Modelling uncertainty in two fibre-orientation estimates within a voxel
Philip A. Cook, Daniel C. Alexander · 2006
i=1 2N x i ;κ 1 ,κ 2 ,µ 1 ,µ 2 ) + 0.5B(x i ;κ 3 ,κ 4 µ 3 ,µ 4 ) with respect to the orientations and concentrations of the two Bingham PDFs. Experiments Experiment 1 illustrates the elliptical contours of the fibre orientation PDF at intermediate anisotropy. We generate 1000 trials of the test function with the fractional anisotropy (FA) of Da and Db set to 0.6. Fig 1 shows the fibre orientation estimates for each of the inversions. The fibre orientations from the restricted inversion are more concentrated but distributed asymmetrically, with greater uncertainty towards the plane defined by the principal directions of Da and Db. The distribution of fibre orientations is similar for the restricted and cylindrical inversion. Fig 2 shows the histogram of the estimated anisotropy. The restricted model gives a much better estimate of the FA in this case. This is important for PICo because the calibration parameterises the PDF as a function of tensor anisotropy. We observe a similar result over a range of FA from 0.4 to 0.8. At FA = 0.9, the fibre orientations are clustered in circular patterns, and the FA estimates are similar from both inversions. Experiment 2 compares the two-Bingham and two-Watson model in a simulated fibre crossing. The crossing consists of two orthogonal fibre paths, which intersect through the centre of the image, forming a 20mm fibre-crossing region where both fibre bundles contribute equally to the signal. The imaging parameters are the same as in the first experiment, and the voxel dimensions are isotropic 2 mm. As a gold standard, we create a probability map using PICo without sampling from a PDF. We add noise to the test function at each of 5000 iterations and calculate the restricted inversion to obtain a fibre orientation estimate. We then run PICo in 50 different noisy images using the Watson and Bingham models, and compute the correlation between the 50 maps from each PDF and the gold standard. We carry out this experiment with two levels of noise, one with SNR=14 (as in experiment 1) and again with SNR=32. Fig. 3 plots the mean and standard error of (CB(i) - CW(i)), where CB(i) is the correlation of the Bingham PICo map to the gold standard for map i and CW(i) is the same statistic for the Watson distribution. As expected, for very anisotropic fibres, both models perform similarly because the distribution of fibre orientation estimates in the fibre crossing is circular. However, at low anisotropy, the Bingham model correlates better to the gold standard. At SNR=14, this effect is not apparent at FA = 0.4 or 0.5, because the fibre crossing is not well resolved at this SNR and FA. Conclusion Our results suggest that the restricted inversion is more useful for PICo, but further work is required to understand the behaviour of this inversion when the mixing parameter in the test function is not 0.5, and when the diffusion tensors in the test function are not cylindrically symmetric. The two-Bingham model can describe a PDF with circular or elliptical contours on the sphere, as appropriate, making it more flexible than the previous two-Watson model proposed by Cook et al (4).