A Polynomial Based Approach to Extract Fiber Directions from the ODF and its Experimental Validation

Introduction: In Diffusion MRI, spherical functions are commonly employed to represent the diffusion information. The ODF is an intuitive spherical function since its maxima are aligned with the dominant fiber directions. Therefore, it is important to correctly determine these maximal directions, as they are the key to tracing fiber tracts. A tractography algorithm will suffer from cumulative error when the maximal directions are incorrectly estimated locally. The goal of this work is to present a polynomial based approach for estimating the maximal directions correctly. The paper will also present a measure of the correctness of the estimation. This approach will be tested on synthetic, phantom, and real data, and will be compared to an existing discrete “mesh-search” approach [2]. It will be shown how this approach naturally overcomes the inherent shortcomings of the discrete search. Finally, although, the approach is demonstrated on the ODF, it can be equally applied to any spherical function. Materials and Methods: Spherical functions can be naturally represented in the Spherical Harmonic (SH) basis. The ODF was written in the SH basis in [1,2] for an analytical q-Ball Imaging (QBI) algorithm. However, as shown in [3,4] the SH basis can be linearly transformed to the Homogeneous Polynomial (HP) basis by computing the coefficients of a HP constrained to the unit sphere from the SH coefficients, when they both describe the same spherical function S. Therefore, an ODF estimated in the SH basis can be rewritten as the polynomial P constrained to the unit sphere. In this form, computing the maximal directions of the ODF turns out to be a constrained optimization problem: 1 || || ) ( max 22= X to subject X P X . This can be solved using Lagrange multipliers: ) 1 || (|| ) ( ) , ( 2 2 − − = X X P X F λ λ , where the