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.

High angular resolution diffusion imaging has recently been of great interest in characterizing non-Gaussian diffusion processes. One important goal is to obtain more accurate fits of the apparent diffusion processes in these non-Gaussian regions, thus overcoming the limitations of classical diffusion tensor imaging. This paper presents an extensive study of highorder models for apparent diffusion coefficient estimation and illustrates some of their applications. Using a meaningful modified spherical harmonics basis to capture the physical constraints of the problem, a new regularization algorithm is proposed. The new smoothing term is based on the LaplaceBeltrami operator and its closed form implementation is used in the fitting procedure. Next, the linear transformation between the coefficients of a spherical harmonic series of order ഞ and independent elements of a rank-ഞ high-order diffusion tensor is explicitly derived. This relation allows comparison of the stateof-the-art anisotropy measures computed from spherical harmonics and tensor coefficients. Published results are reproduced accurately and it is also possible to recover voxels with isotropic, single fiber anisotropic, and multiple fiber anisotropic diffusion. Validation is performed on apparent diffusion coefficients from synthetic data, from a biological phantom, and from a human brain dataset. For the past decade, there has been growing interest in diffusion magnetic resonance imaging (MRI) to understand functional coupling between cortical regions of the brain, for characterization of neurodegenerative diseases, for surgical planning, and for other medical applications. Diffusion MRI is the only noninvasive tool to obtain information about the neural architecture in vivo. It is based on the Brownian motion of water molecules in normal tissues and the observation that molecules tend to diffuse along fibers when contained in fiber bundles (1,2). Using classic diffusion tensor imaging (DTI), several methods have been developed to segment and track white matter fibers in the human brain (3-8). The common way to analyze the data is to fit it to a second-order tensor, which corresponds to the probability distribution of a given water molecule moving by a certain amount during some fixed elapsed time. By diagonalization, the surface corresponding to the diffusion tensor is an ellipsoid with its long axis aligned with the fiber orientation. However, the theoretical basis for this model assumes that the underlying diffusion process is Gaussian. While this approximation is adequate for voxels in which there is only a single fiber orientation (or none). it breaks down for voxels in which there is more complicated internal structure, as seen in Fig. 1, an example of two fibers crossing. This is an important limitation, since the resolution of DTI acquisition is between 1 and 3 mm 3 , while the physical diameter of fibers can be less than 1 m and up to 30 m (9). From anisotropy measure maps, we know that many voxels in diffusion MRI volumes potentially have m...

We propose a simple and straightforward analytic solution for the Q-ball reconstruction of the diffusion orientation distribution function (ODF) of the underlying fiber population. First, the signal is modeled with a high order spherical harmonic series using a Laplace-Beltrami regularization method which leads to an elegant mathematical simplification of the FunkRadon transform using the Funk-Hecke formula. In doing so, we obtain a fast and robust model-free ODF approximation. We validate the accuracy of the estimation quantitatively against synthetic data generated from the multi-tensor model and show that our estimated ODF can recover known multiple fiber regions in a biological phantom and in the human brain.

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