Advanced methods for image reconstruction in fmri

Transcription

1 Advanced methods for image reconstruction in fmri Jeffrey A. Fessler EECS Department The University of Michigan Regional Symposium on MRI Sep. 28, 27 Acknowledgements: Doug Noll, Brad Sutton,

2 Outline MR image reconstruction problem description Overview of image reconstruction methods Model-based image reconstruction Iterations and computation (NUFFT etc.) Regularization Myths about iterative reconstruction Field inhomogeneity correction Parallel (sensitivity encoded) imaging Image reconstruction toolbox: fessler 1

3 Why Iterative Image Reconstruction? Statistical modeling may reduce noise Incorporate prior information, e.g.: support constraints (piecewise) smoothness phase constraints No density compensation needed Non-Fourier physical effects such as field inhomogeneity Incorporation of coil sensitivity maps Improved results for under-sampled trajectories (?)... ( Avoiding k-space interpolation is not a compelling reason!) 2

4 Primary drawbacks of Iterative Methods Choosing regularization parameter(s) Algorithm speed 3

5 Example: Iterative Reconstruction under B 4

6 Introduction to Reconstruction 5

7 Standard MR Image Reconstruction MR k space data Reconstructed Image Cartesian sampling in k-space. An inverse FFT. End of story. Commercial MR system quotes 4 FFTs (256 2 ) per second. 6

8 Non-Cartesian MR Image Reconstruction k-space data y = (y 1,...,y M ) image f( r) k y k x k-space trajectory: κ(t) = (k x (t),k y (t)) = spatial coordinates: d r R 7

9 Textbook MRI Measurement Model Ignoring lots of things, the standard measurement model is: y i = s(t i )+noise i, s(t) = Z i = 1,...,M f( r)e ı2π κ(t) r d r = F( κ(t)). r: spatial coordinates κ(t): k-space trajectory of the MR pulse sequence f( r): object s unknown transverse magnetization F( κ): Fourier transform of f( r). We get noisy samples of this! e ı2π κ(t) r provides spatial information = Nobel Prize Goal of image reconstruction: find f( r) from measurements {y i } M i=1. The unknown object f( r) is a continuous-space function, but the recorded measurements y = (y 1,...,y M ) are finite. Under-determined (ill posed) problem = no canonical solution. All MR scans provide only partial k-space data. 8

10 Image Reconstruction Strategies Continuous-continuous formulation Pretend that a continuum of measurements are available: F( κ) = Z f( r)e ı2π κ r d r. The solution is an inverse Fourier transform: f( r) = Now discretize the integral solution: ˆf( r) = Z F( κ)e ı2π κ r d κ. M F( κ i )e ı2π κ i r w i i=1 M y i w i e ı2π κ i r, i=1 where w i values are sampling density compensation factors. Numerous methods for choosing w i values in the literature. For Cartesian sampling, using w i = 1/N suffices, and the summation is an inverse FFT. For non-cartesian sampling, replace summation with gridding. 9

11 Continuous-discrete formulation Use many-to-one linear model: y = A f +ε, where A : L 2 (R d ) C M. Minimum norm solution (cf. natural pixels ): min ˆf subject to y=a ˆf ˆf ˆf = A (AA ) 1 y = M i=1 c i e ı2π κ i r, where AA c = y. Discrete-discrete formulation Assume parametric model for object: f( r) = N j=1 f j p j ( r). Estimate parameter vector f = ( f 1,..., f N ) from data vector y.

12 Model-Based Image Reconstruction: Overview 11

13 Model-Based Image Reconstruction MR signal equation with more complete physics: s(t) = Z f( r)s coil ( r)e ıω( r)t e R 2 ( r)t e ı2π κ(t) r d r y i = s(t i )+noise i, i = 1,...,M s coil ( r) Receive-coil sensitivity pattern(s) (for SENSE) ω( r) Off-resonance frequency map (due to field inhomogeneity / magnetic susceptibility) R 2 ( r) Relaxation map Other physical factors (?) Eddy current effects; in κ(t) Concomitant gradient terms Chemical shift Motion Goal? (it depends) 12

14 Field Inhomogeneity-Corrected Reconstruction s(t) = Z f( r)s coil ( r)e ıω( r)t e R 2 ( r)t e ı2π κ(t) r d r Goal: reconstruct f( r) given field map ω( r). (Assume all other terms are known or unimportant.) Motivation Essential for functional MRI of brain regions near sinus cavities! (Sutton et al., ISMRM 21; T-MI 23) 13

15 Sensitivity-Encoded (SENSE) Reconstruction s(t) = Z f( r)s coil ( r)e ıω( r)t e R 2 ( r)t e ı2π κ(t) r d r Goal: reconstruct f( r) given sensitivity maps s coil ( r). (Assume all other terms are known or unimportant.) Can combine SENSE with field inhomogeneity correction easily. (Sutton et al., ISMRM 21, Olafsson et al., ISBI 26) 14

16 Joint Estimation of Image and Field-Map s(t) = Z f( r)s coil ( r)e ıω( r)t e R 2 ( r)t e ı2π κ(t) r d r Goal: estimate both the image f( r) and the field map ω( r) (Assume all other terms are known or unimportant.) Analogy: joint estimation of emission image and attenuation map in PET. (Sutton et al., ISMRM Workshop, 21; ISBI 22; ISMRM 22; ISMRM 23; MRM 24) 15

17 s(t) = Z The Kitchen Sink f( r)s coil ( r)e ıω( r)t e R 2 ( r)t e ı2π κ(t) r d r Goal: estimate image f( r), field map ω( r), and relaxation map R 2 ( r) Requires suitable k-space trajectory. (Sutton et al., ISMRM 22; Twieg, MRM, 23) 16

18 Estimation of Dynamic Maps s(t) = Z f( r)s coil ( r)e ıω( r)t e R 2 ( r)t e ı2π κ(t) r d r Goal: estimate dynamic field map ω( r) and BOLD effect R 2 ( r) given baseline image f( r) in fmri. Motion... 17

19 Model-Based Image Reconstruction: Details 18

20 Basic Signal Model y i = s(t i )+ε i, E[y i ] = s(t i ) = Goal: reconstruct f( r) from y = (y 1,...,y M ). Series expansion of unknown object: f( r) N j=1 Z f( r)e ı2π κ i r d r f j p( r r j ) usually 2D rect functions. Substituting into signal model yields Z [ ] N E[y i ] = f j p( r r j ) e ı2π κ i r d r = j=1 = N [ Z p( r r j )e ı2π κ i d r] r f j j=1 N a i j f j, a i j = P( κ i )e ı2π κ i r j, p( r) FT P( κ). j=1 Discrete-discrete measurement model with system matrix A = {a i j }: y = A f + ε. Goal: estimate coefficients (pixel values) f = ( f 1,..., f N ) from y. 19

21 Small Pixel Size Need Not Matter 127 x true N= N= N= N= N=

22 Profiles true N=32 N=64 N=128 N=256 N=512 f(x,) horizontal position [mm] 21

23 Regularized Least-Squares Estimation Estimate object by minimizing a cost function: ˆf = argmin f C N Ψ( f), Ψ( f) = y A f 2 + αr( f) data fit term y A f 2 corresponds to negative log-likelihood of Gaussian distribution regularizing term R( f) controls noise by penalizing roughness, e.g. : R( f) Z f 2 d r regularization parameter α > controls tradeoff between spatial resolution and noise αr( f) Equivalent to Bayesian MAP estimation with prior e Issues: choosing R( f) choosing α computing minimizer rapidly. 22

24 Quadratic regularization 1D example: squared differences between neighboring pixel values: R( f) = N 1 j=2 2 f j f j 1 2. In matrix-vector notation, R( f) = 1 2 C f 2 where C =, so C f = f 2 f 1. f N f N 1 For 2D and higher-order differences, modify differencing matrix C. Leads to closed-form solution: ˆf = argmin y A f 2 + α C f 2 f = [ A A+αC C ] 1 A y.. (a formula of limited practical use for computing ˆf ) 23

25 Choosing the Regularization Parameter Spatial resolution analysis (Fessler & Rogers, IEEE T-IP, 1996): ˆf = [ A A+αC C ] 1 A y ] E[ ˆf = [ A A+αC C ] 1 A E[y] ] E[ ˆf = [ A A+αC C ] 1 A A f }{{} blur A A and C C are Toeplitz = blur is approximately shift-invariant. Frequency response of blur: L(ω) = H(ω k ) = FFT(A Ae j ) (lowpass) R(ω k ) = FFT(C C e j ) (highpass) H(ω) H(ω)+αR(ω) Adjust α to achieve desired spatial resolution. 24

26 Spatial Resolution Example 1 A A e j 1 α C C e j 1 PSF π H(ω) π R(ω) π L=H/(H+R) ω Y ω Y ω Y π π π ω X π π π ω X π π π Radial k-space trajectory, FWHM of PSF is 1.2 pixels ω X 25

27 Spatial Resolution Example: Profiles 1 x 15 H(ω) R(ω) L(ω).8.6 π π ω 26

28 Resolution/noise tradeoffs Noise analysis: } Cov{ ˆf = [ A A+αC C ] 1 A Cov{y} A [ A A+αC C ] 1 Using circulant approximations to A A and C C yields: Var { ˆf j } σ 2 ε k H(ω k ) = FFT(A Ae j ) (lowpass) R(ω k ) = FFT(C C e j ) (highpass) H(ω k ) (H(ω k )+αr(ω k )) 2 = Predicting reconstructed image noise requires just 2 FFTs. (cf. gridding approach?) Adjust α to achieve desired spatial resolution / noise tradeoff. 27

29 Resolution/Noise Tradeoff Example Relative standard deviation α Under sampled radial Nyquist sampled radial Cartesian α PSF FWHM [pixels] In short: one can choose α rapidly and predictably for quadratic regularization. 28

30 Iterative Minimization by Conjugate Gradients Choose initial guess f () (e.g., fast conjugate phase / gridding). Iteration (unregularized): g (n) = Ψ ( f (n)) = A (A f (n) y) gradient p (n) = Pg (n) precondition, n = γ n = g (n), p (n) g (n 1), p (n 1), n > d (n) = p (n) + γ n d (n 1) search direction v (n) = Ad (n) α n = d (n), g (n) / v (n) 2 step size f (n+1) = f (n) + α n d (n) update Bottlenecks: computing A f (n) and A r. A is too large to store explicitly (not sparse) Even if A were stored, directly computing A f is O(MN) per iteration, whereas FFT is only O(M logm). 29

31 [A f] i = Computing A f Rapidly N a i j f j = P( κ i ) j=1 N e ı2π κ i r j f j, j=1 i = 1,...,M Pixel locations { r j } are uniformly spaced k-space locations { κ i } are unequally spaced = needs nonuniform fast Fourier transform (NUFFT) 3

32 NUFFT (Type 2) Compute over-sampled FFT of equally-spaced signal samples Interpolate onto desired unequally-spaced frequency locations Dutt & Rokhlin, SIAM JSC, 1993, Gaussian bell interpolator Fessler & Sutton, IEEE T-SP, 23, min-max interpolator and min-max optimized Kaiser-Bessel interpolator. NUFFT toolbox: fessler/code 1 X(ω) 5 π π/2 ω? π/2 π 31

33 Worst-Case NUFFT Interpolation Error Maximum error for K/N= E max Min Max (uniform) Gaussian (best) Min Max (best L=2) Kaiser Bessel (best) Min Max (L=13, β=1 fit) J 32

34 Further Acceleration using Toeplitz Matrices Cost-function gradient: g (n) = A (A f (n) y) = T f (n) b, where T A A, b A y. In the absence of field inhomogeneity, the Gram matrix T is Toeplitz: [A A] jk = M P( κ i ) 2 e ı2π κ i ( r j r k ). i=1 Computing T f (n) requires an ordinary (2 over-sampled) FFT. (Chan & Ng, SIAM Review, 1996) In 2D: block Toeplitz with Toeplitz blocks (BTTB). Precomputing the first column of T and b requires a couple NUFFTs. (Wajer, ISMRM 21, Eggers ISMRM 22, Liu ISMRM 25) This formulation seems ideal for hardware FFT systems. 33

35 Toeplitz Acceleration Example: image. radial trajectory, 2 angular under-sampling. True CP 1 True CP.6 CG NUFFT CG Toeplitz CG NUFFT CG Toeplitz (Iterative provides reduced aliasing energy.) 34

36 Toeplitz Acceleration Method A Dy b = A y T 2 iter Total Time NRMS (5dB) Conj. Phase % CG-NUFFT % CG-Toeplitz % Toeplitz aproach reduces CPU time by more than 2 on conventional workstation (Xeon 3.4GHz) Eliminates k-space interpolations = ideal for FFT hardware No SNR compromise CG reduces NRMS error relative to CP, but 15 slower... (More dramatic improvements seen in fmri when correcting for field inhomogeneity.) 35

37 NUFFT with Field Inhomogeneity? Combine NUFFT with min-max temporal interpolator (Sutton et al., IEEE T-MI, 23) (forward version of time segmentation, Noll, T-MI, 1991) Recall signal model including field inhomogeneity: s(t) = Z f( r)e ıω( r)t e ı2π κ(t) r d r. Temporal interpolation approximation (aka time segmentation ): e ıω( r)t L a l (t)e ıω( r)τ l l=1 for min-max optimized temporal interpolation functions {a l ( )} L l=1. L Z [ ] s(t) a l (t) f( r)e ıω( r)τ l e ı2π κ(t) r d r l=1 Linear combination of L NUFFT calls. 36

38 Field Corrected Reconstruction Example Simulation using known field map ω( r). 37

39 Simulation Quantitative Comparison Computation time? NRMSE between ˆf and f true? Reconstruction Method Time (s) NRMSE NRMSE complex magnitude No Correction Full Conjugate Phase Fast Conjugate Phase Fast Iterative (1 iters) Exact Iterative (1 iters)

40 Human Data: Field Correction 39

41 Acceleration using Toeplitz Approximations In the presence of field inhomogeneity, the system matrix is: a i j = P( κ i )e ıω( r j)t i e ı2π κ i r j The Gram matrix T = A A is not Toeplitz: [A A] jk = Approximation ( time segmentation ): M P( κ i ) 2 e ı2π κ i ( r j r k ) e ı (ω( r j ) ω( r k ))t i. i=1 e ı L (ω( r j ) ω( r k ))t i b il e ı (ω( r j ) ω( r k ))τ l l=1 T = A L A D D l diag { } e ıω( r j)τ l lt l D l, [T l=1 l ] jk M i=1 P( κ i ) 2 b il e ı2π κ i ( r j r k ). Each T l is Toeplitz = T f using L pairs of FFTs. (Fessler et al., IEEE T-SP, Sep. 25, brain imaging special issue) 4

42 Toeplitz Results 1 Image and support 2.5 Uncorrected Conj. Phase, L= Fieldmap: Brain Hz CG NUFFT L=6 CG Toeplitz L=8 41

43 Toeplitz Acceleration Precomputation NRMS % vs SNR Method L B,C A Dy b = A y T l 15 iter Total Time 5 db 4 db 3 db 2 db Conj. Phase CG-NUFFT CG-Toeplitz Reduces CPU time by 2 on conventional workstation (Mac G5) No SNR compromise Eliminates k-space interpolations = ideal for FFT hardware 42

44 Joint Field-Map / Image Reconstruction Signal model: After discretization: y i = s(t i )+ε i, s(t) = Z f( r)e ıω( r)t e ı2π κ(t) r d r. y = A(ω) f + ε, a i j (ω) = P( κ i )e ıω jt i e ı2π κ i r j. Joint estimation via regularized (nonlinear) least-squares: ( ˆf, ˆω) = argmin f C N,ω R N y A(ω) f 2 + β 1 R 1 ( f)+β 2 R 2 (ω). Alternating minimization: Using current estimate of fieldmap ˆω, update ˆf using CG algorithm. Using current estimate ˆf of image, update fieldmap ˆω using gradient descent. Use spiral-in / spiral-out sequence or racetrack EPI. (Sutton et al., MRM, 24) 43

45 Joint Estimation Example (a) uncorr., (b) std. map, (c) joint map, (d) T1 ref, (e) using std, (f) using joint. 44

46 Activation Results: Static vs Dynamic Field Maps 45

47 Functional results for the two reconstructions for 3 human subjects. Reconstruction using the standard field map for (a) subject 1, (b) subject 2, and (c) subject 3. Reconstruction using the jointly estimated field map for (d) subject 1, (e) subject 2, and (f) subject 3. Number of pixels with correlation coefficients higher than thresholds for (g) subject 1, (h) subject 2, and (i) subject 3. Take home message: dynamic field mapping is possible, using iterative reconstruction as an essential tool. (Standard field maps based on echo-time differences work poorly for spiral-in / spiral-out sequences due to phase discrepancies.)

48 Tracking Respiration-Induced Field Changes 47

49 Choosing α is difficult Myths Sample density weighting is desirable 48

50 Sampling density weighted LS Some researchers recommend using a weighted LS cost function: Ψ( f) = y A f 2 W where the weighting matrix W is related to the k-space sample density compensation factors (DCF). Purported benefits: Faster convergence Better conditioning But, Gauss-Markov theorem from statistical estimation theory states that lowest estimator variance is realized by choosing W = σ 2 ε I, the inverse of the data noise covariance. 49

51 Resolution/Noise Tradeoff: Example with Weighting Relative standard deviation Nyquist sampled radial, DCF weights Nyquist sampled radial Cartesian cond(a WA) e+6 cond(a A) e PSF FWHM [pixels] 5

52 Don t just take it from me... Fig. 5 of Gabr et al., MRM, Dec

53 Acceleration via Weighting? Fig. 8 of Gabr et al., MRM, Dec. 26. Zero initialization! 52

54 Acceleration via Initialization 1 8 Unweighted, initialized Weighted, initialized Unweighted, CP initialized NRMS Error Iteration 53

55 Parallel Imaging 54

56 Sensitivity encoded (SENSE) imaging Use multiple receive coils (requires multiple RF channels). Exploit spatial localization of sensitivity pattern of each coil. Note: at 1.5T, RF is about 6MHz. = RF wavelength is about m/s/6 1 6 Hz = 5 meters RF coil sensitivity patterns 1 Array coil images 1 Regularized sensitivity maps Pruessmann et al., MRM,

57 SENSE Model Multiple coil data: y li = s l (t i )+ε li, s l (t) = Z f( r)s coil l ( r)e ı2π κ(t) r d r, l = 1,...,L = N coil Goal: reconstruct f( r) from coil data y 1,..., y L given sensitivity maps { s coil l ( r) } L l=1. Benefit: reduced scan time. Left: sum of squares; right: SENSE. 56

58 SENSE Reconstruction Signal model: s l (t) = Z f( r)s coil l ( r)e ı2π κ(t) r d r Discretized form: y l = AD l f + ε l, l = 1,...,L, where A is the usual frequency/phase encoding matrix and D l contains the sensitivity pattern of the lth coil: D l = diag { s coil l ( r j ) }. Regularized least-squares estimation: ˆf = argmin f L y l AD l f 2 + βr( f). l=1 Can generalize to account for noise correlation due to coil coupling. Easy to apply CG algorithm, including Toeplitz/NUFFT acceleration. For Cartesian SENSE, iterations are not needed. (Solve small system of linear equations for each voxel.) 57

59 Summary Iterative reconstruction: much potential in MRI Quadratic regularization parameter selection is tractable Computation: reduced by tools like NUFFT / Toeplitz But optimization algorithm design remains important (cf. Shepp and Vardi, 1982, PET) Some current challenges Sensitivity pattern mapping for SENSE Through-voxel field inhomogeneity gradients Motion / dynamics / partial k-space data Establishing diagnostic efficacy with clinical data... Image reconstruction toolbox: fessler 58