of sampling models. We define differ-ent classes of perturbations and quantify the robustness of a modelwith respect to them. We also use the theory of localized frames tostudy the frame algorithm for recovering the original signal fromits samples. Introduction
The sampling and reconstruction problem includes devising efficientmethods for representing a signal (function) in terms of a discrete (finiteor countable) set of its samples (values) and reconstructing the originalsignal from the samples (see e.g., [1, 3, 8, 9, 17, 22] and the referencetherein). In this paper we consider a very general sampling model wherethe signal is assumed to belong to a finitely generated shift invariantspace and the sampling is performed on an irregular separated set andis averaged by finite Borel measures. The main focus of this paper is ondescribing and quantifying “admissible” perturbations of the samplingmodel which may result from altering the sampling set (jitter) (see e.g.[6, 7, 14]), or the averaging sampling measures (measuring devices) orthe generators of the underlying shift-invariant space (see e.g., [5, 18]).As recently became customary in sampling theory (see e.g. [1, 3,11, 19, 20, 21, 22]), we mesh operator theory techniques and those ofshift invariant and Wiener amalgam spaces [13]. The latter provideus with relatively straight-forward proofs while the former allow us tokeep in sight our objective. In section 2 we show that all the propertiesof our sampling model can be encoded in the sampling operator U . Math Subject Classifications.
Key words and phrases.
Irregular sampling, non-uniform sampling, sampling,reconstruction, jitter, measurement error, model error.The second author was supported in part by NSF grants DMS-0504788.
The sampling model admits reconstruction if its sampling operator isbounded both above and below. Our first goal is to show that anyand all of the small perturbations mentioned above result in a smallperturbation of U in the operator norm. This will prove the stabilityof sampling in our model with respect to those perturbations and thecorresponding estimates we obtain will quantify this stability. Oursecond goal is to show how a frame algorithm can be used to reconstructsignals in our sampling model. Finally, our last goal is to show that thereconstruction error due to the perturbations we describe is controlledcontinuously by the perturbation errors.The paper is organized as follows. In section 2 we describe oursampling model, introduce relevant notions and notation, and cite afew preliminary results. The main results are presented in section 3.Perturbation results addressing our first goal are in subsection 3.1.There we prove that a set of sampling remains such under a smallperturbation of the sampling measures and/or the generators of theshift invariant space. It is also shown that sampling remains stable withrespect to a perturbation of the sampling set itself. In subsection 3.2we show that, in case of a signal in a Hilbert space, a frame algorithmcan be used to reconstruct the function from its samples. We also usethe results of the previous subsection and the theory of localized framesto show that under mild additional assumptions a set of sampling fora Hilbert shift invariant space is also a set of sampling for a chain ofBanach shift invariant spaces to which the frame algorithm extends.In subsection 3.3 we study the dependence of the reconstruction errorupon the perturbation errors. The proofs of the results in section 3 arerelegated to section 4.2. Description of the sampling model
This section is primarily devoted to introduction of the samplingmodel we use in this paper. We also present most of the necessarynotation and cite some of the preliminary results that will be usedlater.The signals we are studying in this paper are represented by functions f ∈ L p ( R d ), for some p ∈ [1 , ∞ ] and d ∈ N . Moreover, we assume that f belongs to a shift invariant space(2.1) V p (Φ) = { X k ∈ Z d C Tk Φ k : C ∈ ( ℓ p ( Z d )) ( r ) } . Here Φ = ( φ , . . . , φ r ) T is a vector of functions, Φ k = Φ( · − k ), and C =( c , . . . , c r ) T is a vector of sequences belonging to ( ℓ p ( Z d )) ( r ) . Among N STABILITY OF SAMPLING-RECONSTRUCTION MODELS the equivalent norms in ( ℓ p ( Z d )) ( r ) we choose k C k ( ℓ p ( Z d )) ( r ) = r X i =1 k c i k ℓ p ( Z d ) . In order to avoid convergence issues in (2.1) we assume that the set { φ ( · − k ) , . . . , φ r ( · − k ); k ∈ Z d } generates an unconditional basis for V p (Φ). In particular, we require that there exist constants 0 < m p ≤ M p < ∞ , such that(2.2) m p k C k ( ℓ p ( Z d )) ( r ) ≤ k X k ∈ Z d C Tk Φ k k L p ≤ M p k C k ( ℓ p ( Z d )) ( r ) , ∀ C ∈ ( ℓ p ( Z d )) ( r ) . The unconditional basis assumption (2.2) implies [3] that the space V p (Φ) is a closed subspace of L p ( R d ) . Since we are interested in sampling in V p (Φ) we add an assump-tion that would make all the functions in these spaces continuous and,therefore, pointwise evaluations will be meaningful. To this end, we as-sume that all generators Φ belong to a Wiener-amalgam space ( W ) ( r ) as defined below. For 1 ≤ p < ∞ , a measurable function f belongs to W p if it satisfies k f k W p = X k ∈ Z d esssup x ∈ [0 , d | f ( x + k ) | p ! /p < ∞ . (2.3)If p = ∞ , a measurable function f belongs to W ∞ if it satisfies k f k W ∞ = sup k ∈ Z d { esssup x ∈ [0 , d | f ( x + k ) |} < ∞ . (2.4)Hence, W ∞ coincides with L ∞ ( R d ). It is well known that W p areBanach spaces [13], and clearly W p ⊆ L p . By ( W p ) ( r ) we denote thespace of vectors Ψ = ( ψ , . . . , ψ r ) T of W p -functions with the norm k Ψ k ( W p ) ( r ) = r X i =1 k ψ i k W p . The closed subspace of (vectors of) continuous functions in W p (re-spectively, ( W p ) ( r ) ) will be denoted by W p (or ( W p ) ( r ) ).In this paper we are interested in average sampling performed bya vector of measures. We denote by M ( R d ) = M ( R d ) the Banachspace of finite complex Borel measures on R d . The norm on M ( R d ) isgiven by k µ k = R R d d | µ | ( y ), i.e., the total variation of a measure µ . By( M ( R d )) ( t ) we denote the space of vectors −→ µ = ( µ , . . . , µ t ) of measuresfrom M ( R d ) with the norm k−→ µ k ( M ( R d )) ( t ) = P tj =1 k µ j k . The symbols ERNESTO ACOSTA-REYES, AKRAM ALDROUBI, AND ILYA KRISHTAL M s ( R d ) (( M s ( R d )) ( t ) ), 0 ≤ s < ∞ , will be used for the subspace of M ( R d ) (( M ( R d )) ( t ) ) of all (vectors of) measures µ ∈ M ( R d ) such that(1 + | x | ) s ∈ L ( R d , d | µ | ), i.e., R (1 + | x | ) s d | µ | ( x ) < ∞ . By M ∞ ( R d )(( M ∞ ( R d )) ( t ) ) we denote the space of all (vectors of) measures withcompact support. Clearly M s ( R d ) ⊂ M r ( R d ) for 0 ≤ r ≤ s ≤ ∞ .For µ ∈ M ( R d ) and a measurable function φ on R d , the convolutionof the function φ and the measure µ is defined by( φ ∗ µ )( x ) = Z R d φ ( x − y ) dµ ( y ) , x ∈ R d . When we have a vector of measurable functions Φ = ( φ , . . . , φ r ) T anda vector of finite complex Borel measures −→ µ = ( µ , . . . , µ t ), then theconvolution Φ ∗ −→ µ is the r × t matrix given byΦ ∗ −→ µ = φ ∗ µ . . . φ ∗ µ t ... ... φ r ∗ µ . . . φ r ∗ µ t . Let J be a countable index set and X = { x j : j ∈ J } be a subsetof R d . The reconstruction problem in our sampling model consists offinding the function f ∈ V p (Φ) from the knowledge of its samples( f ∗ −→ µ )( X ) = { ( f ∗ −→ µ )( x j ) = (cid:0) ( f ∗ µ )( x j ) , . . . , ( f ∗ µ t )( x j ) (cid:1) } j ∈ J . When t = 1 and µ = δ , i.e., µ is the Dirac measure on R d concen-trated at zero, then ( f ∗−→ µ )( X ) = { f ( x j ) } j ∈ J and we obtain the classical(ideal) sampling model. When d −→ µ = Ψ dx , where Ψ ∈ ( L ( R d )) ( t ) and dx is the Lebesgue measure on R d , i.e., −→ µ is absolutely continuous withrespect to the Lebesgue measure, then we write ( f ∗ Ψ)( X ) instead of( f ∗ −→ µ )( X ), and our model is reduced to the case analyzed in [5]. Definition 2.1.
Let 1 ≤ p ≤ ∞ and X = { x j : j ∈ J } be a countablesubset of R d . We say that X is a set of sampling for V p (Φ) and −→ µ (or,simply, a −→ µ -sampling set for V p (Φ)) if there exist constants 0 < A p ≤ B p < ∞ such that(2.5) A p k f k L p ≤ k ( f ∗ −→ µ )( X ) k ( ℓ p ( J )) ( t ) ≤ B p k f k L p , for all f ∈ V p (Φ) . If d −→ µ = Ψ dx then a −→ µ -sampling set X will be called a Ψ-samplingset and, if t = 1 and µ = δ , then X will be called an ideal samplingset. To ensure that an upper bound B p in (2.5) always exists (see (4.2))we restrict our attention only to separated sets X . Definition 2.2.
We say that X is separated if there exists δ > i,j ∈ J,i = j | x i − x j | ≥ δ . The number δ is called the separationconstant of the set X . N STABILITY OF SAMPLING-RECONSTRUCTION MODELS It is not hard to extend our results to the case of a finite union ofseparated sets. We do not, however, pursue this relatively trivial butspace consuming generalization.
Definition 2.3.
Let −→ µ ∈ ( M ( R d )) ( t ) , Φ ∈ ( W ) ( r ) satisfy (2.2), and X = { x j , j ∈ J } ⊂ R d be a separated set. The sampling model is thetriple ( X, Φ , −→ µ ). The sampling model ( X, Φ , −→ µ ) is called p -stable if X is a −→ µ -sampling set for V p (Φ), p ∈ [1 , ∞ ].Given a sampling model ( X, Φ , −→ µ ) we proceed to define its samplingoperator. Definition 2.4.
The sampling operator U = U ( X, Φ , −→ µ ) : ( ℓ p ( Z d )) ( r ) → ( ℓ p ( J )) ( t ) is defined by U C = ( f ∗ −→ µ )( X ), where f = P k ∈ Z d C Tk Φ k ∈ V p (Φ).We can think of U as a t × r matrix of operators U = U , . . . U r, ... ... U ,t . . . U r,t , where for each 1 ≤ i ≤ r and 1 ≤ l ≤ t the operator U i,l is defined byan infinite matrix with entries ( U i,l ) j,k = ( φ i ∗ µ l )( x j − k ), j ∈ J , k ∈ Z d .The operator norm of U is given by k U k p,op = P tl =1 P ri =1 k U i,l k .The following proposition shows that all the interesting propertiesof a sampling model ( X, Φ , −→ µ ) are, indeed, encoded in the samplingoperator U . The proof of this result follows immediately from (2.2)and (2.5). Proposition 2.1.
The sampling model ( X, Φ , −→ µ ) is p -stable if and onlyif there exist < η p ≤ β p < ∞ such that for all C ∈ ( ℓ p ( Z d )) ( r ) thesampling operator U satisfies (2.6) η p k C k ( ℓ p ( Z d )) ( r ) ≤ k U C k ( ℓ p ( J )) ( t ) ≤ β p k C k ( ℓ p ( Z d )) ( r ) . The next lemma is, essentially, a nutshell for many of the results inthis paper.
Lemma 2.2.
Let ( X, Φ , −→ µ ) be a p -stable sampling model and U be itssampling operator satisfying (2.6). Let also ( e X, Θ , −→ α ) be a samplingmodel such that its sampling operator U ∆ satisfies k U − U ∆ k < η p .Then ( e X, Θ , −→ α ) is also p -stable.Proof. Let C ∈ ( ℓ p ( Z d )) ( r ) . Then k U ∆ C k ( ℓ p ( J )) ( t ) ≤ k ( U − U ∆ ) C k ( ℓ p ( J )) ( t ) + k U C k ( ℓ p ( J )) ( t ) ≤ k U − U ∆ kk C k ( ℓ p ( Z d )) ( r ) + β p k C k ( ℓ p ( Z d )) ( r ) . ERNESTO ACOSTA-REYES, AKRAM ALDROUBI, AND ILYA KRISHTAL
Therefore, since k U − U ∆ k < η p , then we have(2.7) k U ∆ C k ( ℓ p ( J )) ( t ) ≤ ( η p + β p ) k C k ( ℓ p ( Z d )) ( r ) . On the other hand, since η p k C k ( ℓ p ( Z d )) ( r ) ≤ k U C k ( ℓ p ( J )) ( t ) ≤ k ( U − U ∆ ) C k ( ℓ p ( J )) ( t ) + k U ∆ C k ( ℓ p ( J )) ( t ) ≤ k U − U ∆ kk C k ( ℓ p ( Z d )) ( r ) + k U ∆ C k ( ℓ p ( J )) ( t ) . Hence,(2.8) ( η p − k U − U ∆ k ) k C k ( ℓ p ( Z d )) ( r ) ≤ k U ∆ C k ( ℓ p ( J )) ( t ) . Since k U − U ∆ k < η p , the conclusion of the lemma follows from (2.7),(2.8), and Proposition 2.1. (cid:3) Main Results
In this section we collect the main results of our paper.3.1.
Admissible perturbations of a sampling model.
In practice, shift invariant spaces are used to model classes of signalsthat can occur (or that are allowed) in applications. However often,the functions in a shift invariant space model only give approximationsto the signals of interest. For this reason, we begin with a result wherethe perturbation of a sampling model is due to a small change of thegenetators of the underlying shift invariant space.
Theorem 3.1.