The type 3 nonuniform FFT and its applications

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1 Journal of Computational Phyic 206 (2005) 1 5 Short Note The type 3 nonuniform FFT and it application June-Yub Lee a, *, Lelie Greengard b a Department of Mathematic, Ewha Woman Univerity, 11-1 Daehyundong, Seodaemoongu, Seoul, , Korea b Courant Intitute of Mathematical Science, New York Univerity, New York, NY 10012, USA Received 16 Augut 2004 received in revied form 11 October 2004 accepted 10 December 2004 Available online 22 January Abtract The nonequipaced or nonuniform fat Fourier tranform (NUFFT) arie in a variety of application area, including imaging proceing and the numerical olution of partial differential equation. In it mot general form, it take a input an irregular ampling of a function and eek to compute it Fourier tranform at a nonuniform ampling of frequency location. Thi i ometime referred to a the NUFFT of type 3. Like the fat Fourier tranform, the amount of work required i of the order O(N log N), where N denote the number of ampling point in both the phyical and pectral domain. In thi hort note, we preent the eential idea underlying the algorithm in imple term. We alo illutrate it utility with application to problem in magnetic reonance imagin and heat flow. Ó 2005 Elevier Inc. All right reerved. Keyword: Nonuniform fat Fourier tranform Fourier integral Heat equation Magnetic reonance imaging 1. Introduction The nonuniform or unequipaced fat Fourier tranform (NUFFT) arie in a variety of application area, including imaging proceing and the numerical olution of partial differential equation. In it implet form, one i given an irregular ampling of a function at N data point and eek to compute the coefficient of the correponding Fourier erie. Following Dutt and Rokhlin [2], we refer to thi a a type 1 tranform. The adjoint of thi procedure i that of evaluating a given Fourier erie at a et of nonuniform target point, which we refer to a a type 2 tranform. Like the fat Fourier tranform (FFT) for equipaced data, the NUFFT reduce the cot of the computation from O(N 2 ) operation to O(N log N) operation. Thi hort note i not intended a a review article, and we refer the reader to a ampling of the relevant * Correponding author. Tel.: fax: addree: jylee@math.ewha.ac.kr, jylee@cim.nyu.edu (J.-Y. Lee) /$ - ee front matter Ó 2005 Elevier Inc. All right reerved. doi: /j.jcp

2 2 J.-Y. Lee, L. Greengard / Journal of Computational Phyic 206 (2005) 1 5 literature in the paper [1,3,4,7,8]. Additional citation can be found in our earlier paper [5], where we decribe imple and efficient implementation of thee NUFFT. Here, we conider the mot general uch tranform, where both the phyical and Fourier pace ampling are nonuniform and dicu ome of it application. More preciely, in d dimenion, we conider the computation of F k ¼ XN 1 f j e i kx j ð1þ at N location k. Thi i ometime referred to a a type 3 verion of the NUFFT. The algorithm i not new. Decription can be found, for example, in [1,2,7,8]. Neverthele, it i worth decribing the eential feature of the method in imple term. Moreover, the type 3 NUFFT ha intereting application which do not appear to be widely appreciated. 2. The nonuniform FFT of type 3 We can think of (1) a a dicretization of the continuou Fourier tranform, F ðþ ¼ 1 Z 1 Z 1 f ðxþe ix dx ð2þ ðþ d 1 1 uing nonuniformly ampled dicretization point and evaluated at nonuniformly ampled frequencie. All exiting nonuniform FFT are baed, in eence, on combining a local interpolation cheme with the tandard FFT. The type 3 NUFFT i no exception. For the ake of implicity, however, we limit the detailed dicuion to the one dimenional cae. Note firt that Eq. (1) can be interpreted a the continuou Fourier tranform of the function p f ðxþ ¼ ffiffiffiffiffi X N 1 f j dðx x j Þ ð3þ evaluated at the point = k. We obviouly cannot ample the function f(x) on a uniform meh, o we begin by convolving f(x) with the (one-dimenional) Gauian g ðxþ ¼e x2 =4. Thu, we define f (x) by f ðxþ ¼f g ðxþ ¼ 1 Z 1 ffiffiffi f ðyþg ðx yþdy: ð4þ 1 Since f i now a mooth, infinitely differentiable function, it can be well-reolved by a uniform meh in x whoe pacing i determined by the parameter. For thi, we aume that jx j j 6 X. We define the dicrete equipaced ample of f by f ðnd x Þ¼ XN 1 f j g ðnd x x j Þ: Let u now carry out a le intuitive anti-diffuion tep, whoe purpoe will become clear hortly. For thi, we define pthe (continuou) Fourier tranform of g (x) byg (). A traightforward calculation how that G ðþ ¼ ffiffiffiffiffi 2 e 2. We let ðxþ ¼ f ðxþ G r ðxþ ¼ p 1 ffiffiffiffiffi e rx2 f ðxþ: ð6þ 2r The Fourier tranform of f r, namely ðþ ¼p 1 ffiffiffiffiffi Z 1 1 ðxþe ix dx ð5þ

3 J.-Y. Lee, L. Greengard / Journal of Computational Phyic 206 (2005) can be computed with high accuracy on a uniform grid in with pacing D uing the tandard FFT on a ufficiently fine grid. That i, ðmd Þp Dx ffiffiffiffiffi X n ðnd x Þe imndxd : ð7þ Remark 1. Here, we aume that j k j 6 S. Becaue of the rapid decay of the Gauian in (5), we can ignore contribution to f (nd x ) from point x j more than a certain ditance away with an exponentially mall error. We will denote by m p the number of grid point to which we extend the influence of a point ource in each direction. Given m p, we chooe D x p 1 S R D p 1 X þm p D x R, and M r ¼ D x D, where the overampling parameter R > 1. The actual value of R and m p can be optimized once the accuracy requirement are known. The next tep i to recover the value F ( k ) by convolving ðþ with g r (): i.e., F ð k Þ¼ g r ðk Þ¼ p 1 ffiffiffiffiffi Z 1 1 ðþg r ð k Þd D ffiffiffi X m ðmd Þ g r ð k md Þ: ð8þ Thi follow from the convolution theorem and the fact that we already carried out the deconvolution tep in (6). Once the value F ( k ) are known, it remain only to correct for the initial moothing to obtain the deired value F( k ). An elementary calculation how that F ð k Þ¼p 1 ffiffiffiffiffi e 2 k F ð k Þ: ð9þ 2 Thi follow again from the convolution theorem. The actual implementation require a complete pecification of all detail. We do not repeat the analyi of [2] here, but ummarize the relevant reult a follow: if we let 1 p ffiffiffi 2 p 1 D p ffiffi X þ m p D x 2 D x p ffi D x r ffi D ffi S 2 ð 2 1Þ 2 ð 2 1Þ ð10þ then carrying out the convolution in (5) and (8) with m p = 9 yield about ix digit of accuracy. Carrying out the convolution with m p = 18 yield about twelve digit of accuracy. Efficient implementation of thee tep can be carried out uing the fat Gauian gridding algorithm of [5]. The higher dimenional verion involve more notation but are obviou extenion of the one-dimenional cheme. Appropriate value of M r, D x and D can be choen for each dimenion eparately. m p m p M r ¼ D x D Remark 2. Greater efficiency can be achieved by uing convolution kernel other than Gauian [1,3,7], but at an increaed torage cot compared to the cheme of [5]. 3. Numerical example The NUFFT of type 1, 2 and 3 have been implemented (in Fortran) with fat gridding in one, two and three dimenion. Detailed experiment were preented in [5] for the firt two type in one and two dimenion. Here, we illutrate the performance of the type 3 tranform in the context of two application.

4 4 J.-Y. Lee, L. Greengard / Journal of Computational Phyic 206 (2005) 1 5 (a) 5 Source Point (b) 40 Frequency Sampling Fig. 1. The left-hand figure how a et of ampling point for a function f(x) and the right-hand figure how a et of dicretization point in Fourier pace. Example 1. An important application of the type 3 NUFFT i to heat flow in exterior domain. We have hown [6] that the free-pace heat kernel i well-repreented in the Fourier domain uing a highly nonuniform ampling one that cluter exponentially to the origin (Fig. 1(b)). In order to project a function onto thee Fourier mode, one mut evaluate um of the form F ðk j Þ XN w n f ðx n Þe ik jx n where the point x n are the ampling point for f(x). (The tenor product dicretization hown i eay to contruct but not entirely optimal. More ophiticated quadrature following the general approach of [6] would achieve a modet reduction in the total number of node.) Suppoe for example that f(x) i a ingular heat ource concentrated on the curve depicted in Fig. 1(a). With N = 22,500 point and a pectral dicretization uing node, the type 3 NUFFT with ix digit of accuracy require 0.4 while the direct calculation require about 110. If the data were uniformly paced, the tandard FFT would require only about Example 2. A econd important application of the nonuniform FFT i to magnetic reonance imaging (MRI) [9]. Under the aumption of a perfectly homogeneou magnetic field, the MRI hardware i able to acquire the Fourier tranform of a particular tiue property at elected point in the frequency domain. The ignal produced during the readout phae at time t i given by Z ðtþ ¼ qðxþe ikðtþx dx where x =(x 1, x 2 ) i a point in the two-dimenional image plane. In other word, (t) i preciely the value of the Fourier tranform ^q at the location k(t) =(k 1 (t), k 2 (t)). In the preence of a (known or unknown) field inhomogeneity given by /(x), however, we have Z ðtþ ¼ qðxþe ikðtþx e i/ðxþt dx: ð11þ Suppoe now that one eek to model the ignal (t) from known function of pace q(x) and /(x). Thi require the computation of

5 J.-Y. Lee, L. Greengard / Journal of Computational Phyic 206 (2005) ðt j Þ XN w n qðx n Þe ikðtjþxn e i/ðxnþtj ¼ XN w n qðx n Þe ikjxn where K j =(k 1 (t j ), k 2 (t j ), t j ) and X n ¼ðx 1 n x2 n /ðx nþþ. Thu, by embedding the data point in a higher dimenional pace, one can carry out the tranformation uing a type 3 NUFFT. We cite one timing reult, for a cae where the (x 1, x 2 ) variable are dicretized uing a regular meh on the unit box, and k-pace i travered by a piral trajectory imilar to that hown in Fig. 1(a) up to a maximum frequency of 60. The time interval of the readout phae wa caled i to [0,1], and /(x) wa allowed to vary in the interval [ 5p,5p]. Under thee condition, direct evaluation required about 140, while the NUFFT required 3.8 to obtain ix digit of accuracy. 4. Concluion We have preented a imple verion of the type 3 nonuniform FFT. It can be ued to approximate the continuou Fourier tranform when neither the patial nor the Fourier domain pacing i regular. Further, it allow for the evaluation of more general integral operator uch a (11) by embedding them in a higher dimenional pace. Acknowledgement Thi work wa upported by the Applied Mathematical Science Program of the US Department of Energy under Contract DEFGO200ER Reference [1] G. Beylkin, On the fat Fourier tranform of function with ingularitie, Appl. Comput. Harmonic Anal. 2 (1995) [2] A. Dutt, V. Rokhlin, Fat Fourier tramform for nonequipaced data, SIAM J. Sci. Comput. 14 (1993) [3] J.A. Feler, B.P. Sutton, Nonuniform Fat Fourier tranform uing min max interpolation, IEEE Tran. Signal Proc. 51 (2003) [4] K. Fourmont, Non-equipaced Fat Fourier tranform with application to tomography, J. Fourier Anal. Appl. 9 (2003) [5] L. Greengard, J.-Y. Lee, Accelerating the nonuniform fat fourier tranform, SIAM Rev. 46 (2004) [6] L. Greengard, P. Lin, Spectral approximation of the free-pace heat kernel, Appl. Comput. Harmonic Anal. 9 (2000) [7] A. Nielony, G. Steidl, Approximate factorization of Fourier matrice with nonequipaced knot, Lin. Alg. Appl. 366 (2003) [8] D. Pott, G. Steidl, M. Tache, Fat Fourier Tranform for Nonequipaced Data: A Tutorial, Modern Sampling Theory: Mathematic and Application, in: J.J. Benedetto, P. Ferreira (Ed.), Birkhauer Boton, Applied and Numerical Harmonic Analyi Serie, 2001, Chapter 12, pp [9] B.P. Sutton, D.C. Noll, J.A. Feler, Fat iterative recontruction for MRI in the preence of field inhomogeneitie, IEEE Tran. Med. Imag 22 (2003)