Non-Uniform Fast Fourier Transformation (NUFFT) and Magnetic Resonace Imaging 2008 년 11 월 27 일. Lee, June-Yub (Ewha Womans University)

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1 Non-Uniform Fast Fourier Transformation (NUFFT) and Magnetic Resonace Imaging 2008 년 11 월 27 일 Lee, June-Yub (Ewha Womans University)

2 1 NUFFT and MRI Nonuniform Fast Fourier Transformation (NUFFT) Basics of MRI Comments on Fourier Inversion 2 The NUFFT of Type-1,2 The Type-1 and Type-2 Nonuniform FFT Fast Gaussian gridding method Numerical Results of Type-1,2 NUFFT 3 The NUFFT of Type-3 The non-uniform FFT of Type-3 Numerical examples of Type-3 NUFFT 4 The Fast Sinc-Transform The Fast Sinc-Transform using NUFFT An example of fast Sinc-transformation 5 Imaging Modality for MRI Basic approaches for Fourier Inversion 6 Conclusions

3 Nonuniform Fast Fourier Transformation (NUFFT) DFT and Type 1-3 NUFFT 0. DFT (Discrete Fourier transforation) : O(N 2 ) FFT : O(NlogN) F (k) = 1 N N 1 j=0 f (j) e 2πijk/N, f (j) = k F (k) e2πijk/n (1) F (k) = 1 2π 2π 0 f (x) e ikx j dx, f (j) = k F (k) e ikx j with x j = 2πj N (2)

4 Nonuniform Fast Fourier Transformation (NUFFT) DFT and Type 1-3 NUFFT 0. DFT (Discrete Fourier transforation) : O(N 2 ) FFT : O(NlogN) F (k) = 1 N F (k) = 1 2π N 1 j=0 f (j) e 2πijk/N, f (j) = k F (k) e2πijk/n (1) 2π 0 f (x) e ikx j dx, f (j) = k F (k) e ikx j 1. Type1-NUFFT in 2D : O(NlogN) instead of O(N 2 ) F (k 1, k 2 ) = 1 N with x j = 2πj N (2) N 1 j=0 f j e i(k1,k2) x j, (3) with f j = f (x j ) w j can be considered as a discretization of F (k 1, k 2 ) = 1 2π 2π (2π) f (x) e i(k1,k2) x dx (4) with the quadrature weight w j corresponding to x j.

5 Nonuniform Fast Fourier Transformation (NUFFT) DFT and Type 1-3 NUFFT II 2. Type2-NUFFT in 2D : O(NlogN) instead of O(N 2 ) f (x j ) = F (k 1, k 2 ) e i(k1,k2) xj, (5) k 1 k 2 is simply the evaluation of a finite Fourier series at an arbitrary set of targets.

6 Nonuniform Fast Fourier Transformation (NUFFT) DFT and Type 1-3 NUFFT II 2. Type2-NUFFT in 2D : O(NlogN) instead of O(N 2 ) f (x j ) = F (k 1, k 2 ) e i(k1,k2) xj, (5) k 1 k 2 is simply the evaluation of a finite Fourier series at an arbitrary set of targets. 3. Type3-NUFFT in R d : O(NlogN) instead of O(N 2 ) F k = N 1 j=0 f j e ±is k x j, (6) is a discretization of the continuous Fourier transform, F (s) = 1 (2π) d... f (x) e ±is x dx (7) at nonuniformly sampled frequencies using nonuniformly sampled points.

7 Basics of MRI Structure of Magnetic Resonance Imaging Machine Without Magnetic Field With Magnetic Field H o Main Magnet Coil Shim Coil Gradient coil (Body Coil) Random orientation S N Aligned Spinning tops Head Coil Signal from everywhere M( r, t) = M 0 ρ( r)e iωt, ω = γ H S(t) = M( r, t)d r

8 Basics of MRI Bloch Equation and Signal localization Signal from everywhere M( r, t) = M 0 ρ( r)e iφ(t, r), φ(t, r) = ωt = γ H( r) t S(t) = M( r, t)d r

9 Basics of MRI Bloch Equation and Signal localization Signal from everywhere M( r, t) = M 0 ρ( r)e iφ(t, r), φ(t, r) = ωt = γ H( r) t S(t) = M( r, t)d r MRI using FT : Echo-Planar Imaging(EPI) by Mansfield in 1977 S(t) = M 0 ρ( r)e iφ(t, r) d r, under H(t) = H 0 + G(t) r, φ(t, r) = H 0 t + t 0 (G x x + G y y)dt Let M 0 = M 0 e iω0t = M 0 ρ( r z0 )e iγ t 0 (Gx x+gy y)dt dxdy, Let k(t) = (G x, G y ) dt 0 ( ) = M 0 ρ( r z0 )e iγ k(t) (x,y) dxdy = M 0 FT [ρ( r z0 )] k(t) t

10 Basics of MRI Image Reconstruction Method Real part Imaginary part Frequency K-space Inverse Fourier Transformation Physical XY-space Magnitude Image Phase Image

11 Basics of MRI Classical sampling on uniform grid Classical Uniform Sampling ky Inverse Fourier Transformation In most clinical system, Fourier data ( ) ρ(kq, 1 kq) 2 := FT [ρ( r z0 )] k(tq ) kx are acquired on a uniform cartesian grid (k 1 m 1, k 2 m 2 ). Then the reconstructed image ˆρ(x 1 n 1, x 2 n 2 ) can be computed using backward Fast Fourier Transformation (FFT), C M 1 M 1 m 1=0 m 2=0 e 2πi m 1 n 1 M e 2πi m 2 n 2 M ρ(k 1 m1, k 2 m 2 )

12 Basics of MRI Fast scanning methods Polar Sampling Spiral Sampling Rosette Sampling ky ky ky kx kx kx Advantages and Disadvantages Pros Fast Scanning Time Cons Computationally. Slow reconstruction (FFT is not applicable) Mathematically. Inversion is ill-conditioned (FT is not unitary)

13 Comments on Fourier Inversion Comments on Fourier integral(fourier Inversion) The Fourier integral H(u 1, u 2 ) = h(v) e i(u1,u2) v dv (8) with h(v) at a scatter of points rather than on a regular Cartesian mesh.

14 Comments on Fourier Inversion Comments on Fourier integral(fourier Inversion) The Fourier integral H(u 1, u 2 ) = h(v) e i(u1,u2) v dv (8) with h(v) at a scatter of points rather than on a regular Cartesian mesh. There are three separate issues involved: 1 aquisition of data h(v j ) in the domain, 2 the choice of a quadrature scheme {v j, w j }, and 3 the availability of a fast algorithm for computing the discrete transform itself.

15 Comments on Fourier Inversion Comments on Fourier integral(fourier Inversion) The Fourier integral H(u 1, u 2 ) = h(v) e i(u1,u2) v dv (8) with h(v) at a scatter of points rather than on a regular Cartesian mesh. There are three separate issues involved: 1 aquisition of data h(v j ) in the domain, 2 the choice of a quadrature scheme {v j, w j }, and 3 the availability of a fast algorithm for computing the discrete transform itself. Remarks: Not all schemes for reconstructing Fourier integrals can be represented as a quadrature H(u 1, u 2 ) = 1 N 1 N j=0 w jh j e i(u1,u2) v j, of the type (3), however, if the decision has been made to use a quadrature approach then the remaining task is entirely computational.

16 The Type-1 and Type-2 Nonuniform FFT Type-1 NUFFT: Discrete to Continuum Extension I With x j [0, 2π], the type-1 NUFFT is defined by the calculation of F (k) = 1 N N 1 j=0 f j e ikx j for k = M 2,, M 2 1 (9) is the exact Fourier coefficients of the function f (x) = N 1 j=0 f jδ(x x j ). (10) 1 The convolution of f (x) with g τ (x) = l= e (x 2lπ)2 /4τ, f τ (x) = f g τ (x) = 2π can be well-resolved by a uniform mesh in x. 0 f (y)g τ (x y) dy, (11) N 1 f τ (2πm/M r ) = f j g τ (2πm/M r x j ). (12) j=0

17 The Type-1 and Type-2 Nonuniform FFT Type-1 NUFFT: Discrete to Continuum Extension II 2 The Fourier coefficients of f τ, namely F τ (k) = 1 2π 2π 0 f τ (x)e ikx dx, can be computed using the standard FFT on a over-sampled grid F τ (k) 1 M r 1 M r m=0 f τ (2πm/M r )e ik2πm/mr. (13) 3 Once the values F τ (k) are known, an elementary calculation shows that π F (k) = 2τ τ ek F τ (k). (14) (This is a direct consequence of the convolution theorem and the fact that the Fourier transform of g τ is G τ (k) = 2τ e k2τ ).

18 The Type-1 and Type-2 Nonuniform FFT Discrete Summation using smooth delta functions image xj 0 2π image Figure: In the version of the NUFFT described here, each delta function source at a point such as x j in eq. (10) is replaced by a Gaussian. This smears the source strength to nearby regular grid points. The regular grid must be fine enough to resolve the smeared function f τ in eq. (11). Note that we include 2π-periodic images of the sources in the definition of the heat kernel g τ. They decay sufficiently rapidly that all but the nearest ones can be ignored.

19 The Type-1 and Type-2 Nonuniform FFT The Type-2 Nonuniform FFT The type-2 NUFFT evaluates a regular Fourier series at irregular target points, f (x j ) = M 2 1 F (k) e ikx k= M j. (15) 2 1 We first deconvolve the Fourier coefficients, defining F τ (k) by π F τ (k) = 2τ τ ek F (k) (16) 2 Evaluate f τ (x) on a uniform mesh using the FFT f τ (x) = M r 1 k=0 F τ (k) e ikx. (17) 3 We then compute the desired values f (x k ) from 2π f (x j ) = f τ g τ (x j ) = 1 f τ (x)g τ (x j x) dx 2π 0 1 Mr 1 m=0 M f τ (2πm/M r ) g τ (x j 2πm/M r ). (18) r

20 Fast Gaussian gridding method Fast gridding method using Gaussian Kernel I The dominant task is the calculation of f τ (2πm/M r ) in (12) and f (x j ) in (18). f τ (2πm/M r ) = N 1 j=0 f j l= e (x j 2πm/M r 2lπ) 2 /4τ. An elementary calculation shows that e (x j 2πm/M r ) 2 /4τ = e x 2 j /4τ ( e x j π/m r τ ) m e (πm/m r ) 2 /τ. (19) Fast Gridding Algorithm of Type 1 in two dimension Step I: Initialization 1 Set the oversampling ratio R = M r /M, the spreading parameter M sp, and the Gaussian kernel parameter τ according to the desired precision ɛ. 2 Precompute E 3 (l) = e (πl/mr )2 /τ for 0 l M sp and E 4 (k) = E 4 (M k) = e τk2 for k M 2.

21 Fast Gaussian gridding method Fast gridding method using Gaussian Kernel II Step C: Convolution for each source point (x j, y j ) 1 Find the nearest grid point (ξ 1, ξ 2 ) = 2π M r (m 1, m 2 ) with ξ 1 x j, ξ 2 y j. 2 Compute E 1 = e ((x j ξ 1) 2 +(y j ξ 2) 2 )/4τ, E 2x = e π(x j ξ 1)/M r τ, E 2y = e π(y j ξ 2)/M r τ l and E 2x (l 1 ) = E 1 l 2x, E 2y (l 2 ) = E 2 2y for M sp < l 1, l 2 M sp 3 Convolve the gaussian spreading function with f j as follows: V 0 = f j E 1 for l 2 = M sp +1, M sp V y = V 0 E 2y (l 2 ) for l 1 = M sp +1, M sp Add V y E 2x (l 1 ) to f τ (m 1 + l 1, m 2 + l 2 ). Step D: FFT and deconvolution 1 Compute 2D FFT of f τ (m 1, m 2 ) to obtain F τ (k 1, k 2 ). 2 Set F (k 1, k 2 ) = π τ E 4(k 1 )E 4 (k 2 )F τ (k 1, k 2 ) for M 2 k 1, k 2 < M 2

22 Numerical Results of Type-1,2 NUFFT Example 1. (Verification of Accuracy) We compare the results of the type 1 and 2 transforms with uniformly distributed random data points using direct summation and the NUFFT Type 1: 1024 points in 1D 10 0 Type 1&2: 64*64 grid Type 1&2: points in 2D 0 10 Computed Error Spreading distance Spreading distance Spreading distance Figure: Gridding error in the NUFFT.

23 Numerical Results of Type-1,2 NUFFT Example 2. (Fast gridding compared to gridding) 1.4 1D: Ultra 60(450Mhz) 1 1D: Pentium III(1GHz) 0.7 1D: Pentium IV(2.4Ghz) Time (Sec) Time (Sec) , ,000 Number of Data Points Time (Sec) , ,000 Number of Data Points , ,000 Number of Data Points 10 2D: Ultra 60(450Mhz) 10 2D: Pentium III(1GHz) 6 2D: Pentium IV(2.4Ghz) Time (Sec) 6 4 Time (Sec) 6 4 Time (Sec) , ,000 Number of Data Points , ,000 Number of Data Points , ,000 Number of Data Points Figure: Computing time.

24 Numerical Results of Type-1,2 NUFFT Example 3. (Comparison with direct method) Fast gridding algorithm v.s. direct summation and the FFT D Transformation Cost D Transformation Cost 10 0 Time (Sec) 10 1 Time (Sec) Number of Data Points Number of Data Points Figure: CPU requirements of with tolerance ɛ = 10 3, 10 6, 10 9, D ɛ=10 3 ɛ=10 6 ɛ=10 9 ɛ=10 12 Time (µsec) 3.77 N 4.40 N 4.78 N 5.44 N Breakeven (N) D ɛ=10 3 ɛ=10 6 ɛ=10 9 ɛ=10 12 Time (µsec) 8.45 N N N N Breakeven (N = M 2 ) 10*10 15*15 20*20 26*26

25 Numerical Results of Type-1,2 NUFFT Example 4. (MRI image reconstruction) I The exact reconstruction would obviously be the Fourier integral ˆf (j 1, j 2 ) = 2π 0 0 F (r, θ)e i(j1,j2) (r cos θ,r sin θ) r dr dθ, (20) which can be approximated by a type 1 transformation, ˆf (j 1, j 2 ) = N 1 k=0 F k e i(j1,j2) (sk x,sk y ). (21) Two issues should be considered: (1) the selection of points {(s k x, s k y )} which will certainly affect the image quality, (2) quadrature weights W k so that F k W k F (s k x, s k y ). We simply use (s k x, s k y ) = r j (cos(θ i ), sin(θ i )), r j = πj M, θ i = 2πi 2M (22) and the quadrature weight is r j θ r = (jπ/m) (2π/2M) (π/m).

26 Numerical Results of Type-1,2 NUFFT Example 4. (MRI image reconstruction) II y= y= Figure: Reconstructed image from a radial grid data.

27 The non-uniform FFT of Type-3 The Algorithm of Type-3 NUFFT I Type-3 NUFFT can be interpreted as the continuous Fourier transform of the function f (x) = N 1 2π f j δ(x x j ) (23) evaluated at the point s = s k. j=0 1 We begin by convolving f (x) with g τ (x) = e x 2 /4τ, f τ (x) = f g τ (x) = 1 2π f (y)g τ (x y) dy. (24) Since f τ can be well-resolved by a uniform mesh in x we define the discrete equispaced samples of f τ by f τ (n x ) = N 1 j=0 f jg τ (n x x j ). (25)

28 The non-uniform FFT of Type-3 The Algorithm of Type-3 NUFFT II 2 Using the Fourier transform of g τ (x) by G τ (s) = 2τ e s2τ, we define fτ σ (x) = f τ (x) G σ (x) = 1 e σx 2 f τ (x). (26) 2σ 3 The Fourier transform of fτ σ, namely Fτ σ (s) = 1 2π f σ τ (x)e isx dx, can be computed using the standard FFT on a sufficiently fine grid, Fτ σ (m s ) x 2π n f σ τ (n x )e imn x s. (27)

29 The non-uniform FFT of Type-3 The Algorithm of Type-3 NUFFT III 4 The next step is to recover the values F τ (s k ) by convolving Fτ σ (s) with g σ (s): F τ (s k ) = ( Fτ σ ) g σ (sk ) = 1 2π s 2π m Fτ σ (s)g σ (s k s) ds F σ τ (m s ) g σ (s k m s ). (28) 5 Once the values F τ (s k ) are known, an elementary calculation shows that F (s k ) = 1 2τ e s2 k τ F τ (s k ). (29) This follows again from the convolution theorem.

30 The non-uniform FFT of Type-3 Setting the tuning parameters The actual implementation requires a complete specification of all details. We do not repeat the analysis of [2] here, but summarize the relevant results as follows: if we let τ = 2 x m sp, 2π 2( 2 1) x π S 1 2, s π 1 2, (30) X + m sp x σ = 2 s m sp, M r = 2π, 2π 2( 2 1) x s then carrying out the convolutions in (25) and (28) with m sp = 9 yields about six digits of accuracy. Carrying out the convolutions with m sp = 18 yields about twelve digits of accuracy. Efficient implementation of these steps can be carried out using the fast Gaussian gridding algorithm. The higher dimensional versions involve more notation but are obvious extensions of the one-dimensional scheme. Appropriate values of M r, x and s can be chosen for each dimension separately.

31 Numerical examples of Type-3 NUFFT Example 5. (MRI in inhomogeneous magnetic field) In the presence of a field inhomogeneity given by φ(x), s(t) = ρ(x)e i2πk(t) x e iφ(x)t dx. (31) This requires the computation of s(t j ) N n=1 w nρ(x n )e i2πk(t j ) x n e iφ(xn)t j = N n=1 w nρ(x n )e ik j X n where K j = (k 1 (t j ), k 2 (t j ), t j ) and X n = (2πx 1 n, 2πx 2 n, φ(x n )). Thus, by embedding the data points in a higher dimensional space, one can carry out the transformation using a type-3 NUFFT. For a case with a spiral trajectory up to a maximum frequency of 60 and the regular (x 1, x 2 ) mesh on the unit box, direct evaluation required about 140 seconds and the NUFFT required 3.8 seconds to obtain six digits of accuracy when the time interval is [0, 1] and φ(x) varies in the interval [ 5π, 5π].

32 The Fast Sinc-Transform using NUFFT Definition of Fast Sinc-Transformatin I We define the d-dimensional sinc transforms by G l = N j=1 q j sinc(k j v l ). (32) which can be viewed as the evaluation of the function G(k) = sinc(k k )H(k )dk (33) at the points v l, due to the singular source distribution H(k) = N j=1 q jδ(k k j ). This follows from the elementary properties of the δ-function. From the convolution theorem we have that G(k) is given by G(k) = g(x) e 2πix k dx (34) with g(x) = F 1 sinc(k) F 1 H(k). (35)

33 The Fast Sinc-Transform using NUFFT Definition of Fast Sinc-Transformatin II The latter two functions are easily computed. The inverse Fourier transform F 1 sinc(k) in two dimensions is simply Π(x) = { 0 if x1 > 1/2 or x 2 > 1/2 1 if x 1 < 1/2 and x 2 < 1/2, (36) where x = (x 1, x 2 ). Further, it is easy to see that h(x) = F 1 H(k) = N q j e 2πix k j. (37) j=1 Thus, we can compute G(v l ) from (34)-(37): G(v l ) = 1/2 1/2 1/2 1/2 h(x) e 2πix v l dx. (38)

34 The Fast Sinc-Transform using NUFFT Quadrature considerations Equation (38) is an exact relation, and it remains only to discretize the integral with a collection of exponential functions with maximum frequency given by K max = max j k j L 1. Using a tensor-product rule for the double integral, we have G(v l ) = 1/2 1/2 1/2 M j 1=1 j 2=1 1/2 h(x) e 2πix v l dx M h(x j1, x j2 )e 2πi(x j 1,x j2 ) v l q j1 q j2 (39) The error, as in the one-dimensional case, decays at an exponential rate once M exceeds πk max. In summary, the fast sinc transform requires the adjoint NUFFT to compute h(x) via (37) at the tensor product quadrature points. The amount of work is of the order O((N + M 2 ) log(n + M 2 )) = O((N + K 2 max) log(n + K 2 max)). Since the quadrature used is spectrally accurate, the error in the fast sinc transform is dominated by the tolerance requested of the NUFFT.

35 The Fast Sinc-Transform using NUFFT The fast sinc 2 -transform The he inverse Fourier transform of sinc 2 (k) in two dimensions is t(x 1 ) t(x 2 ) where t(x) = We therefore need to compute G(k) = { 0 if x > 1 1 x if 1 < x < 1. h(x 1, x 2 ) t(x 1 ) t(x 2 ) e 2πi(x1,x2) k dx 1 dx 2 (40) where the integrand is smooth, on the four quadrants. In summary, the fast sinc 2 transform requires the adjoint NUFFT to compute h(x) via (37) at the tensor product quadrature points, followed by the forward NUFFT to compute (40) using tensor product Gaussian quadrature. The amount of work is O((N + K 2 max) log(n + K 2 max)).

36 An example of fast Sinc-transformation Computational Speed To test the utility of the fast algorithm, we constructed an Archimedean spiral sampled at N points according to the formula ( ( ) ( )) k j = K max j N cos 3πK max j N, sin 3πK max j N. Sample timings for K max = 64 and two different values for N are given in Table 1. N T FST T FS 2 T T dir Error < , < < , < 10 5 Table: Timing results for FST and FS 2 T on Archimedean spiral with K max = 64. Calculations were carried out on a laptop computer with a 1.2GHz Pentium processor.

37 Basic approaches for Fourier Inversion Inversion of continuous-to-discrete Fourier Transformation A number of imaging modalities require the inversion of the equation s(n) = ρ(r)e 2πık(n) r dr, (41) V where k(n) denotes the location of the nth measurement in the frequency domain ( k-space ) and r denotes position in the image domain. It will be convenient below to write this in operator form as s n = Hρ (k n ), (42) where H is the continuous-to-discrete Fourier operator which maps the image space to the signal space. We are particularly concerned with non-uniform sampling schemes, where the points {k(n)} do not lie on a regular grid. The inversion of (42) is, of course, an inherently ill-posed problem; the space of all possible densities ρ(r) is infinite dimensional space, while the vector of measurements {s(n)} is finite dimensional.

38 Basic approaches for Fourier Inversion Scheme 1 (Optimal quadrature) The first reconstruction scheme relies on the inverse Fourier transform ρ(r) = s(k)e 2πık r dk, (43) via a quadrature formula or, more precisely, its approximation via ρ(r) n s(n)e 2πık(n) r w n. (44) which can be carried out in O(N log N) and an optimal set of weights {w n } is given by the formula 1 w n = m sinc2 (k(m) k(n)). (45) Here, sinc(k) sin(πk) πk and sinc(k) = sinc(k 1 ) sinc(k d ), where k = (k 1, k 2,..., k d ). While the evaluation of these weights appears to require O(N 2 ) operations, the fast sinc 2 -transform, described below, makes use of the NUFFT to reduce the cost to O(N log N).

39 Basic approaches for Fourier Inversion Scheme 2 (Pseudo-inverse) The minimum-norm least-squares solution, ˆρ(x), ˆρ(x) = H + s = H (HH ) + s, (46) can then be computed in two steps: 1 Solve the system for M mn := (HH ) mn = sinc(k m k n ), Ma = s where (47) 2 Compute ˆρ(x) using a ˆρ(x) = H a. The matrix M may be ill-conditioned, so pseudoinverse construction of a = M + s costs O(N 3 ) work using the SVD. The cost of conjugate gradient method is O(J N 2 ) or O(J N log N) with the fast Sinc-Transformation, where J denotes the number of iterations. Remark: Note that ρ(r) n s(n)e 2πık(n) r w n (44) can be written as ρ(r) H W s and W can be viewed as a diagonal approximation of M +.

40 Basic approaches for Fourier Inversion MR Image Reconstruction Image reconstruction using k-space sampling truncated at K max = 64. Figure: Cartesian (top) and spiral (bottom) Optimal weight reconstruction (left) pseudoinverse approximation using 5 iterations (right)

41 Conclusions We have presented a simple version of the type-3 nonuniform FFT. It can be used to approximate the continuous Fourier transform when neither the spatial nor the Fourier domain spacing is regular. The NUFFT type 3 allows for the evaluation of more general integral operators such as (31) by embedding them in a higher dimensional space. We have constructed a fast algorithm for the (discrete) sinc and sinc 2 transforms which have immediate application in MR image reconstruction. We expect that the algorithms described here will be of fairly broad utility since nufft and sinc-convolution arises naturally in many signal and image processing contexts.

42 References H. Choi and D. C. Munson, Analysis and design of minimax-optimal interpolators, IEEE Trans. Signal Processing 46 (1998), A. Dutt and V. Rokhlin, Fast Fourier tramsforms for nonequispaced data, SIAM J. Sci. Comput. 14 (1993), J. A. Fessler and B. P. Sutton, Nonuniform Fast Fourier transforms using min-max interpolation, IEEE Trans. Signal Proc. 51 (2003), L. Greengard and J.-Y. Lee, Accelerating the Nonuniform Fast Fourier Transform, SIAM Rev., 46 (2004), S. Inati, J.-Y. Lee, L. Fleysher, R. Fleysher, and L. Greengard, Optimal Least Squares Reconstruction of Magnetic Resonance Images from Non-uniform Samples in k-space, in preparation. J.-Y. Lee, and L. Greengard, The type 3 Nonuniform FFT and its Applications, J. Comput. Phys., 206 (2005), 1-5. D. Potts, G. Steidl, M. Tasche, Fast Fourier transforms for nonequispaced data: A tutorial, Modern Sampling Theory, , Birkhauser Boston, Applied and Numerical Harmonic Analysis Series, (2001).