A least squares quantization table method for direct reconstruction of MR images with non-Cartesian trajectory

The direct Fourier transform method is a straightforward solution with high accuracy for reconstructing magnetic resonance (MR) images from nonuniformly sampled k-space data, given that the optimal density compensation function is selected and the underlying magnetic field is sufficiently uniform. The computation however is very time-consuming, making it impractical especially for large-size images. In this paper, the least squares quantization table (LSQT) method is proposed to accelerate the direct Fourier transform computation, similar to the recently proposed methods such as using look-up table (LUT) or equal-phase-line (EPL). With LSQT, all the image pixels are first classified into several groups where the Lloyd-Max quantization scheme is used to ensure the minimal classification error. The representative value of each group is stored in a small-size LSQT in advance to reduce the computational load. The pixels in the same group receive the same contribution, which is calculated only once for each group instead of for each pixel, resulting in the reduction of computation because the number of groups is far smaller than the number of pixels. Finally, each image pixel is mapped into the nearest group and its representative value is used to reconstruct the image. The experimental results show that the LSQT method requires far smaller memory size than the LUT method and fewer multiplication operations than the LUT and EPL methods. Moreover, the LSQT method can perform large-size reconstructions that achieve comparable or higher accuracy as compared to the EPL and gridding methods when the appropriate parameters are given. The inherent parallel structure also makes the LSQT method easily adaptable to a multiprocessor system.