Non-quadratic convex regularized reconstruction of MR images from spiral acquisitions
Introduction
In magnetic resonance imaging (MRI) the acquired data are samples of the Fourier transform of the imaged object [1]. Acquisition is often discussed in terms of location in k-space and most conventional methods collect data on a regular Cartesian grid. This allows for a straightforward characterization of aliasing and Gibbs artifacts, and permits direct image reconstruction by means of 2D-fast Fourier transform (FFT) algorithms. Other acquisition sequences, such as spiral [2], PROPELLER [3], projection reconstruction, i.e. radial [4], rosette [5], collect data on a non-Cartesian (NC) grid. They possess many desirable properties, including reduction of the acquisition time and of various motion artifacts. The gridding procedure associated to an FFT is the most common method for Cartesian image reconstruction from such irregular k-space acquisitions.
Re-gridding data from NC locations to a Cartesian grid has been addressed by many authors. O’Sullivan [6] introduced a convolution-interpolation technique in computerized tomography (CT) which can be applied to MRI [2]. He suggested not to use a direct reconstruction, but to perform a convolution-interpolation of the data sampled on a polar pattern onto a Cartesian k-space. The final image was obtained by FFT. The stressed advantage of this technique was the reduction of computational complexity compared to the filtered back-projection technique. Moreover, it can be applied to any arbitrary trajectory in k-space.
More generally, the reconstruction process is four steps:
- 1.
data weighting for non-uniform sampling compensation,
- 2.
re-sampling onto a Cartesian grid, using a given kernel,
- 3.
computation of the FFT,
- 4.
correction for the kernel apodization.
Jackson et al. [7] precisely discussed criteria to choose an appropriate convolution kernel. This is necessary for accurate interpolation and also for minimization of reconstruction errors due to uneven weighting of k-space. Several authors have suggested methods for calculating this sampling density. Numerical solutions have been proposed that iteratively calculate the compensation weights [3]. But, for arbitrary trajectories, the weighting function is not known analytically and must somehow be extracted from the sampling function itself. A possible solution is to use the area of the Voronoi cell around each sample [8].
The gridding method is computationally efficient. However, convolution-interpolation methods unavoidably introduce artifacts in the reconstructed images [8]. Indeed, for a given kernel the convolution modifies data in k-space and it is difficult to know the exact effect of gridding in the image domain. Moreover, this method tends to correlate the noise in the measured samples and lacks solid analysis and design tools to quantify or minimize the reconstruction errors.
The principle of regularized reconstruction has been described by several authors for parallel imaging: [9], [10] and more recently [11] proposed the use of a general reconstruction method for sensitivity encoding (SENSE) [12] which has been applied with a quadratic regularization term and a Cartesian acquisition scheme. In this paper, we extend this work by: (1) giving a more general formulation of the reconstruction term for NC trajectories, (2) specifically using the exact non-uniform locations of the acquired data in k-space, without the need for gridding the data to a uniform Cartesian grid and, (3) incorporate a non-quadratic convex regularization term in order to maintain edge sharpness compared to a purely quadratic term. The regularization term represents the prior information about the imaged object that improves the signal-to-noise ratio (SNR) of the reconstructed image as well as the spatial resolution.
In Section 2, we recall the basis of MRI signal acquisition and the modelling of the MR acquisition process. Then we address the image reconstruction methods for different acquisition schemes and develop the proposed method, in Section 3. The reconstruction is based on the iterative optimization of a discrete Fourier transform (DFT) regularized criterion. Rewriting this criterion allows to reduce the complexity of the computation and to decrease the reconstruction time. Finally, Section 4 compares the proposed method and the gridding reconstruction for simulated and real sparse data acquired along interleaved spiral trajectories.
Section snippets
Direct model
MRI theory [1] indicates that the acquired signal s is related to the imaged object f through:in a 2D context. D is the field of view, i.e., the extent of the imaged object, is the spatial vector and (“t” denotes a transpose) is the k-space trajectory. Thus, the received signal can be thought as the Fourier transform of the object, along a trajectory determined by the magnetic gradient field : The modulus
Model inversion
A usual inversion method relies on a least squares (LS) criterion, based on Eq. (3):The reconstructed image is the minimizer of : and minimizes the quadratic error between the measured data and the estimated ones generated by the direct model (3). The solution writes if is invertible, property that depends on the acquisition scheme.
Simulation and acquisition results
In this section the proposed reconstruction method is compared to the gridding method on a mathematical model and a real phantom both acquired using a spiral sequence.
Discussion and conclusion
The proposed method differs from more conventional ones insofar as it does not involve any regridding of the acquired data and accounts for edge preserving smoothing penalties. Utilization of only the acquired data and integration of smoothness and edge preservation penalization in the reconstruction opens the way to strong improvement in MRI.
From a computational standpoint, the original formulation leads to the awkward situation of an optimization algorithm permanently shifting from Fourier to
Acknowledgements
The authors express their gratitude to M.J. Graves, University of Cambridge and Addenbrooke's Hospital, Cambridge, UK, for providing the acquisitions, fundamental for proposed evaluations.
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