Quantitative and qualitative experiments using synthetic and real data were performed to evaluate the proposed method. In all experiments, we set r_{s} = 2 voxels, P = 0.1, a_{p} = 30°, a_{s} = 30°, and m = 4. We use peak signal-to-noise ratio (PSNR) as the metric for performance evaluation.

Synthetic Data

A synthetic dataset consisting of one- and two-directional fiber bundles was generated for the quantitative evaluation of the proposed method. The dataset was simulated with b = 1000 s/mm^{2} and 81 non-collinear gradient directions. In order to simulate the dispersion of fiber orientations across subjects, we generate a set of diffusion signal profiles of fiber bundles oriented according to the Watson probability distribution function [10], which in modified form is given as

where 9 is the angle of deviation from the ground truth direction and the concentration parameter к is defined as к = 2(1 — cos^{2}(9_{T}))^{_1}. Parameter 9r determines the degree of dispersion of the orientations of the fiber bundles. We set 9_{T} = 15°, 30°,45°. The fiber orientation distribution functions (ODFs) [11] of some diffusion profiles are shown in Fig. 1. The “atlas” is computed using this distribution of diffusion signal profiles and the outcome is compared with the ground truth without deviation. Four levels of Rician noise (3%, 5%, 7% and 9%) were added to the dataset. Rician noise was simulated by adding Gaussian noise [i.e. N(0, v(p/100))] to the complex domain of the signal with noise variance determined by noise-level percentagep and maximum signal value v (150 in our case).

As shown in Figs. 2 and 3, for the various noise levels, our method improves the PSNRs over simple averaging for both cases of one- and two-directional fiber bundles. The PSNR improvement is over 2 dB and sometimes even up to 3 dB. The fiber ODFs of some representative results, shown in Fig. 4, indicate that simple averaging causes artifacts and that the proposed method yields results that are very close to the ground truth.

Fig. 1 Examples from the synthetic dataset simulating (top) one- and (bottom) two-direction fiber bundles

Fig. 2 PSNR comparison of results given by the mean, computed via simple averaging, and the proposed method for the case of one-direction fiber bundles

Fig. 3 PSNR comparison of results given by the mean, computed via simple averaging, and the proposed method for the case of two-direction fiber bundles

Fig. 4 Comparison of fiber ODFs. (Left) Ground-truth ODFs. (Middle) ODFs given by simple averaging. (Right) ODFs given by the proposed method. The results were generated using synthetic dataset with 9% noise and 0_{T} = 45° and 5% noise and 0_{T} = 15° respectively for the one- and

two-direction cases

Real Data

All images were acquired using a Siemens 3T TRIO MR scanner following a standard imaging protocol: 30 diffusion directions uniformly distributed on a hemisphere, b = 1000 s/mm^{2}, one image with no diffusion weighting, 128 x 128 imaging matrix, voxel size of 2 x 2 x 2 mm^{3}, TE=81 ms, TR=7618 ms.

Fig. 5 Comparisons of white matter fiber ODFs given by the simple averaging method (columns 1 and 3) and our method (columns 2 and 4). The fractional anisotropy images at the top are shown for reference. Visible differences between the methods are marked by arrows and boxes

As shown in Fig. 5, our method obtains ODFs that are more consistent and exhibit stronger directionality. Visible differences are marked using arrows and boxes. For simple averaging, the ODF glyphs are generally shorter, indicating weaker directionality. In contrast, our method gives sharper and longer ODF glyphs, indicating its superiority.