Reordering for improved constrained reconstruction from undersampled k-space data

Abstract

Recently, there has been a significant interest in applying reconstruction techniques, like constrained reconstruction or compressed sampling methods, to undersampled k-space data in MRI. Here, we propose a novel reordering technique to improve these types of reconstruction methods. In this technique, the intensities of the signal estimate are reordered according to a preprocessing step when applying the constraints on the estimated solution within the iterative reconstruction. The ordering of the intensities is such that it makes the original artifact-free signal monotonic and thus minimizes the finite differences norm if the correct image is estimated; this ordering can be estimated based on the undersampled measured data. Theory and example applications of the method for accelerating myocardial perfusion imaging with respiratory motion and brain diffusion tensor imaging are presented.

Figure 1

(a) A fully sampled smoothly varying 1D signal and the corresponding signal reconstructed using IFT from its incomplete Fourier data undersampled by a factor of two (R~2) in a random fashion. (b) A fully sampled nonsmooth varying 1D signal and the corresponding signal reconstructed using IFT from its R~2 Fourier data undersampled in a random fashion. (c) Comparison of the original fully sampled smooth signal and the reconstructed signal from R~2 Fourier data without reordering. (d) Comparison of the original fully sampled nonsmooth signal and the corresponding signal reconstructed from R~2 Fourier data without reordering. (e) Comparison of the original fully sampled nonsmooth signal and the corresponding sorted signal. (f) Comparison of the original fully sampled nonsmooth signal and the signal reconstructed from R~2 Fourier data with reordering.

Figure 2

Comparison of sparsity of the fully sampled original nonsmooth signal in Figure 1(b) and that of the corresponding sorted signal in terms of finite differences.

Figure 3

Comparison of optimal regularization weights without and with reordering. L-curves obtained for reconstruction of the nonsmooth signal in Figure 1(b) from R~2 Fourier data (a) without reordering and (b) with reordering overlaid by the L-curve in Figure 3(a).

Figure 4

Comparison of errors in the reconstruction for the nonsmooth signal in Figure 1(b) without reordering and with reordering as a function of inaccuracies in the ordering.

Figure 5

Reordering method for 2D images. (a) Simulated piecewise constant heart image. (b) Image reconstructed using IFT from ~15% of the full Fourier data, undersampled in a variable density random fashion. (c) Image reconstructed from undersampled data using a TV spatial constraint. (d) Actual MR magnitude image of the short-axis slice of a heart at a single point in a perfusion sequence reconstructed from fully sampled k-space data using IFT. (e) Corresponding IFT reconstruction from R~3 k-space data undersampled in VD random fashion. (f) Reconstruction using a 2D TV constraint without any reordering. (g) Row-reordered image of the real part of the complex MR image of the heart. (h) Reconstructed image with spatial reordering using a TV constraint. Ordering of the data here was obtained using the image reconstructed from fully sampled data.

Figure 6

Comparison of errors in the reconstruction for the actual MR heart image in Figure 5(d) without reordering and with reordering as a function of perturbations in the exact spatial ordering.

Figure 7

Results of multi-image reordering method on dynamic phantom data. Image at a time point reconstructed (a) from full k-space data using IFT, (b) from R~3 data using STCR without any reordering, and (c) from R~3 data using STCR with reordering in time and spatial dimensions. (d) Absolute difference image between Figures 7(a) and 7(b). (e) Absolute difference image between Figures 7(a) and 7(c).

Figure 8

Result of multi-image reordering method on dynamic myocardial perfusion data. (a) Images at two different time points in the sequence reconstructed from full k-space data (first column). (b) Corresponding images reconstructed from R~2.5 k-space data, undersampled in variable density random fashion, using constrained reconstruction method in (4) but without any reordering (second column). The arrows point to the residual artifacts in the images. (c) Corresponding images reconstructed from R~2.5 k-space data using constrained reconstruction method in (4) with reordering (third column).

Figure 9

Result of the reordering method on multi-image brain DTI data. (a) Image of a single diffusion encoding direction reconstructed from full Fourier data. A line for comparison of pixel intensity profiles for different reconstructions is also shown. (b) Corresponding encoding direction reconstructed from R~3 Fourier data, undersampled in variable density random fashion, using constrained reconstruction in (4) but without any reordering. (c) Corresponding direction reconstructed from the incomplete Fourier data using constrained reconstruction in (4) with reordering. (d) Comparison of intensity line profiles for images in Figures 9(a) and 9(b). (e) Comparison of intensity line profiles for images in Figures 9(a) and 9(c).

Figure 10

(a) L-curve obtained for reconstruction of the nonsmooth signal in Figure 1(b) from R~2 Fourier data with reordering, but with large number (~65%) of random perturbations in the exact ordering. (b) L-curve obtained for reconstruction of the nonsmooth signal in Figure 1(b) from R~2 Fourier data with reordering with fewer (~21%) random perturbations in the exact ordering. The L-curve in Figure 10(a) is also overlaid.