A Polynomial Approach for Maxima Extraction and Its Application to Tractography in HARDI

A number of non-parametrically represented High Angular Resolution Diffusion Imaging (HARDI) spherical diffusion functions have been proposed to infer more and more accurately the heterogeneous and complex tissue microarchitecture of the cerebral white-matter. These spherical functions overcome the limitation of Diffusion Tensor Imaging (DTI) at discerning crossing, merging and fanning axonal fiber bundle configurations inside a voxel. Tractography graphically reconstructs the axonal connectivity of the cerebral white-matter in vivo and non-invasively, by integrating along the direction indicated by the local geometry of the spherical diffusion functions. Tractography is acutely sensitive to the local geometry and its correct estimation. In this paper we first propose a polynomial approach for analytically bracketing and numerically refining with high precision all the maxima, or fiber directions, of any spherical diffusion function represented non-parametrically. This permits an accurate inference of the fiber layout from the spherical diffusion function. Then we propose an extension of the deterministic Streamline tractography to HARDI diffusion functions that clearly discern fiber crossings. We also extend the Tensorline algorithm to these HARDI functions, to improve on the extended Streamline tractography. We illustrate our proposed methods using the Solid Angle diffusion Orientation Distribution Function (ODF-SA). We present results on multi-tensor synthetic data, and real in vivo data of the cerebral white-matter that show markedly improved tractography results.