Least squares for diffusion tensor estimation revisited: Propagation of uncertainty with Rician and non-Rician signals

Least Squares (LS) and its minimum variance counterpart, Weighted Least Squares (WLS), have become very popular when estimating the Diffusion Tensor (DT), to the point that they are the standard in most of the existing software for diffusion MRI. They are based on the linearization of the Stejskal-Tanner equation by means of the logarithmic compression of the diffusion signal. Due to the Rician nature of noise in traditional systems, a certain bias in the estimation is known to exist. This artifact has been made patent through some experimental set-ups, but it is not clear how the distortion translates in the reconstructed DT, and how important it is when compared to the other source of error contributing to the Mean Squared Error (MSE) in the estimate, i.e. the variance. In this paper we propose the analytical characterization of log-Rician noise and its propagation to the components of the DT through power series expansions. We conclude that even in highly noisy scenarios the bias for log-Rician signals remains moderate when compared to the corresponding variance. Yet, with the advent of Parallel Imaging (pMRI), the Rician model is not always valid. We make our analysis extensive to a number of modern acquisition techniques through the study of a more general Non Central-Chi (nc-χ) model. Since WLS techniques were initially designed bearing in mind Rician noise, it is not clear whether or not they still apply to pMRI. An important finding in our work is that the common implementation of WLS is nearly optimal when nc-χ noise is considered. Unfortunately, the bias in the estimation becomes far more important in this case, to the point that it may nearly overwhelm the variance in given situations. Furthermore, we evidence that such bias cannot be removed by increasing the number of acquired gradient directions. A number of experiments have been conducted that corroborate our analytical findings, while in vivo data have been used to test the actual relevance of the bias in the estimation.