Maximum likelihood estimation for Rician distributed data in analytical q-ball imaging

Analytical q-ball imaging is widely used for reconstruction of orientation distribution function (ODF) using diffusion weighted MRI data. Estimating the spherical harmonic coefficients is a critical step in this method. Least squares (LS) is widely used for this purpose assuming the noise to be additive Gaussian. However, Rician noise is considered as a more appropriate model to describe noise in MR signal. Therefore, the current estimation techniques are valid only for high SNRs with Gaussian distribution approximating the Rician distribution. The aim of this study is to present an estimation approach considering the actual distribution of the data to provide reliable results particularly for the case of low SNR values. Maximum likelihood (ML) is investigated as a more effective estimation method. However, no closed form estimator is presented as the estimator becomes nonlinear for the noise assumption of the Rician distribution. Consequently, the results of LS estimator is used as an initial guess and the more refined answer is achieved using iterative numerical methods. According to the results, the ODFs reconstructed from low SNR data are in close agreement with ODFs reconstructed from high SNRs when Rician distribution is considered. Also, the error between the estimated and actual fiber orientations was compared using ML and LS estimator. In low SNRs, ML estimator achieves less error compared to the LS estimator.