Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements

Transcription

1 Linear Stabe Samping Rate: Optimaity of D Waveet Reconstructions from Fourier easurements Ben Adcock, Anders C. Hansen, Gitta utyniok, and Jackie a arch, 0 Abstract In this paper we anayze two-dimensiona waveet reconstructions from Fourier sampes within the framework of generaized samping. For this, we consider both separabe compacty-supported waveets and boundary waveets. We prove that the number of sampes that must be acquired to ensure a stabe and accurate reconstruction scaes ineary with the number of reconstructing waveet functions. We aso provide numerica experiments that corroborate our theoretica resuts. Introduction A probem that appears in mutipe discipines is the reconstruction of an object from inear measurements. One specia situation of particuar importance which we wi focus on in this paper are Fourier measurements. This particuar reconstruction probem occurs in numerous appications such as Fourier optics, radar imaging, magnetic resonance imaging RI) and X-ray CT the atter after appication of the Fourier sice theorem). One of the main issues is that we are ony abe to acquire finitey many sampes, since we cannot process an infinite amount of information in practice. The reconstruction of an object from a finite coection of Fourier sampes can be obtained by a truncated Fourier series. However, this typicay eads to undesirabe effects such as the Gibbs phenomenon, which are wid osciation near points of discontinuity. Beside the Gibbs phenomenon, the convergence of the Fourier series in the Eucidean norm) is sow. Conversey, waveets bases are we-known to achieve much better resuts see [, Ch. 9]). Indeed, waveets have much better ocaization properties than the standard Fourier transform, eading to a better detection of image features. For this reason, waveets have found widespread used in compression and denoising. For exampe the agorithm of JPEG 000 is based on waveets. oreover, they come equipped with fast agorithms which are of great importance in today s age of technoogy. Waveets aso pay a pivota roe in biomedica imaging, with an exampe being the technique of waveet encoding in RI see [9,, 9, 0]). The issue, however, is that physica sampers such as an RI scanner naturay yied Fourier measurements, not waveet coefficients. Thus in order to expoit the power of waveets, we need a reconstruction agorithm capabe of producing waveet coefficients given a fixed set of Fourier measurements.. Generaized samping Generaized samping is a framework for this probem deveoped by two of the authors in a series of papers [,,, 5] and based on past work of Unser and Adroubi [7, 8], Edar [5] and Hrycak and Gröchenig [8]. The theory aows for stabe and accurate reconstructions in an arbitrary reconstruction system of choice given fixed measurements with respect to another system. The probem of reconstructing waveet coefficients from Fourier sampes is an important exampe of this abstract framework. athematicay, the reconstruction probem can be modeed in a separabe Hibert space H, resuting in an infinite-dimensiona inear agebra probem. Generaized samping provides a faithfu discretization of such a probem. The stabe samping rate see Section for detais) is fundamenta characteristic within generaized samping that determines how many sampes are needed in order to obtain stabe and accurate reconstructions with a given number of reconstruction eements. It is therefore vita that this rate be determined for important instances of generaized samping.. Our contribution In [7] the respective authors proved the inearity of the stabe samping rate for one-dimensiona compacty supported waveets based on finitey many Fourier sampes. This means, up to a constant, one needs the same

2 number of sampes as reconstruction eements. Our resuts extend the previous one to dimension two, athough higher dimensiona resuts can be obtained in a straightforward manner. This is an important extension, since most of the above appications invove two- or three-dimensiona images. The crucia part that makes our resut non-trivia is the aowance of non-diagona scaing matrices negecting straightforward arguments for separabe two dimensiona waveets from D to D. oreover, we wi not ony prove the inearity for standard twodimensiona separabe waveets, but aso for two-dimensiona boundary waveets which are of particuar interest for smooth images. This case was not considered in [7] but was addressed recenty in [] for the case on D nonuniform Fourier sampes. Here, for simpicity, we consider ony uniform sampes but in the D setting. At this stage we note that other higher dimensiona concepts, such as curveets and shearets, can provide better approximations rates for cartoon-ike-images; a specific cass of functions [0], [0]). However, this paper serves as an extension of known D resuts [7]. It thus provides a necessary first step in the study of reconstructions from Fourier sampes within the context of generaized samping in higher-dimensiona settings. We sha discuss shearets in an upcoming paper. Let us now make one further remark. The reader may at this stage wonder why, given a vector y of Fourier sampes of a D image, one cannot simpy form the vector x = U dft y, and then form z = V dwtx and hope that z woud represent waveet coefficients of the function f to be reconstructed here U dft and V dwt denote the discrete Fourier and waveet transforms respectivey). Unfortunatey, x represents a discretization of the truncated Fourier series of f. Thus, ignoring the waveet crime [5, p. ] for a moment, we find that z represents the waveet coefficients of the truncated Fourier series and not the waveet coefficients of f itsef taking the waveet crime into account, z woud actuay be an approximation to the waveet coefficients of the truncated Fourier series). Thus, z wi typicay have ost a the decay properties of the origina waveet coefficients. oreover, if we map z back to the image domain we get x = V dwtz and thus we do not gain anything as x is the discretized truncated Fourier series. This paper is about getting the actua waveet coefficients of f from the Fourier sampes, thus preserving a the decay properties of the origina coefficients. This is done using generaized samping. We show that the number of sampes that must be acquired to ensure a stabe and accurate reconstruction scaes ineary with the number of reconstructing waveet functions. This means that one can reconstruct a function from its Fourier coefficients yet get error bounds on the reconstruction up to a constant) in terms of the decay properties of the waveet coefficients. Put in short: seeing Fourier coefficients of a function is asymptoticay as good as seeing the waveet coefficients directy. As we wi see in the numerica experiments, the generaized samping reconstruction provides a substantia gain over the cassica Fourier reconstruction. The outine of the remainder of this paper is as foows. In Section we give a more eaborate introduction into generaized samping and present the main resuts of this method. Furthermore, we introduce the stabe samping rate to which our main focus is devoted. Having presented the framework we are deaing with, we introduce the waveet reconstruction systems and the Fourier samping systems in Section. The main resuts are then presented in Section. Finay we demonstrate our theoretica resuts in appication by presenting some numerica experiments in Section 5. Proofs of the main resuts are given in Section 6. Generaized Samping In this section, we reca the main definitions and resuts of the methodoogy of generaized samping from [,,, 5]. For this, we start by introducing a genera mode situation for reconstruction from sampes with associated quaity measures.. Genera Setting Let H be a separabe) Hibert space H with an inner product,, which wi be our ambient space throughout this section. For modeing the acquisition of sampes, et S H be a cosed subspace and {s k } k N H be an orthonorma basis for S. We wi refer to {s k } k N as the samping system and S as the samping space. For a signa f H, we then assume that the associated sampes aso caed measurements) are given by mf) k := f, s k, k N..) Based on the measurements mf) = f, s k ) k N, we aim to reconstruct the origina signa f. To be abe to utiize some prior knowedge concerning the initia signa f, we aso require a carefuy chosen reconstruction system {r i } i N H. The space R = span{r i : i N}, in which f is assumed to ie or be we approximated in, is then referred to as the corresponding reconstruction space.

3 Since in an agorithmic reaization, ony finitey many sampes and ikewise a finite inear combination of reconstruction eements is possibe, we aso introduce the finite-dimensiona spaces S = span{s,..., s }, N and R N = span{r,..., r N }, N N. Thus, the reconstruction probem can now be phrased as foows: Given sampes mf),..., mf) of an initia signa f H, compute a good approximation to f in the reconstruction space R N. Aiming to compare different methodoogies for soving this probem, we next formay introduce the notion of reconstruction method. Definition.. Let a samping system {s k } k N and reconstruction spaces R N, N N be defined as before. Further, et f H and et, N N. Then some mapping F N, : H R N is caed reconstruction method, if, for every f H, the signa F N, f) depends ony on mf),..., mf), where the sampes mf) j are defined in.). We can now strengthen the reconstruction probem in the foowing way: Given the dimension of the reconstruction space N, how many sampes are required to obtain a stabe and optimay accurate reconstruction method? This intuitive phrasing wi be made precise in the next subsections.. Quaity easures for Reconstruction ethods We start by introducing two quaity measures for reconstruction methods that anayze the degree of approximation within the reconstruction space and robustness for reconstruction. For this, throughout this subsection, et {s k } k N be a samping system, and et R N, N N be reconstruction spaces. The first measure quantifies the coseness of the reconstruction to the best reconstruction in the sense of the orthogona projection onto the reconstruction space. In the seque, for the orthogona projection onto a cosed subspace U, we wi aways utiize the notation P U : H U. Definition. [6]). Let F N, : H R N be a reconstruction method. oreover, et µ = µf N, ) > 0 be the east number such that f F N, f) µ f P RN f) for a f H. Then we ca µ the quasi-optimaity constant of F N,. If no such constant exists, then we write µ =. If µ is sma, we say F N, is quasi-optima. The second measure quantifies stabiity in the sense of robustness against perturbations. Definition. [6]). Let F N, : H R N be a reconstruction method. The absoute) condition number κ = κf N, ) > 0 is then given by ) FN, f + g) F N, f) κ = sup im, ε 0 mg) f H sup g H, 0< mg) ε where mg) = mg),..., mg), 0,...). If κ is sma, then we say F N, is we conditioned, otherwise F N, is caed i-conditioned. We now merge both definitions in order to have ony one singe measure for a reconstruction method. Definition. [6, 7]). Let F N, : H R N be a reconstruction method. The reconstruction constant of F N, is defined as CF N, ) = max{µf N, ), κf N, )}, where µf N, ) is the quasi-optimaity constant and κf N, ) is the absoute) condition number.

4 . The Infimum Cosine Ange Intuitivey, the ange between the samping and reconstruction space shoud pay a roe in determining the reconstruction constant for a given reconstruction method. As we wi see, this wi be in particuar the case for the reconstruction method of generaized samping. To prepare those resuts, in this subsection, we first introduce a particuary usefu notion of ange. The concept of principa anges between two Eucidean subspaces is we known in the iterature. However, for arbitrary cosed subspaces of a Hibert space many different notions of an ange exist. We exempariy mention the Friedrichs ange and the Dixmier ange cf. [, ]). For our anaysis, the notion of infimum cosine ange utiized, for instance, in [8, 8] wi be the most appropriate. It is defined as foows. Definition.5. Let U, V be cosed subspaces of H. Then the infimum cosine ange cosωu, V )) between U and V is defined by cosωu, V )) = inf P V u, ωu, V ) [0, π/]. u U, u = We remark that the infimum cosine ange is not symmetric in genera. For exampe, if U and V are two non-trivia cosed subspaces of H and U V with U V, then cosωu, V )) =, whereas cosωv, U)) = 0. The foowing genera characterization of pairs of cosed subspaces for which the infimum cosine ange is not symmetric was proven in [9]. Lemma.6 [9]). For two non-trivia cosed subspaces U, V H, we have cosωu, V )) cosωv, U)) if and ony if one of these quantities is zero and the other is positive. A positive infimum cosine ange between the samping and reconstruction space wi be crucia for enabing reconstruction at a. The next theorem provides a characterization of subspaces which admit a positive infimum cosine ange. Theorem.7 [6]). Let U, V be cosed subspaces of H. Then, we have that cosωu, V )) > 0 if and ony if U V = {0} and U + V is cosed in H. This characterization gives rise to the foowing definition. Definition.8 [6, 8]). If cosωu, V )) > 0 for two cosed subspaces U, V of H, then we say U and V obey the subspace condition. In this case, we define the associated obique) projection with range of P equa to U and kerne of P equa to V. P U,V : U V H We wish to mention that obique projections are customariy used in samping theory, such as, for instance, in [8, 5, 6, 8].. Reconstruction ethod of Generaized Samping We are now ready to introduce the method of generaized samping. To this end, we wi aways assume that the reconstruction space R and the samping space S fufi the subspace condition. In other words, we have cosωr, S)) > 0. For any N, et P S be the orthogona projection given by P S : H S, h h, s k s k. This enabes us to formay define the reconstruction method of generaized samping. Definition.9. For f H and N, N, we define the reconstruction method of generaized samping G N, : H R N by P S G N, f), r j = P S f, r j, j =,..., N..) We aso refer to G N, f) as the generaized samping reconstruction of f. k=

5 We emphasize that this is indeed a reconstruction method in the sense of Definition., since the righthand side of.) ony depends on the given sampes f, s k ) k= and not on f itsef. oreover, generaized samping is a inear reconstruction method. Agorithmicay, to determine G N, f), i.e., soving.), can be phrased as the numerica inear agebra probem of computing the coefficients α [N] = α,..., α N ) C N as the east-squares soution of u... u N U [,N] α [N] = mf) [], where U [,N] =....., u ij = r j, s i. u... u N Note that, if U [,N] is we-conditioned this requires ON) operations in genera. However, in the case of Fourier sampes and waveet reconstruction one can make use of the Fast Fourier Transform and the discrete waveet transform and thus reduce this figure down to ony O og ) operations. The phiosophy of generaized samping is to aow the number of sampes to grow independenty of the fixed number of reconstruction eements N. This fexibiity of and N is crucia for stabe reconstruction. The next theorem guarantees the existence of the reconstruction for any f H provided that the number of sampes is arge enough. Theorem.0 [6]). Let N N. Then, there exists an 0 N, such that, for every f H,.) has a unique soution G N, f) for a 0. In particuar, we then have G N, = P RN,P S R N )). oreover, the smaest 0 is the east number such that cosωr N, S 0 )) > 0. It is a priori not cear how to find N arge enough such that cosωr N, S )) > 0, or even determine the smaest such vaue 0 N. In the next subsection, it wi turn out that this is intimatey reated to the reconstruction constant defined in Definition., and wi ead to the notion of a stabe samping rate..5 Stabe Samping Rate In Subsection., we introduced the reconstruction constant as the main quaity measure for stabe and accurate reconstructions. Intriguingy, we can now reate this notion to the infimum cosine ange in the case of generaized samping as reconstruction method. Theorem. [6]). Retaining the definitions and notations from Subsection., for a f H, we have G N, f) cosωr N, S )) f, and f P RN f f G N, f) cosωr N, S )) f P R N f. In particuar, these bounds are sharp. oreover, the reconstruction constant of generaized samping CG N, ) obeys CG N, ) = µg N, ) = κg N, ) = cosωr N, S )). Hence, in order to obtain stabe and accurate reconstructions, it is both necessary and sufficient to contro the ange between R N and S. This eads us to the definition of the stabe samping rate. Definition.. For N N and θ >, the stabe samping rate is defined as ΘN, θ) = min { N : CG N, ) < θ}. Note that the stabe samping rate is of interest, since it determines the number of sampes required for guaranteed, quasi-optima and numericay stabe reconstructions. We next aim to compare generaized samping to other reconstruction methods. For this, we first introduce a cass of reconstruction methods, which recover signas from the reconstruction space exacty. 5

6 Definition.. A reconstruction method F N, : H R N is caed perfect, if F N, f) = f whenever f R N. Note that any reconstruction method with finite quasi-optimaity constant is automaticay perfect. The next resut proves that generaized samping is in the sense superior to any other perfect reconstruction method that its condition number as we as even its reconstruction constant is smaer. Theorem. [6]). Let N, and et F N, : H R N be a inear or non inear perfect reconstruction method. Then κg N, ) κf N, ), where G N, is the reconstruction method of generaized samping. In particuar, we have CG N, ) CF N, ). Next, we aim to compare the quasi-optimaity constant of generaized samping to other reconstruction methods. For this, assume that the stabe samping rate of generaized samping is inear in N, i.e., ΘN, θ) = ON) as N, and assume that there exist constants C f, D f, γ f > 0 depending on the initia signa f H such that C f N γ f f P RN f D f N γ f, for a N N..) The foowing resut then shows that the error of any reconstruction method F can be ony up to a constant better than the error of generaized samping reconstruction. Theorem.5 [6]). Suppose that the stabe samping rate ΘN, θ) is inear in N, i.e. ΘN, θ) = ON) as N. Let f H be fixed and et F : mf),..., mf) ) F f) R ψf ), be a reconstruction method, where ψ f : N N with ψ f ) λ for some λ > 0. Assume that.) hods. Then, for any θ >, there exist constants dθ) 0, ) and cθ, C f, D f ) > 0 such that f G dθ), f) cθ, C f, D f ) f F f), for a N, where G N, denotes the generaized samping reconstruction method. Linear Samping Rate for Compacty Supported D Waveets As aready eaborated upon in the introduction, the situation of taking the Fourier basis as samping system is of particuar importance to appications. A very common choice for a reconstruction system in imaging sciences are D waveets, predominanty of compact support due to their high spatia ocaization. In this section, we wi state our main resut for the situation of Fourier sampes and reconstruction within a waveet basis using D compacty supported waveets. ore precisey, we wi show inearity of the stabe samping rate for the associated generaized samping scheme, which shows by Theorem.5 the near-optimaity of this reconstruction method. For this, we start by defining first the reconstruction and second the samping space, foowed by the statement of our main resut. Due to its technica nature, we present its proof in a separate section, namey Section 6.. Samping and Reconstruction Spaces.. Compacty Supported D Waveets We start by recaing the notion of scaing matrices. Definition.. Let A be a matrix with non-negative integer entries and eigenvaues greater than one in moduus. Then we ca A a scaing matrix. For the sake of brevity, in the seque, we wi use the notation ) λ λ A =. λ λ 6

7 oreover, for the entries of A j we write ) A j λ j) = λ j) λ j) λ j), j N. Notice that λ j) i λ i ) j in genera. For the sake of competeness, we next give the definition of a D mutiresoution anaysis RA). For more detais, we refer to the existing iterature, e.g [, ]. Definition.. Let A be a scaing matrix. Then a sequence of cosed subspaces V j ) j Z of L R ) is caed a mutiresoution anaysis, if the foowing properties are satisfied. i) {0}... V j V j+... L R ), ii) V j = {0}, j Z iii) V j = L R ), j Z iv) f V j fa ) V j+, v) there exists a function φ L R ) caed scaing function), such that constitutes an orthonorma basis for V 0. {φ 0,m := φ m) : m Z } The associated waveet spaces W j ) j Z are then defined by V j+ = V j W j. It is we known that there exist det A corresponding compacty supported waveets ψ det A,..., ψ such that {ψ p j,m := det A j/ ψ p A j m) : m = m, m ) Z, p =,..., det A } forms an orthonorma basis for W j for each j, see, e.g., [?]. We now consider the decomposition L R ) = V 0 j N W j, where and V 0 := span{φ 0,m : m = m, m ) Z } W j := span{ψ p j,m : m = m, m ) Z, p =,..., det A }, j N... Reconstruction Space Our aim is to reconstruct functions that are supported on [0, a]. To this end, suppose that the scaing function and waveet functions are supported in [0, a]. To mimic the fact that practica appications can ony hande finite systems, we restrict to those functions whose support intersects [0, a], i.e., to the systems and Ω = {φ 0,m : m = m, m ) Z, a < m, m < a} Ω = {ψ p j,m : j N {0}, m = m, m ) Z, a < m < aλ j) + λ j) ), a < m < aλ j) + λ j) ), p =,..., det A }. The reconstruction space R is then defined as the cosed inear span of these functions, which is R = span{ϕ : ϕ Ω Ω }. To define the finite-dimensiona subspaces R N, we require an ordering for this system. The most natura way to order Ω Ω is starting from waveets at coarsest scae and then continue to higher scaes. Within one scae, one might order the transation m, m ) in a exicographica manner starting from the smaest number up the 7

8 argest. ore precisey, we fix m and et m run, increase m by one and repeat. This then eads to the foowing ordering of Ω Ω : {ϕ i } i N = {φ 0, a+, a+),... φ 0, a+,a ), φ 0, a+, a+)..., φ 0, a+,a ),..., φ 0,a,a ), ψ0, a+, a+),..., ψ 0, a+,m 0) ),..., ψ 0,m 0), a+),..., ψ 0,m 0),m0) ),..., det A det A det A det A ψ 0, a+, a+),..., ψ..., ψ..., ), 0,m 0) ψ +), 0,m 0) 0, a+,m 0), m0),m0) ), ψ, a+, m ) +),..., ψ, a+,m ) ), ψ, a+, m ) +),..., ψ, a+,m ) ),...}.) where m j) = aλ j) + λ j) ) and mj) = aλ j) + λ j) ). We emphasize that the presented resuts do not depend on the specific ordering within the scae. Finay, based on the chosen ordering, we define the reconstruction space R N by R N = span{ϕ i : i =,..., N}, N N. In practise, it is much more common to use as approximation spaces those being generated up to a specific scae, say, J. To mimic this approach, we coarsen the choices of N N for which we consider R N suitaby in the foowing way. First, we observe that the number of functions being in R up to a fixed scae J N is J N J = a ) + det A ) aλ j) + λ j) + ) )aλ j) + λ j) + ) )..) j=0 Lemma.. Retaining the definitions and notations above, then N J C a λ J) + λ J) )λj) + λ J) ) for some constant C a depending on a. Proof. The tota number of eements N J in the reconstruction space R NJ up to a scae J is prescribed by the matrix A and the support of the scaing function φ and the waveet ψ, respectivey. However, since the support of both φ and ψ is a square of the form [0, a], the tota number of eements in the reconstruction space N J is the same as if we woud have generated the waveet system with the transpose of A. Hence, it is sufficient to prove N J λ J) + λ J) )λj) + λ J) ), where the constant depends on a. We prove the resut by induction. For J = it is ceary true. Hence, Now, A J = J a ) + det A ) aλ j) + λ j) + ) )aλ j) + λ j) + ) ) ) λ J) λ J) λ J) λ J) which impies = j=0 λ J ) + λ J ) )λ J ) + λ J ) ) + det A )aλ J ) + λ J ) + ) )aλ J ) + λ J ) + ) ) ) λ J ) λ J ) λ ) λ λ J ) λ J ) λ λ λ J) + λ J) = λ J ) λ + λ ) + λ J ) λ + λ ), λ J) + λ J) = λ J ) λ + λ ) + λ J ) λ + λ ). = ) λ J ) λ + λ J ) λ λ J ) λ + λ J ) λ λ J ) λ + λ J ) λ λ J ) λ + λ J ) λ Simiar to the construction in [7], we now define the truncated scaing space by and the truncated waveet spaces by W a) j V a) 0 := span{φ 0,m : m = m, m ) Z, a < m, m < a} := span{ψ p j,m : m = m, m ) Z, a < m < aλ j) + λ j) ), The reconstruction space of interest to us is then defined by R NJ a < m < aλ j) + λ j) ), p =,..., det A }. = V a) 0 W a) 0... W a) J. 8

9 .. Samping Space To define the samping space consisting of eements of the Fourier basis, we first choose T, T > 0 sufficienty arge such that R L [ T, T ] ). Thus, ony functions supported on [ T, T ] are reevant to us. Indeed, choosing T a and T a is enough. Note that λ 0) = λ 0) = and λ 0) = λ 0) = 0. To aow an arbitrariy dense samping, for each ε T +T, we define the samping vectors by s ε) = εe πiε, χ [ T,T ], Z..) Thus, we sampe in each direction with the same samping rate ε. Based on these samping vectors, we now define the samping space S ε) by { S ε) = span s ε) : Z }. The finite-dimensiona subspaces S ε), =, ) N N, are then given by { } S ε) = span s ε) : =, ) Z, i i i, i =,.. ain Resut Our main resuts concerns the stabe samping rate of the generaized samping scheme for the samping spaces S ε) and the reconstruction spaces R N J. By Theorem., for this, we have to contro the infimum cosine ange between the R NJ and S ε). In particuar, we wish to determine =, ) N N such that, for given θ > and J N the term cosωr NJ, S ε) )) can be bounded from beow by θ, where N J denotes the tota number of reconstruction eements up to scae J. For this to work, we wi assume that the scaing matrix does not distort the grid too much. To make this precise, et I = {, ) Z : i i i, i =, }, L = [, ] [, ] for, ) N. Then, we assume that the so-caed mesh norm δ obeys ) og + µl ) δ := max min x y + k < x εa J L ) k εa J Z ) π max{ L, L, L, L },.) for some ε independent of J, where µ denotes the D ebesgue measure and L i, i =,,, are bounds that are obtained by Lemma 6.. We wi aso discuss this assumption in Exampe.5 for better understanding. The foowing resut shows that the stabe samping rate is indeed inear in the considered situation, showing that this scheme is superior to any other reconstruction method in the sense of Theorem.5. Theorem.. Let N J be the number of reconstruction eements up to a fixed scae J, and et R NJ and S ε) be the reconstruction space and samping space, respectivey. Furthermore, assume.) is fufied. If θ >, then there exists a constant S θ) independent of J and ε such that, if then, for =, ), λj) + λ J) S θ), ε cosωr NJ, S ε) )) θ. λj) + λ J) S θ), ε In particuar, the stabe samping rate obeys ΘN J, θ) = ON J ) as N J for every fixed θ >. Summarizing, this resut shows that the required number of Fourier sampes is optimay sma up to a constant when utiizing D compacty supported waveets for the generaized samping reconstruction. This extends the resut of [7] to the two-dimensiona case. We next consider a specia, yet widey used choice for scaing matrices, namey diagona matrices. This incudes two dimensiona dyadic waveets, which are those typicay used in appications. This situation is aso considered in the numerica experiments presented in Section 5. 9

10 Exampe.5. In this exampe we want to demonstrate our resuts and, in particuar, give some better understanding of the construction and assumption.). Furthermore, this exampe sha show that there are arge casses of D waveets that fufi.). For this purpose, et ) 0 A = 0 be the scaing matrix, which as mentioned before gives rise to three waveet generators. Let φ and ψ p, p =,, be two dimensiona scaing and waveet functions with compact support in [0, a], a N, which might be obtained by tensor products of one dimensiona scaing and waveet functions, see [, ]. As in Subsection.., we define and Ω := {φ 0,m : m = m, m ) Z, m i < a, i =, }.5) Ω := {ψ p j,m : j N {0}, m = m, m ) Z, a < m i < j a, i =,, p =,, },.6) since these are the ony functions whose support intersects [0, a]. Again, in ine with our previous approach, we then define the reconstruction space R by R = span{ϕ : ϕ Ω Ω }, order the eements Ω Ω anaogousy to.), and set R N = span{ϕ i : i =,..., N}, N N. Now, each function ϕ in R N can be represented as a inear combination of scaing functions at highest eve J, in particuar, there exist positive integers L, L, L, and L such that ϕ = L =L L =L α, φ J,, ), α, C. This statement wi be proven in Lemma 6.. With a view to.), the expicit expression of the bounds L i, i =,,, are highy important. In fact, we shoud not choose them too arge, otherwise.) might not hod. Since ϕ = L =L L =L ϕ, φ J,, ) φ J,, ), and is compacty supported, shifting by, ) far enough eads to the fact that the coefficients become zero. For the scaing matrix A = diag, ) we have see proof of Lemma 6.) L = L = J a ), L = L = a + J a + ). oreover, the mesh norm δ obeys δ ε. Hence, for ε = J πa ) ) ) og J δ πa ) J ε + πa ) J,.7) so assumption.) is fufied. Thus, we can appy Theorem. to obtain a constant S θ independent of J, such that for J S θ, = ε we have cosωr NJ, S ε) )) > θ. Counting the numbers of eements in the space R NJ gives J N J = a ) + j a+a ) = J )a +6aa ) J )+Ja ) +a ) = O J ),.8) j=0 which means the number of sampes scaes ineary with the number of reconstruction eements. 0

11 Linear Samping Rate for D Boundary Waveets In appications, typicay signas on a bounded domain are considered such as on the interva [0, ]. Aiming to avoid artifacts at the boundaries x = 0,, in [] see aso []) boundary waveets were introduced. The considered waveet system then consists of interior waveets, which do not touch the boundary, and the just mentioned boundary waveets, eading to a system with an associated mutiresoution anaysis and prescribed vanishing moments even at the boundary. Simiar to the cassica waveet construction, this system can be ifted to D by tensor products. In this section, we wi show that in fact aso this boundary) waveet system aows a inear stabe samping rate, though with a proof differing significanty from the one of the cassica waveet case due to the different structure at the highest scae. For this, we start with formay introducing D boundary waveets which we use as an expression for the whoe waveet system foowed by our choice of reconstruction and samping space.. Construction of Boundary Waveets We start with the D construction of waveets on the interva as introduced in []. For this, et φ be a compacty supported Daubechies scaing function with p vanishing moments. It is we known that φ must then have a support of size p. By a shifting argument, we can assume that supp φ) = [ p +, p]. In order to propery define interior waveets and boundary waveets, the scae need to be arge enough i.e., the support of the waveets need to be sma enough to be abe to distinguish between waveets whose support fuy ie in [0, ] and waveets whose support intersect the boundary x = 0 and, ikewise, x =. Therefore, we now et j N such that p j. Then there exist j p interior scaing functions, i.e., scaing functions which have support inside [0, ], defined by φ b j,n = φ j,n = j/ φ j n), for p n < j p. Depending on boundary scaing functions {φ eft n } n=0,...,p and {φ right n } n=0,...,p, which we wi introduce beow, the p eft boundary scaing functions are defined by and the p right boundary scaing functions are φ b j,n = j/ φ eft n j ), for 0 n < p, φ b j,n = j/ φ right j n j )), for j p n < j. We remark that this eads to j scaing functions in tota, which is the number of origina scaing functions φ j,n ) n that intersect [0, ]. We next sketch the idea of the construction of boundary scaing functions {φ eft n } n=0,...,p as we as {φ right n } n=0,...,p foowing [], to the extent to which we require it in our proofs. One starts by defining edge functions φ k on the positive axis [0, ) by φ k x) = p n=0 ) n φx + n p + ), k = 0,..., p, k such that these edge functions are orthogona to the interior scaing functions and such that they together generate a poynomias up to degree p. After performing a Gram-Schmidt procedure one obtains the eft boundary functions φ eft k, k = 0,..., p. The right boundary functions are then after some minor adjustments obtained by refecting the eft boundary functions. This construction from [] aows one to obtain a mutiresoution anaysis. Theorem. []). If j p, then {φ b j,n } n=0,..., j is an orthonorma basis for a space Vj b i.e. V b j V b j+, that is nested, and compete, i.e. j og p V b j = L [0, ].

12 Next, we define an orthonorma basis for the waveet space Wj b, which is as usua defined as the orthogona compement of Vj b in V j+ b. For this, et ψ be the corresponding waveet function to φ with p vanishing moments and supp ψ = [ p +, p]. Simiar to the construction of the scaing functions, we wi obtain interior waveets and boundary waveets, which then constitutes the set of waveets in the interva. Again based on a carefu choice of boundary waveets ψn eft ) n and ψ right k ) k, for which we refer to [], we define j p interior waveets by p eft boundary waveets and p right boundary waveets ψ b j,n = ψ j,n = j/ ψ j n), for p n < j p, ψ b j,n = j/ ψ eft n j ), for 0 n < p, ψ b j,n = j/ ψ right j n j )), for j p n < j. Summarizing, the foowing resut hod for these waveet functions. Theorem. []). Let J p. Then the foowing properties hod: i) {ψ b J,n } n=0,..., J is an orthonorma basis for W b J. ii) L [0, ] can be decomposed as L [0, ] = VJ b WJ b WJ+ b WJ+ b... = VJ b Wj b. iii) { {φ b J,m } m=0,..., J, {ψ b j,n } j J,n=0,..., j } is an orthonorma basis for L [0, ]. iv) If φ, ψ C r [0, ], then { {φ b j,m } m=0,..., J, {ψ b j,n } j J,n=0,..., j } is an unconditiona basis for C s [0, ] for a s < r. As mentioned before, this system gives rise to a D system by tensor products, i.e., by the standard D separabe waveet construction. In particuar, this D system again constitutes an RA, see [].. Samping and Reconstruction Space.. Reconstruction Space For defining the reconstruction space, we wi now assume that the region of interest is [0, ] instead of [0, a] with a N, as previousy chosen. Our starting point is a D compacty supported Daubechies scaing function with p vanishing moments cf. Subsection.). Reca that Daubechies waveets have at east the foowing frequency decay, φξ), as ξ..) + ξ Let ψ be a corresponding waveet with p vanishing moments, and et φ b and ψ b be the waveets on the interva as introduced in the previous subsection. Let J 0 be the smaest number such that J0 p. Then the associated D scaing functions are of the form j=j φ b J 0,n,n ) := φb J 0,n φ b J 0,n, 0 n, n J0 and, for 0 n, n j with j J 0, the D waveet functions are defined by φ b ψ b,k j,n ψj,n b, j J 0, k =, j,n :=,n ) ψj,n b φ b j,n, j J 0, k =, ψj,n b ψj,n b, j J 0, k =. Next, et and Ω = {φ b J 0,n,n ) : 0 n, n J0 } Ω = {ψ b,k j,n,n ) : j = J 0,,..., J, 0 n, n j, k =,, }. Then, for a fixed scae J, and for N J = J, the reconstruction space R NJ is given by R NJ = span{ϕ i : ϕ i Ω Ω, i =,..., N J }..) An ordering can be obtained anaogousy as in Subsection...

13 Numbers for Haar and θ = 0.5 J N J for ε = / for ε = / Numbers for Daubechies and θ = 0.5 J N J for ε = /7 for ε = / Tabe : Number of reconstruction eements and sampes for Haar and Daubechies-. Note that these numbers predict the jumps in Figure... Samping Space The samping space can be chosen simiar to Subsection.., i.e., for ε, we set s ε) = εe πiε, χ [0,], Z, and for =, ) N N { } S ε) = span s ε) : =, ) Z, i i i, i =,..). ain Resut Our main resut of this section states the inearity of the stabe samping rate for boundary waveets, whose proof is presented in Section 7. Theorem.. Let J N, ε, and θ >. Further, et S ε) and R N J be defined as. and.), respectivey. Then inf ϕ R NJ ϕ = P ε) S ϕ θ for = S θ /ε J, S θ /ε J ) N N, where S θ is a constant independent of J. In particuar, the stabe samping rate obeys ΘN J, θ) = ON J ) as N J for every fixed θ > Remark.. This resut wi be proved independenty of Theorem., since boundary waveets do constitute an RA but at highest scaing eve, the space V J contains more than one generating function φ. It uses the refected functions as we. 5 Numerica Experiments In this section we numericay demonstrate the inearity of the stabe samping rate as stated in Theorem.. We wi aso demonstrate how this combines with generaized samping in practice. In particuar, given this inearity, reconstructing from Fourier sampes in smooth boundary waveets wi give an error decaying according to the smoothness and the number of vanishing moments. In this section we consider dyadic scaing matrices ) 0 A =. 5.) 0 Furthermore, our focus are separabe waveets, i.e. waveets that are obtained by tensor products of one dimensiona scaing functions and one dimensiona waveet functions, respectivey. Scaing matrices of the form 5.) preserve the separabiity.

14 Number of sampes Number of sampes N,) Graph of the SSR Sope max/nmax N,) Graph of the SSR Sope max/nmax N Number of reconstruction eements a) ΘN, θ) for Haar. Computed for J = up to N = 0 with ε = / and θ = N Number of reconstruction eements b) ΘN, θ) for Haar. Computed for J = up to N = 0 with ε = / and θ = Number of sampes N,) Graph of the SSR Sope max/nmax Number of sampes N,) Graph of the SSR Sope max/nmax N Number of reconstruction eements c) ΘN, θ) for Daubechies-. Computed for J = up to N = 908 with ε = /7 and θ = N Number of reconstruction eements d) ΘN, θ) for Daubechies-. Computed for J = up to N = 908 with ε = /8 and θ = 0.5. Figure : Stabe samping rate ΘN, θ) for two dimensiona dyadic Haar waveets and two dimensiona dyadic Daubechies- waveets. 5. Linearity exampes with Haar and Daubechies- waveets We use the description of Section. and Exampe.5 in order to perform numerica experiments for some known waveets. This gives the reconstruction space R = span{ϕ : ϕ Ω Ω } where Ω and Ω are defined in.5) and.6) respectivey. We order the reconstruction space R in the same manner as presented at the end of Section. In.8) we counted the number of reconstruction eements up to eve J, which eads to N J = J )a + 6aa ) J ) + Ja ) + a ) 5.) many eements, which is asymptoticay of order J. We test Haar waveets and D Daubechies- waveets. Figure shows the inear behaviour of the stabe samping rate for these two types of waveet generators. By a sma abuse of notation, we aso write for the tota number of sampes. In our anaysis in Section we estimated the ange cosωr N, S ε) )) reca that ɛ is the samping rate) with respect to some fixed θ >. In fact we computed such that inf ϕ R N ϕ = P ε) S ϕ θ

15 Figure : Reconstruction of the function f x, y) = cos x) exp y). The second row shows an 8 times zoomedin version of the upper eft corner. Left: origina function. idde: truncated Fourier series with 56 Fourier coefficients. Right: generaized samping with Daubechies- waveets computed from the same Fourier coefficients. hods. We proved that is up to a constant of the same size as N J. Figure shows the stabe samping rate in bue) { ΘN, θ) = min N, cosωr N, S )) } θ and the inear function f in red) given by fn) = max N max N, where N max is the maximum vaue of N used in the experiment and max = ΘN max, θ). We computed the stabe samping rate up to eve J =. Note the significant) jumps of the stabe samping rate occur whenever N N crosses the scaing eve N J, J = 0,...,. In the Haar case these are N 0 =, N = 6, N = 6, N = 56, N = 0, see 5.). Note that a = in the Haar case. In particuar, the jumps are inear, suggesting a inear stabe samping rate. Figure b), c), and d) are interpreted simiary. However, the theoretica resuts are asymptotic resuts. Therefore, it shoud not be surprising that the stabe samping rate is beow the inear ine in some cases. It aigns asymptoticay. 5. Fourier sampes and boundary waveet reconstruction In this exampe we wi demonstrate the efficiency of generaized samping given the estabished inearity of the stabe samping rate. In particuar, suppose that f is a function we want to recover from its Fourier information. It is smooth, however, not periodic a probem that occurs for exampe in eectron microscopy and aso in RI. This causes the cassica Fourier reconstruction to converge sowy, yet a smooth boundary waveet basis wi give much faster convergence see [, Ch. 9]). As discussed, the issue is that we are given Fourier sampes, not waveet coefficients. However, this is not a probem in view of the inearity of the stabe samping rate. In 5

16 Figure : Reconstruction of the function f x, y) = + x )y ). Upper eft: truncated Fourier series with 5 Fourier coefficients. idde eft: 8 times zoomed-in version of the upper figure. Lower eft: error committed by the truncated Fourier series. Upper right: generaized samping with Daubechies- waveets computed from the same 5 Fourier coefficients. idde right: 8 times zoomed-in version of the upper figure. Lower right: error committed by generaized samping. 6

17 particuar, if f W s 0, ), where W s 0, ) denotes the usua Soboev space, and P RN denotes the projection onto the space R N of the first N boundary waveets see.)), then f P RN f = ON s ), N, provided that the waveet has sufficienty many vanishing moments. Now, if G N, f) R N is the generaized samping soution from Definition.9 given Fourier coefficients, and is chosen according to the stabe samping rate then f G N, f) = ON s ) = O s ), N. Hence, we obtain the same convergence rate up to a constant, by simpy postprocessing the given sampes. To iustrate this advantage we wi consider the foowing two functions: f x, y) = cos x) exp y), f x, y) = + x )y ). In Figure we have shown the resuts for f and compared the cassica Fourier reconstruction with the generaized samping reconstruction. Both exampes use exacty the same sampes, however, note the peasant absence of the Gibb s ringing in the generaized samping case. The same experiment is carried out for f in Figure, however, here we have dispayed the reconstructions in D in order to visuaize the error. 6 Proof of Theorem. The proof of Theorem. is somewhat technica, wherefore we divide the proof into severa steps. First, in Subsection 6., the overa structure of the proof is presented, and the respective detais can then be found in Subsection Structure of the Proof Let ε 0, /T + T )] and θ >. Then we have to prove that inf ϕ R NJ ϕ = P ε) S ϕ θ, 6.) for ) λ J) + λ J) =, ) = S θ), λj) + λ J) S θ), ε ε and some S θ) independent of J. To this end, et ϕ R N be such that ϕ =. Since the samping functions form an orthonorma basis of S ε), we obtain P ε) S ϕ = = = ϕ, s ε), =, ). 6.) By Lemma 6., which reies mainy on the underying RA structure, we can write ϕ in terms of scaing functions at highest scae, i.e., ϕ = L =L L =L α, φ J,, ), for some L, L, L, L Z. Lemma 6., which is proven by direct computations, then shows that where Φ is a trigonometric poynomia of the form Φz) = ϕ, s ε) = ε det A J/ ΦεA J ) T ) φεa J ) T ), L m =L L m =L α m,m e πi z,m, z R, m = m, m ). 7

18 Using 6.), we concude that P ε) S ϕ = = By Theorem 6.0, there exist some ε 0 > 0 and S θ) = that for Since we obtain = = ε det A J ΦεA J ) T ) φεa J ) T ), =, ). ) S θ), Sθ) N N that does not depend on J such ε 0 det A J Φε 0 A J ) T ) φε 0 A J ) T ) Φ θ = = A J ) T ε Sθ) = 0 Φ = L m =L L λ J) +λ J) ε 0 λ J) +λ J) ε 0 S θ) S θ). m = L β m,m = ϕ =, P ε S 0 ) ϕ θ. Finay, Lemma 6. impies that a change of ε ony changes the constant, showing that 6.) is true for any ε 0, /T + T )], thereby proving Theorem.. 6. Auxiiary resuts In this section we wi prove the resuts mentioned in Subsection 6., which are required for competing the proof of Theorem.. The foowing resut is an extension of a resut from [6] to the two dimensiona setting. Lemma 6.. Let J N and ϕ V a) 0 W a) 0... W a) J. Then there exist L, L, L, L Z dependent of J such that ϕ = L =L L =L α, φ J,, ). Proof. Let ϕ V a) 0 W a) 0... W a) J. Then ϕ has an expansion of the foowing form ϕ = m =a m =a m = a+ m = a+ α m,m φ 0,m,m ) + det A p= J j=0 aλ j +λj ) m = a+ aλ j +λj ) m = a+ β p j,m,m ) ψp j,m,m ). 6.) Since V a) 0 V 0 and W a) j W j for j N and the sequence V j ) j Z forms an RA, it foows that V a) 0 W a) 0... W a) J V J. Since an orthonorma basis for V J is given by the functions {φ J,m : m Z }, for each m i < a, i =,, we have φ 0,m,m ) = φ 0,m,m ), φ J, φ J, Z oreover, since φ 0,m,m ) and φ J, are compacty supported, we obtain φ 0,m,m ), φ J, 0, if and a + a + )λ J) + λ J) ) a )λ J) + λ J) ) a + a + )λ J) + λ J) ) a )λ J) + λ J) ). 8

19 This foows by a straightforward computation from the support conditions of φ 0,m,m ) and φ J, together with m i < a, i =,. Simiary, we have ψ p j,m,m ), φ J,, ) 0 if Case I: det A j 0 and a λ J) a + )λ j) + λ j) aλj) + λ j) ) ) det A j < < λ J) aλ j) λ j) ) + aλj) + λ j) ) )λj) + a )λ j) det A j + + λ J) aλ j) λ j) ) + aλj) + λ j) ) )λj) + a )λ j) det A j + λ J) a + )λ j) + aλ j) + λ j) ) )λj) det A j a λ J) a + )λ j) + λ j) aλj) + λ j) ) ) det A j < < λ J) aλ j) λ j) ) + aλj) + λ j) ) )λj) + a )λ j) det A j + + λ J) aλ j) λ j) ) + aλj) + λ j) ) )λj) + a )λ j) det A j. Case II: det A j < 0 + λ J) a + )λ j) + aλ j) + λ j) ) )λj) det A j and a λ J) a + )λ j) + λ j) aλj) + λ j) ) ) det A j > > λ J) aλ j) λ j) ) + aλj) + λ j) ) )λj) + a )λ j) det A j + + λ J) aλ j) λ j) ) + aλj) + λ j) ) )λj) + a )λ j) det A j + λ J) a + )λ j) + aλ j) + λ j) ) )λj) det A j a λ J) a + )λ j) + λ j) aλj) + λ j) ) ) det A j > > λ J) aλ j) λ j) ) + aλj) + λ j) ) )λj) + a )λ j) det A j + + λ J) aλ j) λ j) ) + aλj) + λ j) ) )λj) + a )λ j) det A j. + λ J) a + )λ j) + aλ j) + λ j) ) )λj) det A j inimizing the ower bounds with respect to j {0,..., J } and maximizing the upper bounds with respect to j, respectivey, yieds the caim. The foowing emma is we known see []). Lemma 6.. Let f L R ). Then {f m) : m Z } is an orthonorma system if and ony if m Z fξ + m) = for amost every ξ R. Finay, one more technica emma is needed. Lemma 6.. Let A be a scaing matrix, J Z, and m = m, m ) Z. Further, et ϕ span{φ J,m Z } be compacty supported in [ T, T ], and et L, L, L, L Z be such that : m ϕ = L m =L L m =L α m,m φ J,m,m ), α m C. Then, for a Z, ϕ, s ε) = ε det A J/ ΦεA J ) T ) φεa J ) T ), 9

20 where s ε) is defined in.) and Φ is the trigonometric poynomia given by Φz) = α m e πi z,m, z R, m = m, m ). L m L, L m L, Proof. Since ϕ is supported in [ T, T ], we obtain ϕ, s ε) = ε ϕx)e πiε,x χ [ T,T dx ] This proves the caim. 6. Theorem 6.0 R = ε ϕε) L = ε = ε m =L L L m =L α m,m φ J,m,m ) L α m,m m =L m =L ) ε) ) φj,m,m ) ε) = ε det A J/ ΦεA J ) T ) φεa J ) T ). The proof of Theorem 6.0 requires a particuar estimate Proposition 6.9) for the norm of trigonometric poynomias depending on their evauations on a particuar grid whose mesh norm and associated Voronoi regions come aso into pay. 6.. esh Norm We start with the definition of a mesh norm for the situation we are faced with. A mesh norm can be interpreted as the argest distance between neighboring nodes. Definition 6.. Let Λ Z be an integer grid of the form and et A be a scaing matrix. Set and define a metric ρ on Ω by Λ := {, ) Z : i i i, i =, },, N, The mesh norm of {x Ω : Λ} is then defined as where x := A, Λ denote the nodes in Ω. Ω := Λ A := A[, ] [, ]) R, ρ : Ω Ω R +, x, y) min k AZ ) x y + k. δ := max x Ω min ρx, x), Before we can continue, we require some notions and resuts on Voronoi regions and trigonometric poynomias. Our first resut shows that if the distance between the nodes {x } converges to zero, the mesh norm of the entire grid Ω converges to zero. Lemma 6.5. Let ε > 0 and Λ Z be defined by Λ := {, ) Z : i i i, i =, }, where, N. Furthermore, suppose A is a scaing matrix, and et Ω ε) := Λ εa. If ε 0, then δ ε) 0, where δ ε) denotes the mesh norm of {εa Ω ε) : Λ}. 0

21 Proof. If ε 0, then εaλ) = {εa : Λ} {0, 0)} with respect to the standard Eucidean distance. Furthermore, for x := εa, we have im ε 0 δε) = im ε 0 max min x Ω ε) Since x, x Ω ε) for a Λ, we obtain Inserting this estimate into 6.) yieds min x x + k im k εaz ) max min ε 0 x Ω ε) im max min x x im max x y. ε 0 x Ω ε) ε 0 x,y Ω ε) x x. 6.) im ε 0 δε) im max x y im diam Ω ε) = im diam Λ εa = im diam ελ A = im ε diam Λ A ε 0 x,y Ω ε) ε 0 ε 0 ε 0 ε 0 }{{} = 0, where diamf ) denotes the diameter of a set F R d, i.e., diamf ) = sup d x, y), x,y F and d denotes the Eucidean metric on R d. Since the mesh norm is aways non-negative, the emma is proven. < 6.. Voronoi Regions The next resut studies the voume of the Voronoi regions associated to the previousy considered grid Λ with respect to the metric ρ defined in Definition 6.. We start by formay defining the notion of Voronoi region in our setting. Definition 6.6. Let Ω R, and et x ) be a sequence in R. Then we refer to the sets V ) defined by as Voronoi regions with respect to Λ, Ω, ρ, and x. We can now state the previousy announced resut. Lemma 6.7. Let, N and V := {x Ω : ρx, x ) ρx, x k ) for a k } Λ := {, ) Z : i i i, i =, }. oreover, et Ω = Λ Id, where Id denotes the -identity matrix, and et V ) be the Voronoi regions with respect to Λ, Ω, ρ, and. Then, for a Λ, µv ), where µ denotes the D Lebesgue measure. Proof. Notice that the Voronoi regions V ) are in fact rectanges, since the grid is an integer grid with a constant step-size. Hence, for each Λ, µv ) = a, b,, a,, b, R, 6.5) where a, denotes the width and b, the height of the rectange V. Towards a contradiction, assume that V does contain two different nodes x k and x with k. This impies which is a contradiction. Thus, we can concude that which, by 6.5), proves the caim. 0 ρx k, x ) ρx, x ) = 0, a, ρx +,, x, ) and b ρx,, x, +),

22 We next obtain a sight generaization of the previous resut. Lemma 6.8. Let A be a inear) bijective transformation acting on R with matrix representation ) λ λ A =, λ λ and et Λ and V ) be defined as in Lemma 6.7. Then µav )) det A. Proof. The resut foows from Lemma 6.7 by an integration by substitution. 6.. Trigonometric Poynomias The next theorem is an adapted version of a resut presented in [, Thm..] which again is a reformuation of a resut proven by Gröchenig in [7]. Proposition 6.9. Let J, L, L, L, L Z such that L L and L L, and et Φ a trigonometric poynomia of the form Φz) = L m =L L m =L α m,m e πi z,m, z R, m = m, m ). Further, et the grid Λ be defined as in Lemma 6.7, et A be a scaing matrix, and et Ω ε) := Λ εa J for ε > 0. Set x := εa J ) T, Λ. If the mesh norm δ of {x Ω ε) : Λ} obeys ) og µω + δ < ε) ) π max{ L, L, L, L }, where µ is the D Lebesgue measure, then there exists a positive constant Cδ, L, L, L, L ) such that Cδ, L, L, L, L ) Φ ε det A J ΦεA J ) T ), where Cδ, L, L, L, L ) = ) ) e πδ max{ L, L, L, L } µω ε) ) and ) / f := fx) dx, f L Ω ε) ). Ω ε) Proof. We first observe that by the hypotheses, the constant Cδ, L, L, L, L ) is indeed positive. Second, et V ) be the Voronoi regions with respect to Λ, Ω ε), and ρ. For Λ, we define the weights ω := µv ). As in Lemma 6.8, integration by substitution yieds ω = µv ) ε det A J. Hence it suffices to prove the existence of a constant Cδ, L, L, L, L ) > 0 such that Cδ, L, L, L, L ) Φ ω ΦεA J ) T ), 6.6) For this, we first observe that ω ΦεA J ) T ) = Ω ε) ΦεA J ) T = Φx )χ V. 6.7)