Generalized hardi invariants by method of tensor contraction

Invariants play an important role in diffusion MRI (dMRI). They represent tissue properties, such as diffusion anisotropy, and are used for registration, tissue segmentation and classification, as well as white matter integrity measures in clinical studies of debilitating brain diseases. Importantly, these features need to be rotation invariant to capture orientation independent tissue properties, as well as to enable comparisons across images that are not completely aligned. Modelling diffusion using second-order tensors, as in DTI, enable construction of diverse anatomically meaningful scalars, such as fractional anisotropy (FA), mean diffusivity (MD), and relative anisotropy (RA), which capture tissue microstructure, and are used to indicate anatomical changes [1]. Most of these scalars are derived using the tensor eigenvalues, hence, they are naturally rotation invariant as the eigenvalues remain intact under rotation. In HARDI, however, the situation is more complicated as the diffusion profile is described by higher-order tensors or orientation distribution functions (ODFs), and there is no straightforward generalization of DTI invariants to these models. The generalized FA (GFA), for example, aims to represent anisotropy in HARDI models, but has limited classification power, and is sensitive to noise being directly computed from a discrete representation of orientation distribution function (ODF) [2]. In order to take advantage of the enhanced modelling capabilities of HARDI compared to DTI, it is important to derive new invariants that capture tissue properties, and can be used as white matter biomarkers. Over the last years, researchers have created new invariants for HARDI-based models, for example, the generalized anisotropy (GA) and scaled anisotropy (SE) [3], as well as several approaches that are based on second and fourth-order tensor representations [4, 5]. A recent approach uses the Gaunt coefficients to construct invariants for ODFs and HARDI signals using the more general SH representation [2]. Our proposed technique follows a similar path of using the SH representation. We then leverage the idea of invariants constructed by tensor contraction used in computer vision for 3D pattern recognition [6]. The original idea emerged from the theory of angular momentum addition in quantum physics, and although it relies on deep and complex theoretical foundations, the formulation enables systematic construction of invariants in an elegant and simple way. This method is general, as it enables extraction of invariants from any 3D object represented as a SH series. Therefore, it can be used to construct invariants from the dMRI signal, or from any diffusion modelling object, such as ODF or FOD. In addition, it enables direct construction of invariants for any expansion order, thus, it is not limited to the common second or fourth order expansions. This method generalizes the SH descriptors used in [7] to classify autism spectrum disorder (ASD) patients and controls, and to segment brain tissue [8, 9]. It is based on constructing contravariant rank-1 tensors (vectors) using the SH and Clebsch-Gordan (CG) coefficients, and contracting them with covariant vectors to obtain rank-0 tensors (invariants). This process can be continued repeatedly to build as many invariants as desired regardless of the SH order, therefore, enables the construction of long feature vectors with strong classification capabilities. This is an advantage over the method presented in [2] in which the maximal number of invariants is bounded by the rank of the Toeplitz-like matrix. We demonstrate the strengths of our approach in both synthetic and in vivo experiments. Using simulated data we show that these invariants are robust to noise and can classify voxels based on the number of fiber compartments and their diffusivities. Using in vivo brain data, we show that they capture anatomically meaningful information, and may be used as white matter integrity measures.