Reconstruction of MR images from data acquired on an arbitrary k-space trajectory using the same-image weight

Fast direct fourier reconstruction of radial and PROPELLER MRI data using the chirp transform algorithm on graphics hardware

PURPOSE

To develop and test a new algorithm for fast direct Fourier transform (DrFT) reconstruction of MR data on non-Cartesian trajectories composed of lines with equally spaced points.

THEORY AND METHODS

The DrFT, which is normally used as a reference in evaluating the accuracy of other reconstruction methods, can reconstruct images directly from non-Cartesian MR data without interpolation. However, DrFT reconstruction involves substantially intensive computation, which makes the DrFT impractical for clinical routine applications. In this article, the Chirp transform algorithm was introduced to accelerate the DrFT reconstruction of radial and Periodically Rotated Overlapping ParallEL Lines with Enhanced Reconstruction (PROPELLER) MRI data located on the trajectories that are composed of lines with equally spaced points. The performance of the proposed Chirp transform algorithm-DrFT algorithm was evaluated by using simulation and in vivo MRI data.

RESULTS

After implementing the algorithm on a graphics processing unit, the proposed Chirp transform algorithm-DrFT algorithm achieved an acceleration of approximately one order of magnitude, and the speed-up factor was further increased to approximately three orders of magnitude compared with the traditional single-thread DrFT reconstruction.

CONCLUSION

Implementation the Chirp transform algorithm-DrFT algorithm on the graphics processing unit can efficiently calculate the DrFT reconstruction of the radial and PROPELLER MRI data.

Convolution kernel design and efficient algorithm for sampling density correction

Sampling density compensation is an important step in non-cartesian image reconstruction. One of the common techniques to determine weights that compensate for differences in sampling density involves a convolution. A new convolution kernel is designed for sampling density attempting to minimize the error in a fully reconstructed image. The resulting weights obtained using this new kernel are compared with various previous methods, showing a reduction in reconstruction error. A computationally efficient algorithm is also presented that facilitates the calculation of the convolution of finite kernels. Both the kernel and the algorithm are extended to 3D.

Basis function cross-correlations for Robustk-space sample density compensation, with application to the design of radiofrequency excitations

The problem of k-space sample density compensation is restated as the normalization of the independent information that can be expressed by the ensemble of Fourier basis functions corresponding to the trajectory. Specifically, multiple samples (complex exponential functions) may be contributing to each independent information element (independent basis function). Normalization can be accomplished by solving a linear system based on the cross-correlation matrix of the underlying Fourier basis functions. The solution to this system is straightforward and can be obtained without resorting to discretization since the cross-correlations of Fourier basis functions are analytically known. Furthermore, no restrictions are placed on the k-space trajectory and its point-spread function. Additionally, the linear system can be used to elucidate key trade-offs involved in k-space trajectory design. The approach can be used to compensate samples acquired for image reconstruction or designed for low flip angle radiofrequency (RF) excitation. Here it is demonstrated for the latter application, using reversed spiral trajectories. In this case the linear system approach enables one to easily incorporate additional constraints such as smoothness to the solution. For typical RF excitation durations (<20 ms) it is shown that density compensation can even be achieved without numerical iteration.

Iterative Next-Neighbor Regridding (INNG): improved reconstruction from nonuniformly sampled k-space data using rescaled matrices

The reconstruction of MR images from nonrectilinearly sampled data is complicated by the fact that the inverse 2D Fourier transform (FT) cannot be performed directly on the acquired k-space data set. k-Space gridding is commonly used because it is an efficient reconstruction method. However, conventional gridding requires optimized density compensation functions (DCFs) to avoid profile distortions. Oftentimes, the calculation of optimized DCFs presents an additional challenge in obtaining an accurately gridded reconstruction. Another type of gridding algorithm, the block uniform resampling (BURS) algorithm, often requires singular value decomposition (SVD) regularization to avoid amplification of data imperfections, and under some conditions it is difficult to adjust the regularization parameters. In this work, new reconstruction algorithms for nonuniformly sampled k-space data are presented. In the newly proposed algorithms, high-quality reconstructed images are obtained from an iterative reconstruction that is performed using matrices scaled to sizes greater than that of the target image matrix. A second version partitions the sampled k-space region into several blocks to avoid limitations that could result from performing multiple 2D-FFTs on large data matrices. The newly proposed algorithms are a simple alternative approach to previously proposed optimized gridding algorithms.

Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities

In magnetic resonance imaging, magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast Fourier trasform (FFT) methods. Several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities, but these methods assume that the field map that characterizes the off-resonance frequencies is spatially smooth. Recently, iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off-resonance effects. However, straightforward implementations of such iterative methods suffer from inconveniently long computation times. This paper describes a tool for accelerating iterative reconstruction of field-corrected MR images: a novel time-segmented approximation to the MR signal equation. We use a min-max formulation to derive the temporal interpolator. Speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast Fourier transform with normalized root mean squared approximation errors of 0.07%. The proposed method provides fast, accurate, field-corrected image reconstruction even when the field map is not smooth.