Tomographic image reconstruction using the nonuniform FFT. Jeffrey A. Fessler and Bradley P. Sutton. EECS Department. BME Department. Department of ...
Tomographic image reconstruction using the nonuniform FFT
Jeffrey A. Fessler and Bradley P. Sutton EECS Department BME Department Department of Radiology The University of Michigan June 2002
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Outline • Applications ◦ 1. MRI ◦ 2. Tomography • Min-max framework for nonuniform FFT • Examples • Features / Limitations • Future goals
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MRI Application MRI rosette k-space trajectory ky
kx
2
Simplified MRI Signal Model Ignoring lots of things: yi = s(ti) + noisei,
i = 1, . . . , Nsamples Z s(t) = f (~r) exp −ı2π~k(t) ·~r d~r,
where ~k(t) denotes the “k-space trajectory” of the MR pulse sequence. • MRI measurements are (roughly) samples of the Fourier transform of the object’s transverse magnetization f (~r). • Reconstruction problem: recover f (~r) from measurements {yi}
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Conventional MR Image Reconstruction 1. Interpolate measurements onto rectilinear grid (“gridding”) 2. Apply inverse FFT to estimate samples of f (~r)
k
y
Gridding from Polar to Cartesian
0 0
kx
4
Limitations of MR gridding-based reconstruction 1. Artifacts/inaccuracies due to interpolation 2. Contention about sample density “weighting” 3. Oversimplifications of Fourier transform signal model: • • • •
Magnetic field inhomogeneity Magnetization decay (T2) Eddy currents ...
Statistical MR Image Reconstruction 1. Series expansion of unknown object: np
f (~r) ≈
∑ x j b(~r −~r j) j=1
2. Discrete-discrete measurement model: y = Ax +Zε ~ ai j = b(~r −~r j ) exp −ı2π~k(ti) ·~r d~r = B(~k(ti))e−ı2π k(ti)·~r j 3. ε includes both measurement noise and model error 4. A can also include “non-Fourier” effects (inhomogeneity, decay, etc.) 5. Least-squares formulation (Gaussian noise model): xˆ = arg min Ψ(x), x
Ψ(x) = ky − Axk2
6. Regularization included when needed (depends on ~k(t)) 7. Iterative preconditioned conjugate gradient algorithm for minimization.
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Challenges for iterative MR image reconstruction • Each PCG iteration requires calculation of A0(y − Ax(n)) • A is too large to store explicitly (not sparse) • Even if A were stored, directly computing Ax is O(n2p), per iteration, whereas FFT is only O(n p log n p) ⇒ need fast algorithm for computing Ax, i.e., for computing
∑ ∑ e−ı2π (k1(ti)n1+k2(ti)n2)x(n1, n2), n1 n2
assuming the~r j ’s (basis centers) are unit spaced on a rectilinear grid. Need: fast algorithm for 2D nonuniform Fourier transform
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Tomography Application g (R) θ
Projection
Object
R
50
20
y
x2
100
θ
θ
Sinogram
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Object f(x,y)
150 60
x
20
40 x1
60
−20
0
r
20
Simplified tomography measurement model (sinogram): yi = (h ∗ pθi (·; f ))(ri) + noisei,
i = 1, . . . , nd ,
nd = nr · nθ.
Radon transform degraded by radially shift-invariant blur with PSF h(r). Radon transform (line integrals): pθ(r; f ) =
Z
f (r cos θ − l sin θ, r sin θ + l cos θ) dl
Goal: reconstruct object f (~r) from sinogram measurements {yi} 8
Classical Fourier-transform reconstruction Fourier-slice theorem: 1D FT
2D FT
pθ(r; f ) ←→ Pθ(ρ) = F(ρ, θ) ←→ f (x, y) • • • •
Compute 1D FFT of each row of sinogram. Possibly deconvolve blur h(r) Interpolate from polar samples onto rectilinear frequency samples Compute inverse 2D FFT Limitations
• • • •
Artifacts due to polar-cartesian interpolation Suboptimal treatment of nonuniform-variance noise, e.g., Poisson Over-simplified measurement model Disregards nonnegativity constraint
Proposed approach partially overcomes first two limitations
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Iterative Tomographic Image Reconstruction 1. Series expansion of unknown object: np
f (~r) ≈
∑ x j b(~r −~r j) j=1
2. Discrete-discrete measurement model y = Ax + ε,
ai j = h(r) ∗ pθi (r; b(· −~r j ))
r=ri
3. Penalized weighted least-squares (PWLS) formulation xˆ = arg min Ψ(x), x
Ψ(x) = (y − Ax)0W (y − Ax) + βR(x)
4. Weighting matrix W for nonuniform noise variance (cf Delaney and Bresler, IEEE T-IP, May 1996) 5. Regularization essential due to ill-conditioned nature of tomography 6. Iterative preconditioned conjugate gradient algorithm for minimization
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Challenges for Iterative Tomographic Reconstruction • Each PCG iteration requires calculation of A0W (y − Ax(n)) • A is sparse, but very large for 3D PET, too large to store in 2D X-ray CT • Even if A were stored, directly computing Ax is O(n2p), per iteration, whereas FFT is only O(n p log n p) Proposed approach for reprojection (computing Ax) 1. Apply nonuniform FFT to compute 2D FT on a polar grid accurately 2. Apply shift-invariant blur h(r) in frequency domain 3. Compute inverse 1D FFT to form each row of reprojection • Avoids line-integral calculations! • Routine for A0 is the exact adjoint
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Prior work on NUFFT
• [1] Dutt & Rokhlin, SIAM JSC, 1993 Fast Fourier transforms for nonequispaced data. Gaussian based interpolation
• [2] Beylkin, ACHA, 1995 On the fast Fourier transform of functions with singularities. B-spline based interpolation in multiresolution framework (N-D)
• [3] Dutt & Rokhlin, ACHA, 1995 Fast Fourier transforms for nonequispaced data, II. fast multipole method
• [4] Anderson & Dahleh, SIAM JSC, 1996 Rapid computation of the discrete Fourier transform. Taylor series expansion, requiring multiple FFTs
• [5] Nguyen & Liu, SIAM JSC, 1999 The regular Fourier matrices and nonuniform fast Fourier transforms. least-squares approach to shift-variant Fourier interpolation
• [6] Sutton, Fessler, & Noll, ISMRM, 2001 A min-max approach to the nonuniform N-D FFT for rapid iterative reconstruction of MR images. • [7] Fessler & Sutton, IEEE T-SP, 2001 (Submitted) Nonuniform fast Fourier transforms using min-max interpolation. 12
NUFFT Problem Statement (1D) Given signal xn, n = 0, . . . , N − 1 with (discrete-time) Fourier transform X(ω) =
N−1
∑ xne−ıωn
n=0
and a collection of arbitrary frequencies {ωm : m = 1, . . . , M}, compute ym = X(ωm),
m = 1, . . . , M.
Direct approach is O(NM); impractical for large M.
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NUFFT via linear interpolation 1. Compute K-point FFT of xn (where K ≥ N, possibly oversampled) 2πk , k = 0, . . . , K − 1 Xk = X K 2. Interpolate from set {2πk/K} to set {ωm} yˆm =
K−1
∑ vmkXk
k=0
Design question: how to choose interpolation coefficients {vmk }? Scaled variation 1. Start with “weighted” K-point FFT: Yk =
N−1
−ıωn s x e n n ∑
n=0
2. Design problem includes choosing scaling factors {sn}.
(Important!)
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Interpolators 1. Shift invariant • Rarely precomputed • Less memory • More in-line work
• Gaussian • B-spline • ...
2. Shift variant • Constraint: use the J nearest FFT samples to interpolate onto each ωm J+1 J 2π 4 arg min k ω − K2πk − 2J , J odd yˆm = ∑ u?m j Xk0(ωm)+ j , where k0(ω) = max k : ω ≥ K k − 2 , J even. j=1
X(ω)
100
50
0−π
−π/2
ω
?
π/2
π
• O(JM) memory if interpolation coefficients are precomputed • O(K log K) + O(JM) computation 15
Min-Max Criterion Choose interpolation cofficients {um j } to minimize worst-case error. min
max
um ∈C J x∈C N : kxk≤1
|yˆm − ym|, where um = (um1, . . . , umJ ).
Solution (data independent!):
um
Λ j j (ω) T [C 0C]l, j 0 = Λ (ω)T r(ωm), where: r j (ω) δ0(t)
= = = =
N−1 −ı[ω− 2π K (k0 (ω)+ j)] 2
e [C 0C]−1 ∈ IRJ×J δ0( j − l) δ0(ω/(2π/K) − k0(ω) − j) 4 sin(πtN/K) . = N sin(πt/K)
“Modified truncated-Dirichlet interpolation of oversampled FFT”
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Equivalent interpolator for J=6, K/N=2 1
Min−max Sinc
Interpolator
0.8 0.6 0.4 0.2 0 −0.2 −3
−2
−1
0 ω/γ
1
2
3
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Equivalent interpolator for J=7, K/N=2 1
Min−max Sinc
Interpolator
0.8 0.6 0.4 0.2 0 −0.2 −3
−2
−1
0 ω/γ
1
2
3 18
Accuracy √ p Error for worst-case unit-norm signal x ∈ C is Emax(ω) = N 1 − r 0(ω)T r(ω). Worst-case error for ω usually at midpoint between two nearest FFT neighbors. N
Maximum error for α = (1)
0
10
K/N=1.5 K/N=2 K/N=2.5 K/N=3 K/N=4 K/N=5
−2
10
−4
E
max
10
−6
10
−8
10
−10
10
2
5
8
11 J
14
17
20
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Comparison with Dirichlet Maximum error for K/N=2 −1
10
−2
10
−3
E
max
10
−4
10
−5
10
−6
10
2
Truncated Dirichlet Tapered Dirichlet Linear (J=2) Min−Max (uniform) Min−Max (best L=2) 4 6
8
10
12
J 20
Comparison with Gaussian (Dutt/Rokhlin) Maximum error for K/N=2
−1
10
−2
10
−3
Emax
10
−4
10
Gaussian (best σ) Min−Max (uniform) Min−Max (L=5 LS fit) Min−Max (best L=2)
−5
10
−6
10
2
4
6
8
10
J 21
Extensions • Multidimensional NUFFT Use J × J neighborhood (in 2D, e.g.) around each spatial frequency location of interest. Straightforward generalization. • Adjoint operator 1. Hermitian transpose of interpolation matrix 2. K-point inverse FFT • Adaptive neighborhoods Neighborhood size J vs distance between ωm and nearest neighbor. • Free software: http://www.eecs.umich.edu/∼fessler
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Kaiser-Bessel Interpolator s 2 κ I (α f (κ)) 4 m J 1− , |κ| < J/2 , where fJ (κ) = F(κ) = fJm(κ) J/2 Im(α) 0, otherwise. • Optimality properties? • Usually m = 2 so continuous and differentiable on boundaries. Optimized NUFFT Interpolation Functions, J=10 1 Kaiser−Bessel Gaussian 0.8
F(κ)
0.6
0.4
0.2
0 −5
0 κ
5
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Kaiser-Bessel: Optimizing Order Kaiser−Bessel Error for K/N=2 and α=2.34⋅J −4
E
max
10
−5
10
−6
10
−2
J=6 J=7 −1
0 1 m (Kaiser−Bessler order)
2 24
Kaiser-Bessel: Optimizing Width −3
Kaiser−Bessel Error for K/N=2 and m=0
10
−4
E
max
10
−5
10
−6
10
1.5
J=5 J=6 J=7 2 2.5 α/J (Kaiser−Bessler width)
3
25
Kaiser-Bessel: Optimizing Scaling Factors Kaiser−Bessel Error for K/N=2, α=2.34 ⋅ J, and m=0
−5
E
max
10
−10
10
Numerical FT scaling factors Analytical FT scaling factors
−15
10
4
6
8
10 J
12
14
16
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Kaiser-Bessel: Scaling Factors Tradeoff Kaiser−Bessel Error for K/N=2, α=2.34 ⋅ J, and m=0 Numerical FT scaling factors Analytical FT scaling factors
−5
E
max
10
0
0.2
0.4 0.6 ω / (2π/K)
0.8
1 27
Kaiser-Bessel vs Min-Max Interpolators Maximum error for K/N=2 −2
10
−4
Emax
10
−6
10
Min−Max (uniform) Gaussian (best) Min−Max (best L=2) Kaiser−Bessel (best) Min−Max (L=13, β=1 fit)
−8
10
−10
10
2
4
6
8
10
J 28
Fourier-Based Tomographic Projection (Radon Transform) 1. Compute 2× oversampled 2D FFT of object 2. Min-max interpolation onto polar coordinates (5 × 5 neighborhood) 3. Multiply spectrum by effects of • shift-invariant detector blur • and (square) pixel basis. 4. 1D inverse FFT for each sinogram row
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Forward Projector Simulation
• • • •
128 × 128 Shepp-Logan digital phantom 160 bins × 192 angles sinogram 1-bin rectangular detector PSF Exact DSFT-based Fourier projector (no interpolation) vs NUFFT based on min-max interpolator • 6.3s precompute time on 1GHz Pentium III / Linux Exact DSFT
NUFFT/KB(J=4, K/N=2)
Shepp−Logan image 8 −60 7 −40 6
−20
y
5
4
0
3 20 2 40 1
60 0 −60
−40
−20
0 x
20
40
60
cpu = 101.5 s cpu = 0.15 s Sinograms max diff = 0.04%
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Bilinear Interpolation (“Gridding”) Comparison
K/N=2
NUFFT/KB(J=4)
K/N=2
NUFFT/Bilinear
Exact DSFT
cpu = 101.5 s
cpu = 0.11 s
Shepp−Logan
Bilinear |Error|
3.2% max
cpu = 0.15 s 15
0
KaiserB |Error|
0.18
0.04% max 31
Back-projector (Adjoint) Test
NUFFT/KB
Exact DSFT J=4, K/N=2
Sinogram
cpu = 144.0 s
cpu = 0.34 s
max |error| = 0.08%
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NUFFT Projector Time/Accuracy Tradeoff NUFFT Reprojector vs Exact FT Reprojector 3 J=2
2.5
RMS % Error
NUFFT (K/N=1) NUFFT (K/N=2) NUFFT (K/N=3) NUFFT (K/N=4)
2 1.5
J=3 J=2
1 J=3
0.5
J=3 0
J=5 0.2 NUFFT CPUtime
0.4 33
QPWLS Iterative Reconstruction 5
5
0
NUFFT(J=5)
5
20 iter of CG
Exact DSFT 20 iter of CG
Phantom
4726.6 sec
0
13.0 sec
0
−3
FBP
5
0
Exact − NUFFT
x 10 6
−4
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Summary Min-max interpolation approach for NUFFT: minimizes worst-case interpolation error. Accurate and fast projector/backprojector for 2D tomography. Future Applications • MRI with field inhomogeneity • MRI with multiple coils • 3D PET Limitations / Challenges • Slightly negative ai j ’s (in tomography) • Shift-invariant PSF • Parallel-beam geometry • Non-uniform radial sampling in ring PET geometry • Numerical conditioning for large J • Ordered-subsets
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Iterative MRI Reconstruction with Field Inhomogeneity
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References
[1] A. Dutt and V. Rokhlin. Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Comp., 14(6):1368–93, November 1993. [2] G. Beylkin. On the fast Fourier transform of functions with singularities. Applied and Computational Harmonic Analysis, 2(4):363–81, October 1995. [3] A. Dutt and V. Rokhlin. Fast Fourier transforms for nonequispaced data, II. Applied and Computational Harmonic Analysis, 2:85–100, 1995. [4] C. Anderson and M. D. Dahleh. Rapid computation of the discrete Fourier transform. SIAM J. Sci. Comp., 17(4):913–9, July 1996. [5] N. Nguyen and Q. H. Liu. The regular Fourier matrices and nonuniform fast Fourier transforms. SIAM J. Sci. Comp., 21(1):283–93, 1999. [6] B. P. Sutton, J. A. Fessler, and D. Noll. A min-max approach to the nonuniform N-D FFT for rapid iterative reconstruction of MR images. In Proc. Intl. Soc. Mag. Res. Med., page 763, 2001. [7] J. A. Fessler and B. P. Sutton. Nonuniform fast Fourier transforms using min-max interpolation. IEEE Tr. Sig. Proc., ?, 2001. Submitted to IEEE T-SP on 2001-12-19.
