A weighted least-squares method for nuclear magnetic resonance velocity imaging

The phase method for velocity measurements in NMR imaging with more than two velocity encoding steps is discussed. The weighted least-squares method takes into account the variation in the accuracy of phase calculations at a voxel with the size of the velocity encoding gradients. We choose the weights so that the method is equivalent to the method of maximum-likelihood for high signal-to-noise ratios. We propose a method of implementation to minimize the problem of phase wrapping. We also discuss the number of velocity encoding steps, the choice of step size, and signal averaging, to improve the reproducibility of velocity measurements. Standard deviation images for the velocity have been calculated and used to reduce velocity noise by thresholding the velocity image.