Fig.1

The acquisition strategy of stack of golden angle variable density spiral. The acquisition loop is in slice direction first. After spiral interleaves with a fixed angle being sampled in all the slices, in the time dimension, the next spiral interleaves which are rotated by a golden angle will be sampled

Fig.2

The reconstruction process of the proposed method with l0 minimization. In part (1), sensitivity maps are acquired from the full-sampled spiral data through time. In part (2), the acquired k-space data is sorted into several undersampled groups. In part (3), the sorted data is reconstructed to dynamic image series with the minimization problem in the middle. In the right black box, the approximation of l0 norm is expressed with a limitation, and the image series after the sparse transform (temporal total variation) are shown

Table 1

Algorithm for homotopic l0 minimization

Fig.3

Reconstruction results of simulated image (Grouping 4 interleaves for 1 frame, corresponding to 48 interleaves of full sampled track, undersampling rate = 12) and their difference maps (7×) with reference. The four columns show frame 12, frame 21, the difference map of frame 12 and the difference map of frame 21, respectively. Figure (a) shows the reference images of the simulation phantom. Figures (b)~(d) show the reconstruction results using TRACER, CS with l1 minimization and l0 minimization respectively. The boxes indicate the regions of the detailed structure in the phantom, and the regions are zoomed in and placed on the top left side of each image

Fig.4

Reconstruction results of simulated image (Grouping 2 interleaves for 1 frame, corresponding to 48 interleaves of full sampled track, undersampling rate=24) and their difference maps (7×) with reference. The four columns show frame 24, frame 42, the difference map of frame 24 and the difference map (7×) of frame 42 respectively. Figure (a) shows the reference images of the simulation phantom. Figures (b)~(d) show the reconstruction results using TRACER, CS with l1 minimization and l0 minimization respectively. The boxes indicate the regions of the detailed structure in the phantom, and the regions are zoomed in and placed on the top left side of each image

Fig.5

The simulated signal intensity curves in the small discs. These figures show the performance of different algorithms in a fixed undersampling rate (temporal resolution). (a) and (b) show the results with 12-fold undersampling rate and 24-fold undersampling rate respectively. The blue curve represents the reference of the signal intensity curve. The green, red, black curves represent the signal intensity curves resulting from TRACER, CS with l1 minimization and l0 minimization respectively

Fig.6

RMSE values of the reconstructed signal intensity curves relative to the reference image in the small discs with 12-fold and 24-fold undersampling rate. The results from TRACER, CS with l1 minimization and l0 minimization are shown respectively

Fig.7

Reconstruction results of the non-contrast-enhanced liver image series (undersampling rate = 24). The first three columns show frame 20, frame 50, frame 80 and the fourth column shows the regions in the red boxes in frame 50 which are zoomed in. Rows (a), (b), (c) show the reconstruction results using TRACER, CS with l1 minimization and l0 minimization respectively