Mohammad Golbabaee^{1}, Carolin M. Pirkl^{2,3}, Marion I. Menzel^{3}, Guido Buonincontri^{4}, and Pedro A. Gómez^{3,5}

Deep learning (DL) has recently emerged to address the heavy storage and computation requirements of the baseline dictionary-matching (DM) for Magnetic Resonance Fingerprinting (MRF) reconstruction. Fed with non-iterated back-projected images, the network is unable to fully resolve spatially-correlated corruptions caused from the undersampling artefacts. We propose an accelerated iterative reconstruction to minimize these artefacts before feeding into the network. This is done through a convex regularization that jointly promotes spatio-temporal regularities of the MRF time-series. Except for training, the rest of the parameter estimation pipeline is dictionary-free. We validate the proposed approach on synthetic and in-vivo datasets.

Dictionary-matching
(DM) approaches proposed for MR Fingerprinting (MRF)^{1-4} do not scale
well to the growing complexity of the emerging multi-parametric quantitative
MRI problems^{5,6}. Deep learning (DL) methodologies are recently
introduced to overcome this problem^{7-9}. Time-series of Back-Projected
Images (BPI) are fed into a compact neural network which temporally processes voxel
sequences and approximates the DM step to output the parametric maps. Trained
by independently corrupted (i.i.d. Gaussian) noisy fingerprints, the network is
unable to correct for dominant spatially-correlated (aliasing) artefacts appearing
in BPIs in highly undersampled regimes. Also larger DL models aiming to learn spatio-temporal
data structures^{10,11 }are prone to overfitting due to the limited
access to properly large datasets of ground-truth parametric maps (i.e. spatial joint distributions
of the quantitative MR parameters) in practice. Further, such approaches build customized
de-noisers which require expensive re-training by changing sampling parameters
i.e. the forward model.

This abstract aims
to address these shortcomings by taking a dictionary-free compressed sensing
approach to spatio-temporally process data before feeding into a compact and
easily-trained network of the first type. Casted in a convex problem, the
spatial regularities of the MRF time-series are promoted by Total Variations^{12}
(TV) shrinkage and temporal structures are relaxed to low-rank subspace
constraints.

The undersampled
k-space measurements $$$Y\in \mathbb{C}^{m\times L}$$$ acquired across $$$L$$$ timeframes are first processed by solving the
following convex and dictionary-free regularized inverse problem:$$\hat{X} = \arg\min_X \| Y-\mathcal{A}(XV_s^H)\|_F^2 +\lambda \sum_{i=1}^S \|X_i\|_{TV} \,\,\,\, (P1)$$ in order to find $$$S\ll L$$$ principal/subspace images $$$X\in \mathbb{C}^{n\times S}$$$. Subspace bases $$$V_s\in \mathbb{C}^{L\times S}$$$ are the $$$S$$$ leading (left) SVD components of a large-size
MRF dictionary $$$D\in \mathbb{C}^{L\times d}, d\gg L$$$ used here for unsupervised
training^{2,13,14}. The forward operator $$$\mathcal{A}$$$ models the multi-coil sensitivities and the
per-frame subsampled 2D Fourier Transforms.

The subspace model is a
convex (in fact linear) relaxed representation of the temporal dictionary responses and when
accurate enough, it is computationally advantageous over the full image
representation $$$X^{full}\approx XV_s^H\in\mathbb{C}^{n\times L}$$$ because it reconstructs smaller objects and
promotes temporally low-rank structures. This prior alone is, however, insufficient
to obtain artefact-free solutions e.g. when using spiral readouts^{15,16}.
We additionally use TV regularization by choosing $$$\lambda>0$$$ to promote spatial (piecewise) smoothness
across recovered subspace images. Optimization (P1) can be efficiently solved
using the iterative shrinkage Algorithm 1 with momentum acceleration and
backtracking step-size^{17-19}.

Parameter maps
are estimated using the MRF-Net^{9} that is a 3-layer fully-connected
network (Figure 2). The MRF dictionary is only used for training and not during
parameter recovery. Fed with the iteratively reconstructed images $$$\hat{X}$$$, the MRF-Net processes each (normalized) voxel sequence and
outputs per-voxel quantitative parameters. The
MRF-Net has implicitly 4-layers by including the subspace projection that is
incorporated in solving (P1). Three other layers include nonlinearities in
order to approximate subspace dictionary matching. Thanks to this dimensionality-reduction,
MRF-Net requires far less units and training resources compared to the uncompressed
DL approaches^{7,8}.

Methods are
tested on a numerical brain phantom^{20}, and a healthy human brain (in-vivo
data was acquired on a 3T GE MR750w system, GE Healthcare, Milwaukee, WI,
using 8-channel receive-only head RF coil). Acquisition follows the Steady
State Precession (FISP) sequence that jointly encodes T1 and T2 values using
sinusoidal flip angle variations^{3}, fixed TR=10msec, TE=1.908msec, Tinv=18msec,
L=1000 repetitions, variable-density spiral sampling, 377 interleaves, 22.5x22.5cm2
FOV, 256x256 spatial resolution, 1.3mm in-plane resolution and 5mm slice
thickness).

We use the Extended Phase-Graph
formalism^{21} and simulate a dictionary of $$$d=113'640$$$ fingerprints for combinations of T1=[100:10:4000],
T2
=
[20:2:600] msec. Clean fingerprints are used for unsupervised subspace model learning
of sufficiently low-rank ($$$S=10$$$ here^{14}).
Further, fingerprints corrupted by additive white Gaussian noise (data augmentation
by factor 100) supervisedly train the dimension-reduced MRF-Net on a standard
CPU desktop^{9}. We adopt
a practical phase-alignment heuristic^{15} to de-phase dictionary atoms
(for training) and subspace images (the inputs). Complex-valued extensions^{8}
are also possible by extra training for randomly-generated complex phases. We find this treatment unnecessary in our experiments.

We
compare three methods for reconstructing subspace images before fed to the
MRF-NET: i) non-iterative BPI i.e., $$$\hat{X}:=\mathcal{A}^H(Y)V_s$$$ and iterative reconstructions
incorporating ii) only the low-rank (LR) subspace prior by solving P1 with $$$\lambda=0$$$, and iii) joint TV and subspace
spatio-temporal priors (LRTV) by solving P1 with an experimentally tuned $$$\lambda=2\times10^{-5}$$$. Note that the BPIs are the
first iteration of the LR. Figures 3 and 4 show reconstructed maps for
synthetic and in-vivo data. Undersampling
artefacts are visible in BPI+MRF-Net. The subspace iterations of LR+MRF-Net also
admit undesirable solutions with high-frequency artefacts due to the insufficient
measurements collected from the k-space corners in spiral readouts (for details
see^{15,16}). By adding
sufficient spatial regularization, the proposed LRTV+MRF-Net outputs
artefact-free maps within 8-12 iterations.

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