Research On The NUFFT Algorithm For The Spiral Magnetic Resonance Reconstruction

With the development of medical level and computer technology, magneticresonance imaging technique is becoming one indispensable part of the diseasediagnosis. Compared with computer tomography (CT), MRI can provide images withhigh soft tissue contrast and spatial resolution and without harmful radiation, andmeasure the blood changes of the heart and the vessels. Unfortunately, MR imagingtime is too long to satisfy the need for some special patients, such as seriously ill orincoordinate patients. This prevents the development and application of MRItechnology in the medical clinical tomography.In order to enhance the clinical application of MRI technique and reduce thescan-time, Meyer etc. successfully acquired the spiral trajectory of K-space samplingdata in 1992. This sampling mode is extremely valuable for functional brain imagingand cardiac imaging because of its high efficiency in the use of power gradient andfast speed imaging and its immune to motion and flow artifacts.In the traditional MRI, FFT imaging technique is comprehensively applied. Butthe K-space data are non-uniform for the spiral trajectory of K-space sampling, andthus FFT technique can not be used directly in reconstruction. In order to deal withthis problem, the conventional method is to resample the data in the Cartesiancoordinate, and then the new data are reconstructed by 2D FFT. Consequently, how toresample the data is the key problem during reconstruction. One of famousapproaches is gridding, which using the acquired spiral data convoluted with thekernel function to interpolate and resample in the Cartesian coordinate.Non-uniform fast Fourier (NUFFT) algorithm is one of the gridding algorithmsproposed by Dutt and Rokhlin in 1993. The main idea of this algorithm is that anypoint value in the non-uniform frequency space can be approximately expressed bythe adjacent points in the uniform space. The process of the approximation is the interpolation by the convolution kernel function. The NUFFT algorithm in the leastsquares sense, proposed by Liu and Nguyen, was performed to analyze unequallyspaced electromagnetic spectral wave in one dimension. In this algorithm, a regularFourier matrix was used to generate a convolution kernel matrix for each non-uniformpoint. The optimal sampling was obtained in the least squares sense at a highprecision of the computation.Scaling factor is one of the important parameters of the kernel matrix. It decidesthe size, the shape and the property of the convolution kernel matrix, which in turndecides the quality of the reconstructed image. However, the optimal selection of thescaling factor is not determined at present. Liu and Nguyen presented three types ofscaling factors that are Gaussian, cosine and trivial scaling factor, respectively. Fromthe experimental results, the authors drew a conclusion that the images reconstructedby the kernel matrix based on cosine scaling factor have the least errors. In this paper,based on a careful review of the articles, we found that the kernel matrix based on thetrivial scaling factor was derived mistakenly by Liu and Nguyen and the conclusionthat the cosine was the optimal scaling factor of the three was wrong. On theframework of the original algorithm, the mistaken kernel matrix (based on the trivialscaling factor) has been corrected in this article and the accurate and strict deviation ispresented. Compared with the kernel matrix based on cosine scaling factor, thecomputation operations are simplified further and the computational complexity arereduced further.The NUFFT reconstruction algorithm based on trivial scaling factor is proposedin this article. In conclusion, simulated data as well as phantom show that significantimprovements image quality applying the corrected kernel matrix based on the trivialscaling factor, compared with the wrong kernel matrix. Furthermore, thereconstructed images have lower reconstruction errors than the NUFFT algorithmbased on the cosine factor. In addition, the computational speed is faster due toutilizing the simplified kernel matrix. This method is particularly advantageous that itcan be applicable for both the single-shot or multi-shot spiral data and the othernon-Cartesian K-space data. But, it indispensable to correspondingly changed thedensity compensation functions for the other types of data.