Regularized, fast, and robust analytical Q-ball imaging

We propose a regularized, fast, and robust analytical solution for the Q‐ball imaging (QBI) reconstruction of the orientation distribution function (ODF) together with its detailed validation and a discussion on its benefits over the state‐of‐the‐art. Our analytical solution is achieved by modeling the raw high angular resolution diffusion imaging signal with a spherical harmonic basis that incorporates a regularization term based on the Laplace–Beltrami operator defined on the unit sphere. This leads to an elegant mathematical simplification of the Funk–Radon transform which approximates the ODF. We prove a new corollary of the Funk–Hecke theorem to obtain this simplification. Then, we show that the Laplace–Beltrami regularization is theoretically and practically better than Tikhonov regularization. At the cost of slightly reducing angular resolution, the Laplace–Beltrami regularization reduces ODF estimation errors and improves fiber detection while reducing angular error in the ODF maxima detected. Finally, a careful quantitative validation is performed against ground truth from synthetic data and against real data from a biological phantom and a human brain dataset. We show that our technique is also able to recover known fiber crossings in the human brain and provides the practical advantage of being up to 15 times faster than original numerical QBI method. Magn Reson Med 58:497–510, 2007. © 2007 Wiley‐Liss, Inc.