Radial and Temporal Order Fitting In this noiseless experiment, we find the optimal choice of radial and temporal order to accurately fit the diffusion signal with the lowest number of coefficients. We fit our multi-spherical basis to the Camino data using different radial and temporal orders and calculate the mean squared error (MSE) of the fitted signal to the original signal. We show the result in Fig. 2a. We find that the mean absolute error of the signal over all distributions falls below 1 % at a radial order of 6 and temporal order of 2, resulting in 150 coefficients. We will use this combination in our next experiments.

Comparison with DTI Approximation In Fig. 2b we compare the MSE of fitting DTI, the basis of Fick et al. [4] and our multi-spherical approach to subsets of the noiseless data with increasing maximum b-values. As the maximum b-value increases, data with higher gradients strengths and diffusion times are included (see Fig. 1). Our approach fits diffusion restriction over q and r best of the three methods regardless of b-value.

Multi-Spherical Signal Reconstruction and q-Space Index Estimation To

reduce the number of measurements, we regularize the basis fitting with a combination of imposing smoothness in the fitted signal and sparsity in the basis coefficients. To study its effectiveness, we first add Rician noise to the Camino data such that the signal-to-noise (SNR)-ratio is 20. We then randomly subsample, fit and then recover the data from our model and estimate the MSE with the noiseless data. The experiment for every chosen number of samples is repeated 50 times for all 44 voxels with each a different noise instance. The result can be seen in Fig. 3a, where

Fig. 2 (a) Noise free fitting of Camino data set using different radial and time orders using our multi-spherical basis. The color intensity shows the mean squared error and the green dots indicate orders for which the mean absolute error of the reconstruction is smaller than 1% of the b0 value. (b) Comparison of the fitting error between DTI, the approach of Fick et al. [4] and our multispherical approach over maximum b-value

Fig. 3 Effect of random subsampling at SNR=20 on (a) mean squared error (MSE) for different regularization techniques, (b) the time-dependent Return-To-Origin Probability (RTOP) and (c) Mean Squared Displacement (MSD). (a) Our combined sparsity and Laplacian regularization (yellow) has lower MSE than only Laplacian (green) and least squares (red). (b) and (c) show the MSD and RTOP using 600 samples (green) to 100 samples (blue)

our combined approach (yellow) has the lowest MSE, followed by using only the Laplacian (green) and the worst is least squares (red). We also show the effects of using between 600 samples (green) and 100 samples (blue) on the estimation of the Mean Squared Displacement (MSD) and the q-space index Return-To-Origin probability (RTOP) in Fig. 3b, c. We see that MSD increases as time increases, while its profile does not change much until the profile flattens for 100 samples. In contrast, we see that RTOP decreases over time and as the number of samples reduces, the overall RTOP values decrease. Again for 100 samples, the profile flattens out.

Application to In-Vivo Mouse Acquisition Finally, we apply our method to in vivo acquired data from a C57Bl6 wild-type mouse. The results are shown in Fig. 4. First, we estimate MSD and RTOP for the whole data and show their values for

Fig. 4 (top-left) Region of interest in mouse corpus callosum. (top-right) Maps ofRTOP and MSD for different diffusion times. (bottom) Histograms of the MSE (left), MSD (middle) and RTOP (right) for different numbers of fitted points. The RTOP and MSD were calculated for x = 14 ms

different diffusion times on the top left. RTOP decreases as time increases, which corresponds with the in-silico experiments. In MSD we first find an overall increase, after which a small decrease is seen. The latter phenomenon does not correspond with what we previously found. We then again randomly subsample the data for all voxels in the ROI and estimate the MSE, together with the MSD and RTOP for a chosen diffusion time of x = 14 ms. The trends for all markers correspond with the synthetic data: MSE increases, RTOP decreases, and MSD stays the same as the number of samples decreases.