Declaration of Relevant Financial Interests or Relationships David Atkinson: I have no relevant financial interest or relationship to disclose with regard to the subject matter of this presentation.

Image Reconstruction: Motion Correction David Atkinson Centre for Medical Imaging, Division of Medicine, University College London

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.

Any Motion Can Corrupt Entire Image k-space acquired in time image s(x) Fourier Transform The sum over all k in the Fourier Transform means that motion at any time can affect every pixel.

K-Space Corrections for Affine Motion Image Motion Translation (rigid shift) Rotation Expansion General affine K-Space Effect Phase ramp Rotation (same angle) Contraction Affine transform

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 “Encoding” matrix with motion, coil sensitivities etc Measured data Artefact-free Image Least squares solution: Conjugate gradient techniques such as LSQR.

The Forward Model as Image Operations motion- free patient motion coil sensitivity sample shot = Measured k-space for shot FFT k i 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 spin echo 1 readout = 1 shot single-shot EPI multi-shot

Forward Model as Matrix-Vector Operations motion- free patient motion coil sensitivity sample shot = Measured k-space for shot FFT k i *

Converting Image Operations to Matrices The trial motion-free image is converted to a column vector. n n n2n2 motion-free patient image

Expressing Motion Transform as a Matrix ? = motion coil FFT sample Measured image k i = 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 = motion coil FFT sample Measured image k i =

Stack Data From All Shots, Averages and Coils *

Conjugate Gradient Solution Efficient: does not require E to be computed or stored. User must supply functions to return result of matrix-vector products We know the correspondence between matrix- vector multiplications and image operations, hence we can code the functions.

The Complex Transpose E H 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... motion FFT HHHH coil sample

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 artefact- free 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.

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 probes spirometers accelerometers Power deposition Field Probes

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. Coil consistency. Iterative methods. Motion models. Predict and compare k-space lines. 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).

Golden Angle Sampling For Motion Detection and High Resolution Imaging Determine motion from registration of images (each from wide temporal range). Use motion in CG high resolution reconstruction. [Hansen et al #749]

Outlook Prospective corrections limited to affine motion. 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.