A Majorize-Minimize Framework for Rician and Non-Central Chi MR Images

Abstract

The statistics of many MR magnitude images are described by the non-central chi (NCC) family of probability distributions, which includes the Rician distribution as a special case. These distributions have complicated negative log-likelihoods that are nontrivial to optimize. This paper describes a novel majorize-minimize framework for NCC data that allows penalized maximum likelihood estimates to be obtained by solving a series of much simpler regularized least-squares surrogate problems. The proposed framework is general and can be useful in a range of applications. We illustrate the potential advantages of the framework with real and simulated data in two contexts: 1) MR image denoising and 2) diffusion profile estimation in high angular resolution diffusion MRI. The proposed approach is shown to yield improved results compared to methods that model the noise statistics inaccurately and faster computation relative to commonly-used nonlinear optimization techniques.

Diffusion‐relaxation correlation spectroscopic imaging: A multidimensional approach for probing microstructure

“…The data consistency constraint is used to enforce the fact that our estimated spectra should match the measured data reasonably well. We have measured data consistency using the ‐norm to simplify the optimization algorithm, and while the ‐norm implicitly assumes that the noise distribution is Gaussian (which, for magnitude images, is approximately valid at high SNR), it would be straightforward to more accurately model the Rician or non‐central chi signal distributions associated with low‐SNR magnitude images using the ‐norm optimization strategy described in . The data consistency constraint also uses the scalars t i to avoid fitting multidimensional correlation spectra to noise‐only voxels of the image.…”

Model‐based denoising in diffusion‐weighted imaging using generalized spherical deconvolution

“…However, none of them reflects the underlying noise distribution of typical diffusion MR data, that is, Rician or noncentral chi distributions. Although algorithms capable of handling such distributions exist, namely PML methods , we did not include them in our analysis due to their practical limitations (eg, difficulties in precisely estimating the noise level of the data, nonconvexity of the cost function, or excessive computation times). Moreover, when comparing the results of the three algorithms used with genSD, we did not observe a significant change in bias.…”

Greedy algorithms for nonnegativity-constrained simultaneous sparse recovery

“…It should be noted that P Ω is still relevant to such cases. Specifically, we have recently introduced a quadratic majorize-minimize procedure [50] that allows Rician and noncentral chi log-likelihoods to be maximized by iteratively solving a sequence of NNLS problems, where each NNLS problem has the same form as P Ω . This type of approach is easily adapted to our proposed NNS greedy algorithms.…”