Compressed sensing for MRI

Variable density sampling with continuous trajectories. Application to MRI.

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Reducing acquisition time is a crucial challenge for many imaging techniques. Compressed Sensing (CS) theory offers an appealing framework to address this issue since it provides theoretical guarantees on the reconstruction of sparse signals by projection on a low dimensional linear subspace. In this paper, we focus on a setting where the imaging device allows to sense a fixed set of measurements. We first discuss the choice of an optimal sampling subspace (smallest subset) allowing perfect reconstruction of sparse signals. Its standard design relies on the random drawing of independent measurements. We discuss how to select the drawing distribution and show that a mixed strategy involving partial deterministic sampling and independent drawings can help breaking the so-called “coherence barrier”. Unfortunately, independent random sampling is irrelevant for many acquisition devices owing to acquisition constraints. To overcome this limitation, the notion of Variable Density Samplers (VDS) is introduced and defined as a stochastic process with a prescribed limit empirical measure. It encompasses samplers based on independent measurements or continuous curves. The latter are crucial to extend CS results to actual applications. Our main contribution lies in two original continuous VDS. The first one relies on random walks over the acquisition space whereas the second one is heuristically driven and rests on the approximate solution of a Traveling Salesman Problem. Theoretical analysis and retrospective CS simulations in magnetic resonance imaging highlight that the TSP-based solution provides improved reconstructed images in terms of signal-to-noise ratio compared to standard sampling schemes (spiral, radial, 3D iid…).

TSP

(a): Target distribution π to be approximated. Continuous random trajectories reaching distribution π based on Markov chains (b) and on a TSP solution (c). The latter is much more accurate.

Variable density sampling based on physically plausible gradient waveform. Application to 3D MRI angiography

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Performing k-space variable density sampling is a popular way of reducing scanning time in Magnetic Resonance Imaging (MRI). Unfortunately, given a sampling trajectory, it is not clear how to traverse it using gradient waveforms. In this paper, we actually show that existing methods can yield large traversal time if the trajectory contains high curvature areas. Therefore, we consider here a new method for gradient waveform design which is based on the projection of unrealistic initial trajectory onto the set of hardware constraints. Next, we show on realistic simulations that this algorithm allows implementing variable density trajectories resulting from the piecewise linear solution of the Traveling Salesman Problem in a reasonable time. Finally, we demonstrate the application of this approach to 2D MRI reconstruction and 3D angiography in the mouse brain.

cs1

Full k-space acquisition with an EPI sequence (a) and corresponding reference image (f). Comparison between an exact parameterization of the TSP trajectory (b) and projection from Traveling Salesman Problem trajectory onto the set of constraints (c),(d). In experiments (b,c), the number of measured locations is fixed to 9\% (r = 11.2), whereas in (b,d), the time to traverse the curve is fixed to 62 ms. (e): Spiral trajectory with acquisition of the k-space center. (g-j): Reconstructed images corresponding to sampling strategies (b-e).

A projection method on measures sets.

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We consider the problem of projecting a probability measure π on a set MN of Radon measures. The projection is defined as a solution of a variational problem. To motivate and illustrate our study, we show that this problem arises naturally in various practical image rendering problems such as stippling (representing an image with N dots) or continuous line drawing (representing an image with a continuous line). We provide a necessary and sufficient condition on the sequence of measure that ensures weak convergence of the projections to π. We then provide a numerical algorithm to solve a discretized version of the problem and show several illustrations related to computer-assisted synthesis of artistic paintings/drawings.

 lionProjection of a lion image onto a set of N = 8, 000 measures. The figure depicts the resulting line with several values
of the iterates of our Algorithm.

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