Fig. 1

Illustration of the well-known Swiss roll manifold: 2-manifold embedded in 3D space

Table 1

Several important symbols and their descriptions

Fig. 2

Illustration of the proposed manifold structured data regularization scheme

Fig. 3

(a) and (b) are the flip angles and repetition time patterns that were used in the experiment. (c), (d), (e) shows the spiral undersampling trajectory, pseudo radial Cartesian sampling mask, and the variable density spiral undersampling trajectory, respectively, used in one repetition time in the experiments

Fig. 4

Reconstructed parameter maps from 5% noisy undersampled measurements with the acquisition length of L=500

Fig. 5

Error maps between the reconstructed parameter maps and the reference maps corresponding to Fig. 4

Fig. 6

The SNRs (dB) of the reconstructed $\mathsf{\mathcal{X}}$ data by different methods using various acquisition length L and 5% noisy undersampled data

Table 2

The NMSEs of the reconstructed parameter maps by different methods using various acquisition lengths L

Fig. 7

Reconstructed parameter maps from in vivo undersampled measurements with the acquisition length of L=500 collected from a healthy volunteer

Fig. 8

Reconstructed parameter maps from in vivo undersampled measurements with the acquisition length of L=500 collected from a healthy volunteer

Table 3

T1 and T2 values (ms, mean ± standard deviation) of several brain tissues measured on the corresponding regions (blue boxes) in Fig.7

Table 4

T1 and T2 values (ms, mean ± standard deviation) of several brain tissues measured on the corresponding regions (blue boxes) in Fig.8

Fig. 9

SNR values of the reconstructed MRF data as functions of the parameters. Left: radial Cartesian trajectories; right: spiral trajectories

Table 5

The NMSEs of the reconstructed parameter maps using different patch sizes

Table 6

The NMSEs of the reconstructed parameter maps using different methods under various sampling patterns

Fig. 10

Analysis of consistency between fingerprint manifold and parameter manifold based on Laplacian matrix. Visualization of the Laplacian matrix L

Table 7

The computational time of different methods with non-Cartesian sampling patterns (in minutes)