Magnetic resonance fingerprinting (MRF) is a novel technique, which enables simultaneous multi-parameter mapping and tissue characterization [1]. A pulse sequence with varying acquisition parameters is applied to generate unique signal evolution for each tissue type, which is matched to a dictionary of simulated signal evolutions to recover the MR parameters of interest. MRF usually requires the acquisition of images at many time points to gain sufficient parameter encoding capabilities. Often undersampled data are acquired for each time point and matching is directly performed on these data, relying on the spatio-temporal incoherence of the sampling pattern. However, this can reduce the accuracy of the estimated parameter maps, especially in case of high undersampling. An iterative reconstruction alternately applying MRF matching and data consistency steps was proposed in [2], which shows significant improvement in the accuracy of MRF maps but is computationally expensive. In this work, we present an alternative reconstruction for undersampled MRF data based on matrix completion, which is performed entirely in k-space and has lower computational cost.

MRF data are inherently compressible due to the high correlation between images acquired at different time points. This compressibility has been previously utilized to improve the computational efficiency of MRF matching [3]. In other words, if we form a $$$t\times n$$$ matrix containing $$$n$$$ MR fingerprints with $$$t$$$ time points (e.g. the MRF dictionary), this matrix has a low rank. Therefore, we can define a compression matrix that projects the data to a lower dimensional space, without significant signal loss. Due to the linearity of the Fourier transform this compression matrix can be equivalently formulated in k-space. Since each point in k-space is a linear combination of all points in image space, the signal evolution of a single k-space location contains information about all tissues in the object. Furthermore, the low rank of the MRF dictionary indicates that the data can be described by a small number of temporal basis vectors, therefore our hypothesis is that only a small portion of k-space is sufficient to obtain a good compression matrix. This compression matrix enables data reconstruction from incomplete measurements using matrix completion. First we form a $$$t\times n$$$ calibration matrix $$$M_c$$$ containing the temporal signals for a fully sampled central part of k-space. The SVD of the matrix $$$M_c = U\Sigma V^H$$$, where $$$U$$$and $$$V$$$ are unitary matrices of sizes $$$t\times t$$$ and $$$n\times n$$$, respectively and $$$\Sigma$$$ contains the singular values of $$$M_c$$$. If $$$r$$$ is the rank of $$$M_c$$$, the matrix $$$M_c$$$ can be compressed to a size $$$r\times n$$$ by projecting it onto the subspace spanned by the first $$$r$$$ rows of $$$U$$$. Our main premise is that the complete data matrix $$$M$$$ also lives in the subspace defined by $$$U_r$$$ . We can recover the missing k-space data by iteratively applying two steps until convergence:

1) Projection onto the $$$r$$$ dimensional subspace spanned by $$$U_r$$$ :

$$M_{i+1} = U_r U_r^H M_i $$

2) Data consistency step:

$$M_{i+2} = M_0+M_{i+1}(1-R),$$

where $$$M_0$$$ is the measured undersampled k-space data and $$$R$$$ indicates the sampling positions.

To demonstrate the feasibility of the proposed method, MRF data were acquired in a phantom with a FISP-based MRF sequence [4] with 1000 time points and fully sampled Cartesian trajectory on a 1.5T Philips Achieva scanner. The k-space data were retrospectively undersampled by a factor of 5, using a uniform density Poisson disk sampling pattern in $$$ky-t$$$ space with 7 fully sampled central k-space lines. The compression matrix $$$U_r$$$ is computed from the 7x7 central part of k-space. As a first test, the fully sampled data is compressed using $$$U_r$$$ to investigate the accuracy of the low rank approximation. The undersampled MRF data were recovered by the matrix completion scheme described above. The image series from the fully sampled, undersampled, and reconstructed data were used as an input to an MRF parameter mapping, estimating T1, T2, and proton density maps, using a dictionary with 24000 elements (T1 range 100:2100ms, T2 range 10:610).

Fig.1 shows the excellent agreement between the measured signal evolution and the approximated signal using low rank matrix approximation for a single pixel of the phantom. Fig.2 shows the parameter maps obtained from the fully sampled, undersampled, and low rank reconstructed data. The proposed reconstruction shows very good agreement with the fully sampled data, while direct matching from undersampled data leads to significant deviation in the matched parameter values, especially for T1.The proposed method shows significant improvement in the parameter maps compared to the direct matching proposed in [1].