Dynamic MRI using model-based deep learning and SToRM priors: MoDL-SToRM.

PURPOSE: To introduce a novel framework to combine deep-learned priors along with complementary image regularization penalties to reconstruct free breathing & ungated cardiac MRI data from highly undersampled multi-channel measurements. METHODS: Image recovery is formulated as an optimization problem, where the cost function is the sum of data consistency term, convolutional neural network (CNN) denoising prior, and SmooThness regularization on manifolds (SToRM) prior that exploits the manifold structure of images in the dataset. An iterative algorithm, which alternates between denoizing of the image data using CNN and SToRM, and conjugate gradients (CG) step that minimizes the data consistency cost is introduced. Unrolling the iterative algorithm yields a deep network, which is trained using exemplar data. RESULTS: The experimental results demonstrate that the proposed framework can offer fast recovery of free breathing and ungated cardiac MRI data from less than 8.2s of acquisition time per slice. The reconstructions are comparable in image quality to SToRM reconstructions from 42s of acquisition time, offering a fivefold reduction in scan time. CONCLUSIONS: The results show the benefit in combining deep learned CNN priors with complementary image regularization penalties. Specifically, this work demonstrates the benefit in combining the CNN prior that exploits local and population generalizable redundancies together with SToRM, which capitalizes on patient-specific information including cardiac and respiratory patterns. The synergistic combination is facilitated by the proposed framework.

Convex recovery of continuous domain piecewise constant images from nonuniform Fourier samples.

We consider the recovery of a continuous domain piecewise constant image from its non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities/edges of the image are localized to the zero level-set of a bandlimited function. This assumption induces linear dependencies between the Fourier coefficients of the image, which results in a two-fold block Toeplitz matrix constructed from the Fourier coefficients being low-rank. The proposed algorithm reformulates the recovery of the unknown Fourier coefficients as a structured low-rank matrix completion problem, where the nuclear norm of the matrix is minimized subject to structure and data constraints. We show that exact recovery is possible with high probability when the edge set of the image satisfies an incoherency property. We also show that the incoherency property is dependent on the geometry of the edge set curve, implying higher sampling burden for smaller curves. This paper generalizes recent work on the super-resolution recovery of isolated Diracs or signals with finite rate of innovation to the recovery of piecewise constant images.

MODEL-BASED FREE-BREATHING CARDIAC MRI RECONSTRUCTION USING DEEP LEARNED & STORM PRIORS: MODL-STORM.

We introduce a model-based reconstruction framework with deep learned (DL) and smoothness regularization on manifolds (STORM) priors to recover free breathing and ungated (FBU) cardiac MRI from highly undersampled measurements. The DL priors enable us to exploit the local correlations, while the STORM prior enables us to make use of the extensive non-local similarities that are subject dependent. We introduce a novel model-based formulation that allows the seamless integration of deep learning methods with available prior information, which current deep learning algorithms are not capable of. The experimental results demonstrate the preliminary potential of this work in accelerating FBU cardiac MRI.

Subspace aware recovery of low rank and jointly sparse signals.

We consider the recovery of a matrix X, which is simultaneously low rank and joint sparse, from few measurements of its columns using a two-step algorithm. Each column of X is measured using a combination of two measurement matrices; one which is the same for every column, while the the second measurement matrix varies from column to column. The recovery proceeds by first estimating the row subspace vectors from the measurements corresponding to the common matrix. The estimated row subspace vectors are then used to recover X from all the measurements using a convex program of joint sparsity minimization. Our main contribution is to provide sufficient conditions on the measurement matrices that guarantee the recovery of such a matrix using the above two-step algorithm. The results demonstrate quite significant savings in number of measurements when compared to the standard multiple measurement vector (MMV) scheme, which assumes same time invariant measurement pattern for all the time frames. We illustrate the impact of the sampling pattern on reconstruction quality using breath held cardiac cine MRI and cardiac perfusion MRI data, while the utility of the algorithm to accelerate the acquisition is demonstrated on MR parameter mapping.

STRUCTURED LOW-RANK RECOVERY OF PIECEWISE CONSTANT SIGNALS WITH PERFORMANCE GUARANTEES.

We derive theoretical guarantees for the exact recovery of piecewise constant two-dimensional images from a minimal number of non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities of the image are localized to the zero level-set of a bandlimited function, which induces certain linear dependencies in Fourier domain, such that a multifold Toeplitz matrix built from the Fourier data is known to be low-rank. The recovery algorithm arranges the known Fourier samples into the structured matrix then attempts recovery of the missing Fourier data by minimizing the nuclear norm subject to structure and data constraints. This work adapts results by Chen and Chi on the recovery of isolated Diracs via nuclear norm minimization of a similar multifold Hankel structure. We show that exact recovery is possible with high probability when the bandlimited function describing the edge set satisfies an incoherency property. Finally, we demonstrate the algorithm on the recovery of undersampled MRI data.