The fiber orientation density function (fODF) provides detailed information about the microgeometry of axons within individual white matter imaging voxels [1,2]. It can be estimated with diffusion MRI (dMRI) by using any of a variety of methods [[1], [2], [3], [4], [5], [6], [7], [8], [9]] and plays an important role in fiber tractography [10] and microstructural modeling [2,8,9,[11], [12], [13], [14]]. The fODF is quantified as a function F(u) over a spherical surface, where u is a unit direction vector, and it can be interpreted as the probability of a randomly selected water molecule from the intra-axonal compartment being contained inside an axon oriented parallel to u [2,9]. Since negative probabilities are meaningless, physically realistic fODFs must be nonnegative. However, several dMRI methods generate fODFs that can take on negative values in some directions [[1], [2], [3],5,6,8,9]. This is particularly true when the fODF is represented, as is commonly done, with a spherical harmonic expansion, in which case Gibbs ringing due to truncation of the expansion and signal noise may both cause negative fODF values to occur. For this reason, it is often desirable to rectify an fODF, as the final step in its calculation, to guarantee nonnegative values for all directions.

A general method for constructing a rectified fODF that is as close as possible to an original unrectified fODF, in the mean square sense, has been described in prior work [15]. The method is straightforward to implement, simply requiring that a single parameter be determined numerically from the root of an equation constructed from the original fODF. An added benefit of this optimized rectification scheme is that fODF artifacts, such as spurious peaks due to ringing and noise, are attenuated. Nevertheless, significant artifacts may remain even after rectification that can be mistaken for true biological features and hamper comparison of fODFs for voxels from different brain regions or different subjects.

In this paper, we describe an extension of this prior rectification procedure that allows fODF artifacts to be further suppressed. The key idea is to impose an additional constraint that requires the rectified fODF for all directions in which the original fODF is less than a specified threshold, η, to be absorbed into a constant background. In this way, peaks and other fODF features smaller than η are eliminated. Here an exact solution of this extended optimization problem is given for minimizing the mean square distance between the rectified and original fODFs so that the fODF is both nonnegative and satisfies the added threshold condition. The qualitative features of this solution are similar to those for the prior rectification problem, which is recovered by setting η to zero. The solution is simple to calculate and preserves the position and shapes of all features not absorbed into the background. This rectification method constitutes a form of fODF preprocessing similar in spirit to the type of image preprocessing routinely applied to dMRI data, with the fODF itself playing the role of an MRI image.

The main motivation for this work is to support investigation of the fine structure of fODFs rather than to aid fiber tractography, for which it is mainly the peak directions that are employed. While fiber tractography has heretofore been the predominant application of fODFs [10], improvements in scanner hardware and dMRI data processing together with recent theoretical insights are making it feasible to obtain more accurate and detailed representations for fODFs [2], thereby opening up new avenues of exploration of white matter microstructure. For such applications, simple expedients, such as setting all fODF values below a given threshold to zero, would distort fODFs more than the optimized rectification procedure proposed here.

We illustrate optimized rectification with a background threshold for an analytic fODF model based on a Watson distribution and for human dMRI data acquired at 3 T using a b-value of 8000 s/mm2. For the human data, the original fODFs are obtained with fiber ball imaging (FBI), in which the fODF is found from an inverse generalized Funk transform of the dMRI signal [2,8,16]. The effects of different choices of the background threshold are demonstrated for the number of fODF peaks and for an axon-specific diffusion anisotropy.