1
Fast Iterative Reconstruction for MRI from Nonuniform k-space Data Tobias Knopp, Stefan Kunis and Daniel Potts
Abstract— In magnetic resonance imaging (MRI), methods that use a non-Cartesian (e.g. spiral) sampling grid in k-space are becoming increasingly important. Reconstruction is usually performed by resampling the data onto a Cartesian grid. In this paper we compare the standard approach (gridding) and an approach based on an implicit discretisation. We show that both methods can be solved efficiently with the fast Fourier transform for nonequispaced knots.
be handled by our NFFT software library. Section V presents numerical examples for a 3d-Shepp-Logan phantom and for real life data emphasising the role of certain density compensation weights for different sampling schemes. We conclude by a discussion of our results in Section VI.
EDICS Categorie BMI-MRIM (Biomedical Imaging, Magnetic resonance imaging)
Given a trajectory k(t), the relation between the MR signal sMR during the readout and the object p can be modelled by Z sMR (t) = p (r) c (r, t) e2πirk(t) dr , (II.1)
Index Terms— 2d/3d image reconstruction, iterative methods, magnetic resonance imaging, Fourier transform, FFT, NFFT, gridding, spiral scan.
II. T HEORY
R3
I. I NTRODUCTION In magnetic resonance imaging (MRI) the raw data is measured in k-space, the domain of spatial frequencies. NonCartesian sampling schemes have received much attention because of its fast acquisition, efficient use of the hardware and low motion and flow sensitivity. In contrast, sampling the data on a Cartesian grid allows the use of the computationally efficient fast Fourier transform (FFT) to reconstruct the image. In the case of non-Cartesian sampling trajectories, e.g., spiral scans, radial scans or other acquisitions schemes, one can apply the fast Fourier transform for nonequispaced knots (NFFT). Iterative image reconstruction algorithms play an important role in modern tomographic systems [1]. Recently, the iterative image reconstruction for MRI in combination with the nonuniform fast Fourier transform has been applied to data on spiral k-space trajectories [2] and in the presence of field inhomogeneities [3]. In this paper we compare the gridding approach and an approach based on an implicit discretisation [4], both to be solved in a simple fashion by our NFFT software [5]. In fact, it turns out that the iterative solution of the latter approach resembles the gridding method in its first iteration. Furthermore, we compare different weights arising from the discretisation of the underlying integrals. We present numerical results, based on the Shepp-Logan phantom as well as on real life data. The outline of this paper is as follows: A brief introduction to the theory of Fourier transform image reconstruction is given in Section II. Hereafter, Section III reviews the algorithms developed for the fast Fourier transform for nonequispaced knots. In Section IV we unify the two considered Fourier approaches in MRI and show that they can efficiently University of L¨ubeck Institute of Mathematics, D–23560 L¨ubeck, Germany
[email protected] [email protected] [email protected]
where c(r, t) mimics the known sensitivity of the receiver coil and field inhomogeneities, see [3] for a low rank approximation of these effects. For simplicity we focus on the simplified signal equation Z s (k) = p (r) e2πirk dr . (II.2) R3
In the following we describe two different approaches [4]. For convenience let the available samples in k-space be contained in the shifted unit cube, i.e. k ∈ [− 21 , 12 )3 , and the field of view be restricted to ΩN ⊂ [− N21 , N21 )×[− N22 , N22 )× [− N23 , N23 ), where N = (N1 , N2 , N3 )> ∈ 2N3 . Then, the discretisation of integral (II.2) on equispaced points leads to X s (k) ≈ s˜ (k) := p (r) e2πirk , (II.3) 3 r∈IN
3 where IN := {− N21 , . . . , N21 − 1} × {− N22 , . . . , N22 − 1} × N3 {− 2 , . . . , N23 − 1}. Thus, the unknown object p is given implicit by (II.3). The authors of [4] call this inverse model. A second possible discretisation uses the Fourier inversion theorem first, i.e., Z p (r) = s (k) e−2πirk dk . (II.4) R3
The discretisation of the integral (II.4) leads to p (r) ≈ p˜ (r) :=
M −1 X
s (kj ) e−2πirkj wj ,
(II.5)
j=0
where wj are weights, which compensate different sampling densities. Here, the unknown object p ≈ p˜ can be computed explicit. We use different methods in order to compute the sampling density compensation values wj . It was pointed out, see [6], that the use of the area respectively the volume of the Voronoi cell around each sample, as a measure of the local
2
sampling density is a very reliable and general technique for the estimation of the weights. The important difference between (II.3) and (II.5) is that the former is discretised in the image domain with pixels on a uniform grid and hence with unit weighting coefficients and the later is an integral discretised in the k-space domain with non-uniform samples and specific weights. III. NFFT AND iNFFT If the data are sampled on a Cartesian grid in k-space the reconstruction can be done with the FFT. For non-Cartesian trajectories in k-space the reconstruction can be done via (II.3) or (II.5), where the FFT has to be replaced by the NFFT. In the following we briefly describe the NFFT, which has, for a fixed accuracy, the same arithmetical complexity as the FFT. One can use the software package, which is available from the NFFT homepage [5]. The first problem to be addressed here, can be regarded as a matrix vector multiplication. For a finite number of given d Fourier coefficients fˆr ∈ C, r ∈ IN := {− N21 , . . . , N21 − Nd N2 N2 1} × {− 2 , . . . , 2 − 1} × · · · × {− 2 , . . . , N2d − 1} with N := (N1 , . . . , Nd )> one wants to evaluate the multivariate trigonometric polynomial X f (x) := fˆr e2πirx (III.1) d r∈IN
at given non equispaced knots xj ∈ [− 12 , 12 )d , 0, . . . , M − 1. Thus, our concern is the evaluation of X fj = fˆr e2πirxj , j = 0, . . . , M − 1.
j =
In matrix vector notation this reads as (III.2)
where A := e2πirxj
d j=0,...,M −1; r∈IN
f := (fj )j=0,...,M −1 ,
fˆ := fˆr
,
d r∈IN
fˆr :=
M −1 X
fj e−2πirxj ,
(III.3)
j=0
d r∈IN
f = Afˆ
[17], based on minimising the Frobenius norm of certain error matrices [18], or based on min-max interpolation [10] are proposed. However, the numerical results in [10], [18] show that these approaches are not superior to the approach based on the Kaiser-Bessel window function, see also [13]. The relation between speed and accuracy of the algorithm was initially investigated in [11], [12]. In [7] a unified approach was suggested and the error estimates from [11] were improved, which led to criteria for the choice of the parameters of the algorithm. Our algorithms are based on the approach in [8] where one can change the window function ϕ in a very simple way. Note, that i) For a fixed oversampling factor σ > 1 the approximation error introduced by the NFFT decays exponentially with the window-width m. ii) The arithmetical complexity of the NFFT is O((σN )d log N + md M ). Beyond these estimates, we have shown that our software package [5] computes accurate results in a fast fashion for small input sizes, already. Furthermore, the library adapts over a wide class of available memory through so called flags, see e.g. the Numerical Example 4. The manual [9] and the appendix in [19] present details for a suitable choice of the window function, the window-width, and the oversampling factor with respect to accuracy, speed and memory usage. The adjoint algorithm, i.e. the evaluation of the sums
.
The fast approximate matrix vector multiplication with A and A`a , is know as NFFT. A unified approach to the efficient computations of (III.1) was suggested in [7], [8]. In matrixvector notation our approach to a fast evaluation of (III.1) can be written as A ≈ BF D with a diagonal matrix D = diag(ϕ(r)), ˆ an oversampled Fourier matrix F , and a sparse matrix B with nonzero entries bj,l = ψ(xj − l/(σN )), where ψ approximates the window function ϕ and σ > 1 denotes the oversampling factor. Details concerning the NFFT can be found for example in [7]–[9]. Common names for these algorithms are non-uniform fast Fourier transform [10], generalised fast Fourier transform [11], unequally-spaced fast Fourier transform [12], non-equispaced fast Fourier transform [13]. In various papers, different window functions ϕ for the NFFT were considered, e.g. a Gaussian pulse tapered with a Hanning window in [14], Gaussian kernels combined with sinc kernels in [15], and special optimised windows in [14], [16]. Furthermore, special approaches based on scaling vectors
can be computed by the matrix vector multiplication with A`a , where we obtain a fast algorithm by using A`a ≈ D > F `a B > . It was already pointed out in [8], [20], that the gridding method is simply a fast algorithm for the multiplication of the matrix A`a with a vector. Including a sampling density compensation yields the following gridding algorithm, see also [15], [16], [21]: i) sampling density compensation, i.e., multiplication with a diagonal matrix W , ii) approximation to an oversampled Cartesian grid, i.e., multiplication with the matrix B > , iii) inverse fast Fourier transform, i.e., mult. with F `a , iv) roll-off correction, i.e., multiplication with D > . Note, that the role of the window function, appearing in the matrices D and B, and the sampling density compensation, appearing in W , should not be mixed up. We remark further that the NFFT and its adjoint NFFT`a can be realized with a broad class of suggested window functions yielding the same accuracy by changing the truncation parameter m. Thus, the obtained reconstruction quality in MRI is fairly independent of the particularly chosen window function. In our numerical test we use the Kaiser-Bessel window since it provides the best accuracy for a fixed truncation parameter m, i.e., sparseness of the matrix B. This truncation parameter is chosen as m = 6 and the used oversampling factor fulfils σ ≥ 1.5 in all tests. We stress again, that the NFFT and the NFFT`a are algorithms that realise the matrix vector multiplications in a
3
fast and efficient way. As a result one may compare our adjoint NFFT`a in [5] with other well-implemented gridding algorithms. The aim of the inverse NFFT (iNFFT) is to construct a d-variate trigonometric polynomial f , respectively its Fourier d coefficients fˆr , r ∈ IN , such that f (xj ) approximates given values yj ∈ C, i.e., X yj ≈ f (xj ) = fˆr e2πirxj , j = 0, . . . , M − 1. (III.4) d r∈IN
In matrix vector notation this reads as Afˆ ≈ y. We propose to solve this system of linear equations by weighted least squares with sampling density compensating weights wj > 0, i.e., M −1
2 X
2 fˆ wj |yj − f (xj )| → min,
y − Afˆ = W
(III.5)
j=0
where W := diag(wj )j=0,...,M −1 . This problem is equivalent to the weighted normal equation of first kind A`a W Afˆ = A`a W y.
(III.6)
The iterative solution of problem (III.6) is addressed in [22], [23] by solving the explicit formed Toeplitz system, whereas the iNFFT solves the problem by a factorised variant of conjugated gradients (CGNR, N for ’Normal equation’ and R for ‘Residual minimisation’). Both variants generate the same sequence of approximations in exact arithmetic, but the CGNR approach is considered to be more stable with respect to round-off errors, cf. [24, Sec. 7.1]. In summary, we suggest the following algorithm. Algorithm 3.1: (CGNR) Input M ∈ N number of given samples N ∈N total number of unknowns (xj )j=0,...,M −1 sampling knots W ∈ RM ×M diagonal matrix with weights y ∈ CM given sampled values N ˆ f0 ∈ C initial guess r 0 = y − Afˆ0 pˆ0 = zˆ0 = A`a W r 0 for l = 0, 1, . . . v l = Apˆl αl = zˆ`la zˆl / v `la W v l fˆl+1 = fˆl + αl pˆl r l+1 = r l − αl v l zˆl+1 = A`a W r l+1 a βl = zˆ`l+1 zˆl+1 / zˆ`la zˆl pˆl+1 = zˆl+1 + βl pˆl end for Output: approximate solution fˆl Similar methods can be derived in the case of an underdetermined reconstruction problem, yielding a general iNFFT, see [25] for details. IV. R ECONSTRUCTION M ETHODS From the mathematical point of view, the reconstruction of MR data acquired on non-Cartesian trajectories in k-space is
straightforward by using the NFFT. We compute the solution of problem (II.5) by the NFFT and the solution of problem (II.3) by solving a system of linear equations, i.e., apply the iNFFT. To be more precise we restate the problems (II.5) and (II.3) in matrix vector notation: We denote the vector of the given values by s := (s(kj ))j=0,...,M −1 ∈ CM , the reshaped vector of the unknown 3 object by p := (p(r))r∈IN ∈ CN1 ×N2 ×N3 and the nonuniform Fourier matrix again by A. We state problem (II.3) as the weighted approximation problem p ks − s˜kW → min, (IV.1) where s˜ = Ap, see (II.3), and the density compensation weights are collected in the diagonal matrix W = diag(wj )j=0,...,M −1 . In contrast to [4], we include density compensation weights also for the implicit discretisation. This is due to the fact that the weighted problem is better conditioned, see [23]. Furthermore, the minimisation of (IV.1) is more natural with respect to the ’continuous residual’ in k-space. We would like to emphasise that the CGNR method minimises (IV.1) in each iteration over a certain Krylov space. Thus, iterate for only one step, we obtain the ’optimised’ gridding solution. More formally, let p˜ = A`a W s denote the gridding solution, see (II.5), and p0 = 0 be the initial guess in the CGNR-method. Then the first iterate fulfils 2
p1 =
kpk ˜ 2 2
kApk ˜ W
p˜ = arg min ks − ApkW , p=αp ˜
(IV.2)
since for its residual r 1 = s − Ap1 holds the perpendicular proposition 2 `a kpk ˜ 2 ` a r `1a W Ap˜ = A`a W s p˜ − ˜ W Ap˜ = 0. 2 (Ap) kApk ˜ W In other words, the gridding solution p˜ is scaled such that its residual is minimised. Hence, we do not consider a separate gridding approach furthermore. V. N UMERICAL RESULTS All algorithms were implemented in Matlab&C and tested on a AMD Athlon(tm) XP 2700+, 2GB memory, SuSeLinux, kernel 2.4.20-4GB-athlon, FFTW3.0.1, and NFFT2. We are concerned with the application to real life data, the reconstruction quality, and the usage of time and memory resources. The reconstruction algorithm was tested with MR measurements of a physical phantom by the Philips Achieva 1.5T device. Here, the sampling scheme consists of 36 equidistant planes with 48 interleaved spiral arms with 1625 data points each, i.e., 2.808.000 k-space samples in total. Our reconstruction with Voronoi weights is shown in Figure V.1 after one iteration. Furthermore, we use two 3d-Shepp-Logan phantoms, cf. Figure V.2, of size 256 × 256 × 36 and of size 128 × 128 × 128 for comparison of our results with respect to the number of iterations, sampling schemes, and density compensation weights. The simulated MR data are computed by a 3d-NFFT using the following k-space trajectories. Note, that all but the
4 0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5 −0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Fig. V.1. A subset of the k-Space trajectory, here with only 3 interleaved spiral arms with 25 knots each (left) and two slices of the reconstruction with Voronoi weights after one iteration (middle and right). The measurements of the k-space data was done by the Philips Achieva 1.5T device.
Fig. V.2.
Slice plot of our 3d-Shepp-Logan phantom
first trajectory are of a special 2d ⊗ 1d type, i.e., they consist of a stack of 36 equidistant planes where the M1 nonuniform samples within each are distributed accordingly. In these cases, the first two coordinates of a k-space knot k = (k1 , k2 , k3 )> ˜ = (k1 , k2 )> ∈ [− 1 , 1 )2 . form the 2d-knot k 2 2 i) (3d-RADIAL) The only 3d-k-space trajectory, which is not of 2d ⊗ 1d type, is given by r+1 > kp,q,r = √ (cos φp sin θq , sin φp sin θq , cos θq ) 2R π(2q+1) where φp = 2πp , P , p = 0, . . . , P − 1, θq = 2Q q = 0, . . . , Q − 1, and r = 0, . . . , R − 1. Furthermore, we restrict this set to the unit cube (− 12 , 12 )3 . Choosing P = Q = R = 160 in our experiments yields a total number of M = 3.398.033 knots in k-space. ii) (SPIRAL) This 2d-k-space trajectory is given by one Archimedean spiral, i.e., √ ˜j = √ j (cos ωj , sin ωj )> , k 2 M1 √ where ωj = 8π j, and j = 0 . . . M1 − 1. We have 5 chosen M1 = 65536 yielding a total number of M = 2.359.296 knots in k-space.
iii) (RADIAL) Also very popular within computer tomography is the radial trajectory > ˜p,r = (−1)r r − 0.5 cos πp , sin πp k , R P P with p = 0 . . . P − 1, r = 0 . . . R − 1, and P R = M1 . We set P = 410 and R = 512 yielding a total number of M = 7.557.120 knots in k-space. iv) (INTERLEAVED SPIRAL) The last k-space trajectory, having 2.808.000 knots in total, from the Philips Achieva 1.5T device, consists of 48 interleaved spiral arms with 1625 knots each. Examples of the first three trajectories are shown in Figure V.3, an example of the fourth is shown in Figure V.1 (left). In the following different weights W in (IV.1) are used to take into account the local sampling density in k-space. We propose formulations i) with no weights, i.e. W = I, ii) with analytic weights in the case of the 3d-RADIAL trajectory, given by wp,q,r =
π (2q + 1) r+1 sin , R 2Q
iii) with approximate weights obtained by counting the number of sample knot in a regular partition of size 256 × 256 in the k-space, and iv) with weights obtained as the area of the Voronoi cell around each sample knot, see also [6], [26]. Numerical Example 1: First of all, we compare the reconstruction quality with respect to different k-space trajectories and sampling density compensation weights. Using the 2d ⊗ 1d type of the trajectories, we reconstruct each slice separately and compute the overall reconstruction by regular 1d-FFTs of size N3 = 36 within the third component. Table V.1 shows the signal-to-noise-ratio SNR(p, ˜ p) := 20 log10 (kpk2 / kp − pk ˜ 2 ) where p˜ is our reconstruction and p denotes the original image of size 128 × 128 × 128 for the 3d-RADIAL trajectory and of size 256 × 256 × 36 for all others. We stress again, that after one iteration we obtain an optimal gridding solution, cf. (IV.2). In our second and third test, we present some reconstructions by our algorithms.
5
Fig. V.3.
k-Space trajectories, 3d-RADIAL for R=3, P=Q=32 (left), SPIRAL for M1 = 1024 (middle), and RADIAL for P = R = 32 (right).
weights \ iter.
1
2
5
10
good results in Table V.1 and in Figure V.6, V.8, and V.10. However, using no weights gives reasonable results only if the trajectory covers the k-space uniform, cf. Figure V.6, V.8, and V.10 (left) and Table V.1 ’weights: none’. Including approximate weights improves the reconstruction quality but causes an artefact that the intensity varies also in regions where it should be constant, e.g., outside the object as seen in V.6 and V.10 (middle). The best gridding results are obtained using the Voronoi weights, these results are improved substantially by more iterations in the case of the SPIRAL trajectory, cf. Table V.1 ’weights: Voronoi’. Note the same behaviour, i.e., a substantial improvement in the reconstruction quality for all used trajectories if the weights are only approximated or neglected at all. Furthermore, only using the iterative method with weights gives a reasonable reconstruction for the 3dRADIAL, cf. Figures V.4 and V.5.
3d-RADIAL none
1.63
2.51
3.76
5.18
analytic
3.12
4.95
10.32
13.37
SPIRAL none
15.60
20.84
22.28
22.30
approximation
15.46
21.26
22.28
22.28
Voronoi
17.32
21.80
22.14
22.16
RADIAL none
2.56
4.98
9.98
18.34
approximation
19.36
22.24
22.30
22.34
Voronoi
22.20
22.20
22.24
22.28
CPU-time in seconds N \#
1
INTERLEAVED SPIRAL none
2.36
5.20
11.80
21.88
approximation
11.88
21.98
22.22
22.22
Voronoi
22.02
22.20
22.20
22.20
FOR DIFFERENT WEIGHTS AND K - SPACE TRAJECTORIES AFTER
5,
AND
5
MByte 10
2d ⊗ 1d trajectory
TABLE V.1 SNR
2
512
118.95
178.04
355.08
650.55
691
256
29.24
43.55
87.21
159.71
172
128
6.56
9.84
19.73
35.78
43
64
1.45
2.16
4.31
8.00
10
1, 2,
3d trajectory
10 ITERATIONS .
Numerical Example 2: Our aim with the 3d-RADIAL example is to demonstrate, that the NFFT-based reconstruction is straightforward. Figure V.4 and Figure V.5 show the 64th slice and the profile of the 64th row of this slice after one, five and ten iterations for no weights and analytic weights, respectively. Numerical Example 3: Furthermore, we show the 18th slice and the profile of the 128th row of this slice after one and ten iteration for different weights and the 2d ⊗ 1d k-space trajectories in Figure V.6 – Figure V.11, respectively. We see that already the gridding method may lead to very
512
11422
17114
34189
62650
1952
256
1865.9
2800.4
5603.3
10274
1244
128
452.31
679.28
1344.9
2454.4
311
64
89.24
133.72
267.18
489.59
77
TABLE V.2 CPU- TIME AND MEMORY USAGE FOR DIFFERENT ITERATION NUMBERS # AND SIZES N OF THE K - SPACE DATA AND THE RECONSTRUCTED PHANTOM .
Numerical Example 4: In the last test, we give CPU-time and used memory of our algorithms with respect to the size of
6
Fig. V.4. 64th slice of the phantom (top) and the profile of the 64th row of this slice (bottom) with 3d-RADIAL data and without weights: from left to right after one, five and ten iterations.
Fig. V.5. 64th slice of the phantom (top) and the profile of the 64th row of this slice (bottom) with 3d-RADIAL data and analytic weights: from left to right after one, five and ten iterations.
the reconstruction problem N , i.e., the size of the reconstructed image is N ×N ×36 and the total number of SPIRAL k-space samples is 36N 2 . We compare the algorithm exploiting the 2d ⊗ 1d type of the trajectory with the full 3d-iNFFT, where both algorithms produce almost the same reconstructions. The
optimisation of the computing time is done by different NFFTflags which affect the storage of the precomputed values for the third step within the NFFT, i.e., the left factor B, see Section III. We were able to store all nonzero entries of the matrix B, flag PRE FULL PSI, in the 2d ⊗ 1d tests, a
7
Fig. V.6. 18th slice of the phantom (top) and the profile of the 128th row of this slice (bottom) after one iteration with SPIRAL data: from left to right, without weights, with approximative weights and with Voronoi weights.
Fig. V.7. 18th slice of the phantom (top) and the profile of the 128th row of this slice (bottom) after ten iterations with SPIRAL data: from left to right, without weights, with approximative weights and with Voronoi weights.
lossless compressed form, flag PRE PSI, in the 3d-tests up to N = 256, and a lossy compressed, trajectory independent form, PRE LIN PSI, in the 3d-tests for N = 512, see [5] for details. Table V.2 shows our results for the Shepp-Logan phantom, the SPIRAL trajectory, and Voronoi weights. The
measured CPU-times grow as expected like N 2 log N . Clearly, the method based on the 2d × 1d model is faster, however our algorithms can handle arbitrary scattered data in d dimensions.
8
Fig. V.8. 18th slice of the phantom (top) and the profile of the 128th row of this slice (bottom) after one iteration with RADIAL data: from left to right, without weights, with approximative weights and with Voronoi weights.
Fig. V.9. 18th slice of the phantom (top) and the profile of the 128th row of this slice (bottom) after ten iterations with RADIAL data: from left to right, without weights, with approximative weights and with Voronoi weights.
VI. D ISCUSSION Magnetic resonance signals are measured in the Fourier domain, also called k-space. Samples of the MR signal lie along k-space trajectories determined by the magnetic field gradients. We have shown that the reconstruction from non-
Cartesian grids can be computed by the NFFT, comparing the explicit discretisation, i.e. gridding, and an implicit discretisation where we have solved a system of linear equations. It has been shown, that one step of the inverse NFFT is nothing else but the well known gridding method, cf. Equation IV.2.
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Fig. V.10. 18th slice of the phantom (top) and the profile of the 128th row of this slice (bottom) after one iteration with INTERLEAVED SPIRAL data: from left to right, without weights, with approximative weights and with Voronoi weights.
Fig. V.11. 18th slice of the phantom (top) and the profile of the 128th row of this slice (bottom) after ten iterations with INTERLEAVED SPIRAL data: from left to right, without weights, with approximative weights and with Voronoi weights.
A major complication in non-Cartesian sampling schemes is the compensation of the non-uniform sampling density. The implicit discretisation and its iterative solution generalises the gridding approach and improves the image quality substantially. Especially, the gridding approach fails if the sampling
geometry is highly non-uniform or the density compensation weights are only approximated. In contrast, iterate for only a few steps and approximate the density compensation in a cheap way leads to very good results. Note further, that an iterative scheme without weights may fail. Finally, we have
10
compared the CPU-times and memory usage of our algorithms for the 2d ⊗ 1d model and a full 3d-model. Acknowledgement. The second author is grateful for partial support of this work by the German Academic Exchange Service (DAAD) and the warm hospitality during his stay at the Numerical Harmonic Analysis Group, University of Vienna. We would like to thank the referees for their valuable suggestions. Especially, we would like to thank Michael Schacht Hansen, Steffen Ringgaard (MR Research Centre at the University of Aarhus, Denmark), and Holger Eggers (Philips Research, Sector Technical Systems, Hamburg, Germany) for numerous fruitful discussions and for providing datasets. R EFERENCES [1] S. Matej, J. Fessler, and I. Kazantsev, “Iterative tomographic image reconstruction using Fourier-based forward and back- projectors,” IEEE Trans. Med. Imag., vol. 23, pp. 401 – 412, 2004. [2] B. Sutton, J. Fessler, and D. Noll, “A min-max approach to the nonuniform N -dimensional FFT for rapid iterative reconstruction of MR images,” in Proc. ISMRM 9th Scientific Meeting, 2001, p. 763. [3] B. Sutton, D. Noll, and J. Fessler, “Fast, iterative, field-corrected image reconstruction for MRI in the presence of field inhomogeneities,” IEEE Trans. Med. Imag., vol. 22, pp. 178 – 188, 2003. [4] B. Desplanques, D. Cornelis, E. Achten, R. van de Walle, and I. Lemahieu, “Iterative reconstruction of magnetic resonance images from arbitrary samples in k-space,” IEEE Transactions on Nuclear Science, vol. 49, pp. 2268 – 2273, 2002. [5] S. Kunis and D. Potts, “NFFT, Softwarepackage, C subroutine library,” http://www.math.uni-luebeck.de/potts/nfft, 2002 – 2004. [6] V. Rasche, R. Proksa, R. Sinkus, P. B¨ornert, and H. Eggers, “Resampling of data between arbitrary grids using convolution interpolation,” IEEE Trans. Med. Imag., vol. 18, pp. 385 – 392, 1999. [7] G. Steidl, “A note on fast Fourier transforms for nonequispaced grids,” Adv. Comput. Math., vol. 9, pp. 337 – 353, 1998. [8] D. Potts, G. Steidl, and M. Tasche, “Fast Fourier transforms for nonequispaced data: A tutorial,” in Modern Sampling Theory: Mathematics and Applications, J. J. Benedetto and P. J. S. G. Ferreira, Eds. Boston: Birkh¨auser, 2001, pp. 247 – 270. [9] S. Kunis and D. Potts, “NFFT2.0,” Preprint B-04-07, Universit¨at zu L¨ubeck, http://www.math.uni-luebeck.de/potts/nfft, 2004. [10] J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process., vol. 51, pp. 560 – 574, 2003. [11] A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Stat. Comput., vol. 14, pp. 1368 – 1393, 1993. [12] G. Beylkin, “On the fast Fourier transform of functions with singularities,” Appl. Comput. Harmon. Anal., vol. 2, pp. 363 – 381, 1995. [13] K. Fourmont, “Non equispaced fast Fourier transforms with applications to tomography,” J. Fourier Anal. Appl., vol. 9, pp. 431 – 450, 2003. [14] A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophysics, vol. 64, pp. 539 – 551, 1999. [15] J. Pelt, “Fast computation of trigonometric sums with applications to frequency analysis of astronomical data,” in Astronomical Time Series, D. Maoz, A. Sternberg, and E. Leibowitz, Eds., Kluwer, 1997, pp. 179 – 182. [16] J. I. Jackson, “Selection of a convolution function for Fourier inversion using gridding,” IEEE Trans. Med. Imag., vol. 10, pp. 473 – 478, 1991. [17] N. Nguyen and Q. H. Liu, “The regular Fourier matrices and nonuniform fast Fourier transforms,” SIAM J. Sci. Comput., vol. 21, pp. 283 – 293, 1999. [18] A. Nieslony and G. Steidl, “Approximate factorizations of Fourier matrices with nonequispaced knots,” Linear Algebra Appl., vol. 266, pp. 337 – 351, 2003. [19] D. Potts and G. Steidl, “Fast summation at nonequispaced knots by NFFTs,” SIAM J. Sci. Comput., vol. 24, pp. 2013 – 2037, 2003. [20] G. E. Sarty, R. Bennett, and R. W. Cox, “Direct reconstruction of noncartesian k-space data using a nonuniform fast Fourier transform,” Magn. Reson. Med., vol. 45, pp. 908 – 915, 2001. [21] R. A. Scramek and F. R. Schwab, “Imaging,” in Astronomical Society of the Pacific Conference, R. Perley, F. R. Schwab, and A. Bridle, Eds., vol. 6, 1988, pp. 117 – 138.
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