Signal and Noise in Complex-Valued SENSE MR Image Reconstruction

Transcription

1 Signal and Noise in omplex-valued SENSE M mage econstruction Daniel B. owe, Ph.D. (Joint with ain P. Bruce) Associate Professor omputational Sciences Program Department of Mathematics, Statistics, and omputer Science Adjunct Associate Professor Department of Biophysics March 22, 11 1

2 OUTLNE 1. Motivation 2. Background 3. Methods 4. esults 5. Discussion 2

3 Motivation n M it is not voxel values that are measured. The actual measurements are spatial frequencies (k-space). The k-space measurements are not acquired instantaneously. n parallel imaging, k-space is subsampled and measured in parallel then combined to form a single image. mage and volume measurement time is decreased at the expense of increased image reconstruction difficulty and time. One popular parallel imaging method is SENSE. Pruessmann et al.: SENSE: Sensitivity Encoding for Fast M. MM 42: ,

4 Background mage inverse Fourier econstruction for single coil. ( y i y ) ( F if) ( ) T ( V iv) x ix i i i i owe, Nencka, Hoffmann,Signal and noise of Fourier reconstructed fm data. JNSM 159: ,

5 Background n parallel imaging there is more than one receive coil. Each coil measures a k-space array where every A th line is skipped. k y k y k x k x Full k-space. Skipped k-space. Bruce, Karaman, and owe: n Submission, 11. 5

6 5 15 Background The k-space arrays where every A th line is skipped are reconstructed into an aliased image to be combined to form a single image. coil 1 Skipped k-space. coil 1 Aliased images. k y ombined image. k x coil 2 coil coil 2 coil coil 4 coil Bruce, Karaman, and owe: n Submission, 11. 6

7 Background The combination of aliased images to form a single image utilizes coil sensitivities. k y k x Aliased images. a oil sensitivities. S coil 1 ombined image. v coil 2 coil 3 coil 1 coil 4 coil Bruce, Karaman, and owe: n Submission, 11. coil 2 coil 3 7

8 Methods The SENSE model for aliased voxel values from n coils is a S v, ~ N(, ) n1 na A1 n1 where for each voxel a is a vector of the n complex-valued aliased voxel values a a ia ν is a vector of the A unaliased voxel value v v iv S is an nxa matrix of complex-valued coil sensitivities S S is ε is a vector of the n complex-valued error values Pruessmann et al.: SENSE: Sensitivity Encoding for Fast M. MM 42: , Bruce, Karaman, and owe: n Submission, 11. i i 8

9 Methods 1 n 1 coil 2 coil 4 coil a S v n1 na A1 n1 coil 1 1 n A A coil 1 coil 4 * n coil 2 coil 3 Bruce, Karaman, and owe: n Submission, 11. 9

10 Methods 1 1 coil 4 coil 3 coil 2 coil 1 coil coil 4 * The SENSE process uses the complex-valued normal distribution f a S v ~ N(, ) n1 na A1 n1 ( ) (2 ) n and for n coil measurements 1 1/2 e coil 2 coil 3, H 1, i H is the conjugate transpose (Hermetian) f ( a ) (2 ) n 1 H 1 1/2( a Sv ) ( a Sv ) e Pruessmann et al.: SENSE: Sensitivity Encoding for Fast M. MM 42: , Wooding The multivariate distribution of complex normal variables. Biometrika 43: , Bruce, Karaman, and owe: n Submission, 11.

11 Methods 1 1 coil 4 coil 3 coil 2 coil 1 coil coil 4 * From the distribution for the n coil measurements 1 f ( a ) (2 ) n the voxel values can be estimated as ( S S ) S a H 1 1 H 1 1 H 1 1/2( a Sv ) ( a S v ) e coil 2 coil 3 with knowledge of S and Ψ. Pruessmann et al.: SENSE: Sensitivity Encoding for Fast M. MM 42: , Wooding The multivariate distribution of complex normal variables. Biometrika 43: , Bruce, Karaman, and owe: n Submission,

12 Methods nstead of writing the model with complex numbers as a S v n1 na A1 n1 a a ia S S is v v iv i we can write the model using an isomorphism as a S v 2nx1 2nx2A 2Ax1 2nx1, a a a S S S S S v v v. Wooding The multivariate distribution of complex normal variables. Biometrika 43: , Bruce, Karaman, and owe: n Submission,

13 Methods Then the distribution for n coil measurements is n f( a) (2 ) e 1/2 1/2( )' 1 a Sv ( a Sv ), with a a a S S S S S v v 2nx1 2nx2A 2Ax1 2nx1 and the imposed skew-symmetric covariance structure v,. Wooding The multivariate distribution of complex normal variables. Biometrika 43: , Bruce, Karaman, and owe: n Submission,

14 Methods The SENSE voxel values can be estimated by or in terms of an isomorphism ( S S ) S a H 1 1 H T 1 T v S S S S S S a v S S S S S S a 2A1 2A2n 2n2n 2n2A 2A2n 2n2n 2n1 v a U * v a 2A1 2A2n 2n1 n=4 real / n=4 imaginary true aliased voxel values v 1 v 1 v 2 v SENSE unfolding U a a a 1 a 1 a 2 a 2 Acceleration A A=3 real / A=3 imaginary un-aliased fold values Bruce, Karaman, and owe: n Submission,

15 Methods SENSE unfolding v a U * v a v 1 v U 1 U 2 a a 1 a v a 2 a a 2 a 2 v * 2 4 p a v p 8 a 2 p a q a 12 v p 12 U q a Bruce, Karaman, and owe: n Submission,

16 real Methods eal-valued isomorphism imaginary f f f owe, Nencka, Hoffmann,Signal and noise of Fourier reconstructed fm data. JNSM 159: , 7. 16

17 Methods n f image1 aliased coil1 k-space image n aliased coil n k-space Bruce, Karaman, and owe: n Submission,

18 Methods y P u U here is a for each voxel a PP S U 1 image1 aliased P U PP S U p image n aliased permute unfold matrix U s have S and Ψ Single image vector Bruce, Karaman, and owe: n Submission, 11. permute to by folded voxel 18

19 owe Methods y P u processing on unfolded image vector U here is a for each voxel a PP S processing on each coil image vector ( n ) insert processing on each coil k-space vector f U 1 P U PP S U p permute Single image vector unfold matrix U s have S and Ψ permute to by folded voxel Bruce, Karaman, and owe: n Submission, 11. reconstruct n=4 images k-space vector of n images 19

20 Methods y U PP ( O ) O P u S n a k f where f ( f1,..., f n )' O k a Pu U O P S O are coil k-space is k-space preprocessing is adj. inverse Fourier matrix,, P, permutation matrices SENSE unfolding matrix is image space preprocessing f P O = P P a O 1 1 k row row S m adjusted for and for T 2 k-space vector censor blip points row reverse permute B mage smoothing Bruce, Karaman, and owe: n Submission, 11.

21 Methods Statistical Expectation and ovariance. f E(f)=f, then for Mf, E(Mf)=Mf. f cov(f)=γ, then for Mf, cov(mf)=mγm'. This means that with y Of, E( y) Of and cov( y) OO ' co r( ) D D 1/2 1/2 2 2 So even if, it is not necessarily true that! This has H fm noise and fcm connectivity implications! 2px2p Nencka, Hahn, owe: JNSM, 181: , 9. Nencka, owe: OHBM, 7. 21

22 esults Since y Of, we inverted and made the n coil spatial frequencies from ( ) T 1 T O O O v f where O and v are known v is true/noiseless Shepp-Logan phantom (scaled by 5) O S P UP P ( ) m U S n The number of coils, n, and the reduction factor, A, are specified in the dimensions of operators, O. Bruce, Karaman, and owe: n Submission,

23 y Of esults 1 Noiseless data ( T T f O O) O v generated for N X =N Y =96, n=4, A=3 O S P UP P ( ) m U S n O had diagonal blocks Sensitivities, S coil 1 coil 4 coil 2 coil 3 Bruce, Karaman, and owe: n Submission, 11. U ( S S ) S T 1 1 T 1 j j j j Markovian coil covariance, Ψ Not skew-symmetric

24 esults Gaussian Smoothing applied in image-space - FWHM = 3 voxels, -Normalized to leave variance unaffected (Scales mean by 4.516) By definition, smoothing induces a covariance and correlation between voxels and their neighbors. This effect is in turn transferred to the correlated voxels from each fold in SENSE. Gaussian smoothing kernel, S m, was applied in image-space to reconstructed images. Bruce, Karaman, and owe: n Submission, 11. Nencka et al.: A Mathematical Model for Understanding the STatistical effects of k-space (AMMUST-k) Preprocessing Operators on Observed Voxel Measurements in fcm and fm. Journal of Neuroscience Methods 181: , 9. 24

25 y-axis y-axis owe 5 esults Magnitude Phase x-axis Ghosting because symmetric coil cov Ψ used Alternatice symmetric coil cov Ψ proposed. Phase is important in complex-valued fm! x-axis -3 25

26 esults x5 image cor = 25x25 correlation matrix x5 correlation image 26

27 esults N X =96 N Y =96 n = 4 A = Fold enter voxel orrelations induced about the center voxel. real correlation SE imag correlation SE real imag Functional connectivity implications 96 Fold real/imag correlation SE mage 1 96 mag 2 correlation SE mage 1 real/imag mag TH= mage 1 Bruce, Karaman, and owe: n Submission, mage 1 27

28 esults Phantom Human Fold Fold enter voxel enter voxel Fold Fold underlay artificially expanded Extrapolate to human, mistakenly conclude regions correlated! 28

29 Discussion The SENSE image reconstruction method was described. Wrote SENSE reconstruction with an isomorphism y O P UP P ( O ) f Of. U S n k The new mean E( y) Of and covariance of complex-valued SENSE described. OO' Theoretical results of SENSE reconstruction presented. Ghosting present in SENSE magnitude and phase images. nduced correlation between folds of no biological origin. 29

30 Thank You Acknowledgements: ain Bruce, Marquette University Muge Karaman, Marquette University Sweet 16! vs. Go Marquette! 3