To estimate the orentation of fibers in each voxel of a HARDI brain dataset, orientation distribution functions (ODFs) are estimated from the HARDI signal. ODFs are probability distributions that live on the unit sphere and therefore must be nonnegative. Our work on ODF estimation enforces nonnegativity not only computationally on discrete grid points [1] but theoretically on the entire continuous spherical domain [2,3]. Enforcing nonnegativity is important for estimating accurate ODFs used for downstream processes of tractography, registration, segmentation and disease classification. In our most recent work, we consider that to enforce nonnegativity everywhere on a continuous domain, this equates to an infinite number of optimiztion contraints which is intractable. But even with a large number of constraints, many will be redudant. In fact, if we knew the minimum of the function, a priori, only one nonnegativity constraint at this point would suffice to enforce nonnegativiety everywhere. The work of [3] constructs an algorithm to select the optimal constraint needed to enforce ODF nonnegativity everywhere.

Figure 1: Illustration of nonnegativity enforcement on discrete set of points may leave continous functions negative. This calls for algorithms to enforce nonnegativity on the coninuous domain.

Reducing the amount of information stored in diffusion MRI (dMRI) data to a set of meaningful and representative scalar values is a goal of much interest in medical imaging. Such features can have far reaching applications in segmentation, registration, and statistical characterization of regions of interest in the brain. In fact, extracting scalar features from dMRI data has become an integral part of group/longitudinal studies of changes in brain connectivity related to development, neurodegeneration, or disease. Common features used in DTI are FA and MD and GFA for HARDI. However, existing features discard much of the information inherent in dMRI and embody several theoretical shortcomings and physical ambiguities. In this work [4] we propose a new framework for extracting a large set of rotation invariant features from the spherical harmonic (SH) representation of HARDI signals and ODFs. The advantage of this framework is its generality. In fact, we derive a family of rotation invariant features that can be extracted from any spherical function written in an SH basis. Numerous HARDI reconstruction methods such as Spherical Deconvolution (SD), Diffusion Orientation Transform (DOT), Spherical Polar Fourier Imaging (SPFI), Bessel Fourier Orientation Reconstruction (BFOR) model spherical functions like the Ensemble Average Propagator (EAP), Fiber Orientation Distribution (FOD), and Apparent Diffusion Coefficient (ADC) using an SH basis. Our framework can be applied to any of these spherical functions to extract a new set of scalar values. In addition, any continuous function of these scalar values can be used to generate additional features that can be significant for a specific experiment or application.

Figure 2: ISBI 2013 HARDI Phantom. First Row: The left image is the ground truth fiber segmentation of a slice of the phantom dataset, where we've identified an intricate region of crossing fibers. The right image is a count of the number of fibers that cross in a given voxel, ranging from 0 to 3. Second Row: GFA and eigenvalue variance of the phantom slice. We notice here the striking similarity between the plot of crossing fibers and the eigenvalue variance whereas the GFA is unable to reveal this information. Third/Fourth Row: Close up of the ROI with ODFs.