Declaration of Conflict of Interest or Relationship David Atkinson: I have no conflicts of interest to disclose with regard to the subject matter of this presentation. Image Reconstruction: Motion Correction David Atkinson [email protected] Centre for Medical Image Computing, University College London with thanks to Freddy Odille, Mark White, David Larkman, Tim Nielsen, Murat Askoy, Johannes Schmidt. Problem: Slow Phase Encoding • Acquisition slower than physiological motion. – motion artefacts. • Phase encode FOV just large enough to prevent wrap around. – minimises acquisition time, – Nyquist: k-space varies rapidly making interpolation difficult. Motion and K-Space Fourier Transform sx Sk e image ikx k k-space acquired in time The sum in the Fourier Transform means that motion at any time can affect every pixel. K-Space Corrections for Affine Motion Image Motion K-Space Effect Translation (rigid shift) Phase ramp Rotation Rotation (same angle) Expansion Contraction General affine Affine transform [Guy Shechter PhD Thesis] Rotation Example Time Example rotation mid-way through scan. Ghosting in phase encode direction. Interpolation, Gridding and Missing Data FFT requires regularly spaced samples. Rapid variations of k-space make interpolation difficult. K-space missing in some regions. Prospective Motion Correction Motion determined during scan & plane updated using gradients. • • • • • Prevents pie-slice missing data. Removes need for interpolation. Prevents through-slice loss of data. Can instigate re-acquisition. Reduces reliance on post-processing. • Introduces relative motion of coil sensitivities, distortions & field maps. • Difficult to accurately measure tissue motion in 3D. • Gradient update can only compensate for affine motion. Non-Rigid Motion • Most physiological motion is non-rigid. • No direct correction in k-space or using gradients. • A flexible approach is to solve a matrix equation based on the forward model of the acquisition and motion. Forward Model and Matrix Solution Eρ m “Encoding” matrix with motion, coil sensitivities etc Measured data Artefact-free Image min Eρ m ρ 2 Least squares solution: Conjugate gradient techniques such as LSQR. The Forward Model as Image Operations Measured k-space for shot = sample shot FFT k i coil sensitivity motion motionfree patient Image transformation at current shot Multiplication of image by coil sensitivity map Fast Fourier Transform to k-space Selection of acquired k-space for current shot Shots single-shot EPI multi-shot spin echo 1 readout = 1 shot Forward Model as Matrix-Vector Operations Measured k-space for shot = sample shot m FFT k i coil sensitivity motion motionfree patient E * ρ Converting Image Operations to Matrices • The trial motion-free image is converted to a column vector. motion-free patient image ρ n n n2 Expressing Motion Transform as a Matrix Measured = image = sample FFT k i coil motion ? • Matrix acts on pixels, not coordinates. • One pixel rigid shift – shifted diagonal. • Half pixel rigid shift – diagonal band, width depends on interpolation kernel. • Shuffling (non-rigid) motion - permutation matrix. Converting Image Operations to Matrices • Pixel-wise image multiplication of coil sensitivities becomes a diagonal matrix. • FFT can be performed by matrix multiplication. • Sampling is just selection from k-space vector. patient Measured = image = sample FFT k i coil motion Stack Data From All Shots, Averages and Coils m E * ρ Conjugate Gradient Solution min Eρ m ρ • Efficient: does not require E to be computed or stored. • User must supply functions to return result of matrix-vector products Ev and EH w • We know the correspondence between matrixvector multiplications and image operations, hence we can code the functions. 2 The Complex Transpose EH H motion H coil H FFT H sample • Reverse the order of matrix operations and take Hermitian transpose. • Sampling matrix is real and diagonal hence unchanged by complex transpose. • FFT changes to iFFT. • Coil sensitivity matrix is diagonal, hence take complex conjugate of elements. • Motion matrix ... Complex transpose of motion matrix Options: • Approximate by the inverse motion transform. • Approximate the inverse transform by negating displacements. • Compute exactly by assembling the sparse matrix (if not too large and sparse). • Perform explicitly using for-loops and accumulating the results in an array. Example Applications of Solving Matrix Eqn averaged cine ‘sensors’ from central k-space lines input to coupled solver for motion model and artefactfree image. multi-shot DWI example phase correction artefact free image Summary: Forward Model Method • Efficient Conjugate Gradient solution. • Incorporates physics of acquisition including parallel imaging. • Copes with missing data or shot rejection. • Interpolates in the (more benign) image domain. • Can include other artefact causes e.g. phase errors in multi-shot DWI, flow artefacts, coil motion, contrast uptake. • Can be combined with prospective acquisition. • Often regularised by terminating iterations. • Requires knowledge of motion. Alternative Iterative Reconstruction uncorrected • Fourier-transform each interleave. • Initialize image: I=0 rotate +shift weight with the fold coil sensitivities coil 1 motion compensated measurement data - unfold combine updates rotate +shift coil 2 coil 3 + [Nielsen et al. #3048] Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models. Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models. ECG, respiratory bellows, optical tracking, ultrasound (#3961), spirometer (#1553), accelerometer (#1550). Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models. pencil beam navigator, central k-space lines, orbital navigators, rapid, low resolution images, FID navigators. Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models. repeated acq near k-space centre, PROPELLER, radial & spiral acquisitions, spiral projection imaging, Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. Predict and compare k-space lines. • Coil consistency. • Iterative methods. • Motion models. Detect and minimise artefact source to make multiple coil images consistent. Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models. Find model parameters to minimise cost function e.g. image entropy, coil consistency. Estimating Motion • External measures. • Explicit navigator measures. • Self-navigated sequences. • Coil consistency. • Iterative methods. • Motion models. Link a model to scan-time signal. Solve for motion model and image in a coupled system (GRICS). Example combined prospective and retrospective methods at ISMRM 2010 5 mm gating window 34.7 min 2x SENSE 20 mm window Motion corrected 2xSENSE 20 mm window 9 min 9 min Retrospective correction: motion model from low res images, LSQR solution, 6 iterations. Implicit SENSE allows undersampling [Schmidt et al #492] Example combined prospective and retrospective methods at ISMRM 2010 no correction prospective optical tracking additional entropy-based autofocus [Aksoy et al #499] Outlook • Reconstruction times, 3D and memory still challenging. • Expect intelligent use of prior knowledge: sparsity, motion models, atlases etc. • Optimum solution target dependent. Power in combined acquisition and reconstruction methods. www.ucl.ac.uk/cmic Physiological Motion can be useful • Functional information – cardiac wall motion, bowel motility. • Elastography. • Randomising acquisition for compressed sensing reconstruction.