SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $\alpha_{l,m}$  and  $\beta_{l,m}$  denote the SH coefficients for two FODs with degree  $l$  and order  $m$ .
A.9 Additional NODDI Analysis
Table 11 presents results from the NODDI experiment within the  $q_{\mathrm{in}} = 10$  sub-sampling scheme.
A.10 Ablation Studies
This section describes results from various ablation studies performed with the PCCNN model as a baseline. Each of the models were used to infer multi-shell data from single-shell input data across three separate shells. The following ablations were performed:
- No Fourier features: The Fourier feature mapping  $\gamma(p_m)$ , as described in Equation (3.2) were removed from the PCConv operation.
- No b-vectors: The b-vector coordinates  $\mathbf{e}_b$  were removed from the PCConv operation, thus reducing the operation to only convolve using spatial based co-ordinates and the b-value difference  $\rho_i - \rho_j$ .
-  $d_{\mathrm{max}} = \frac{1}{4}\pi$ : Only points on the q-space sphere at a spherical distance of  $d_r \leq \frac{1}{4}\pi$  away from the centre of the kernel were included within the PCConv operation.
Table 9: Peak signal-to-noise ratio (PSNR) of dMRI intensity in eight subjects with data derived from various models. Models use single-shell  $b = 2000\mathrm{s} / \mathrm{mm}^2$  data with angular dimension size  $q_{\mathrm{in}}$  as input, and produce inferred  $b = 2000\mathrm{s} / \mathrm{mm}^2$  data with angular dimension size  $90 - q_{\mathrm{in}}$ .
qin=6	qin=10	qin=20
SH Interpolation	27.49 ± 1.27	35.94 ± 1.40	36.40 ± 1.43
RCNN	33.25 ± 1.21	36.94 ± 1.35	37.79 ± 1.36
PCCNN	32.17 ± 1.23	37.95 ± 1.36	40.03 ± 1.46
PCCNN-Bv	32.43 ± 1.23	38.20 ± 1.42	39.75 ± 1.52
PCCNN-Sp	32.22 ± 1.23	38.22 ± 1.38	40.50 ± 1.45
PCCNN-Bv-Sp	32.26 ± 1.22	38.38 ± 1.36	40.51 ± 1.45
Table 10: Mean structural similarity index (MSSIM) of dMRI intensity in eight subjects with data derived from various models. Models use single-shell  $b = 2000\mathrm{s} / \mathrm{mm}^2$  data with angular dimension size  $q_{\mathrm{in}}$  as input, and produce inferred  $b = 2000\mathrm{s} / \mathrm{mm}^2$  data with angular dimension size  $90 - q_{\mathrm{in}}$ .
qin=6	qin=10	qin=20
SH Interpolation	0.869 ± 0.008	0.972 ± 0.003	0.975 ± 0.003
RCNN	0.950 ± 0.004	0.978 ± 0.002	0.982 ± 0.002
PCCNN	0.936 ± 0.006	0.983 ± 0.002	0.990 ± 0.001
PCCNN-Bv	0.940 ± 0.006	0.984 ± 0.002	0.990 ± 0.001
PCCNN-Sp	0.936 ± 0.006	0.983 ± 0.002	0.990 ± 0.001
PCCNN-Bv-Sp	0.937 ± 0.006	0.984 ± 0.002	0.990 ± 0.001
Table 11: Absolute error (AE) of orientation dispersion index (OD),  $f_{\mathrm{intra}}$ , and  $f_{\mathrm{iso}}$  derived from neurite orientation dispersion and density imaging using both input data and inferred data from various models. Models use single-shell data ( $b = 1000\mathrm{s/mm}^2$ ) with angular dimension size  $q_{\mathrm{in}} = 10$  as input, and produce  $80b = 1000\mathrm{s/mm}^2$ ,  $90b = 2000\mathrm{s/mm}^2$ , and  $90b = 3000\mathrm{s/mm}^2$  inferred volumes.
OD	fiso	fintra
SR-q-DL	0.089 ± 0.004	0.162 ± 0.015	0.128 ± 0.010
RCNN	0.054 ± 0.006	0.026 ± 0.003	0.048 ± 0.004
PCCNN	0.048 ± 0.006	0.027 ± 0.003	0.049 ± 0.006
PCCNN-Bv	0.044 ± 0.005	0.025 ± 0.003	0.048 ± 0.006
PCCNN-Sp	0.048 ± 0.007	0.025 ± 0.004	0.048 ± 0.010
PCCNN-Bv-Sp	0.050 ± 0.008	0.024 ± 0.004	0.049 ± 0.012
-  $d_{\mathrm{max}} = \frac{1}{8}\pi$ : Only points on the q-space sphere at a spherical distance of  $d_r \leq \frac{1}{8}\pi$  away from the centre of the kernel were included within the PCConv operation.
Each modification was applied to all layers within the respective model. The baseline PCCNN model had a  $d_{\mathrm{max}} = 1$ . The results of these ablations are shown in Table 12.
Results within Table 12 clearly demonstrate that including b-vector information within the PCConv operation is essential for the model to perform well, as the AE significantly increases within the 'No b-vectors' experiment, as compared to the baseline PCCNN. The two  $d_{\mathrm{max}}$  ablation results also show that removing information pertaining to the angular co-ordinate system readily reduces performance of the model. It is noteworthy that a  $d_{\mathrm{max}} = \frac{1}{8}\pi$  has significantly higher AE compared to the 'No b-vectors' variant. This is likely due to the greatly reduced number of points available for each kernel operation within the  $d_{\mathrm{max}} = \frac{1}{8}\pi$  experiment. This is in contrast to the 'No b-vector' variant which uses all available points within the operation, but with no angular distance information.
The inclusion of Fourier features within the PCConv operation improves performance in two of the shells, but decreases performance in one. This suggests that the inclusion of the Fourier features may improve the ability of the model to generalise to different shells, however this effect is not as apparent as the inclusion of b-vector information. The limited improvement in this case may be twofold.
Table 12: Absolute error of dMRI intensity in eight subjects with data derived from ablation models. Models use single-shell data with angular dimension size  $q_{\mathrm{in}}$  as input, and infer 80 volumes within the input shell, and 90 volumes within the other two shells. b-values are quoted in units of  $\mathrm{s} / \mathrm{mm}^2$ .
b = 1000	b = 2000	b = 3000
Baseline PCCNN	51.994 ± 2.719	54.553 ± 3.427	74.946 ± 6.399
No Fourier features	49.681 ± 3.386	55.619 ± 5.190	76.210 ± 8.119
No b-vectors	86.074 ± 3.589	89.058 ± 5.064	110.495 ± 10.384
dmax= 1/4π	73.863 ± 4.738	85.552 ± 9.441	99.232 ± 8.272
dmax= 1/8π	205.432 ± 8.988	222.977 ± 13.554	259.969 ± 20.038
First, due to the increased complexity of the model, as the co-ordinate embedding dimensionality would increase by a factor of  $2L$ . Second, whilst Fourier features were shown to vastly improve performance when predicting image intensity from co-ordinates in MLP networks [35], the Fourier feature mapping here is only applied to the co-ordinates within the domain of the kernel, which is a much smaller subset of the entire image domain.
A.11 Noisy Data
This section provides additional results from an experiment using dMRI data that were not denoised prior to training or inference. This analysis includes an additional model, denoted the 'Q-Space CGAN' as detailed within Ren et al. [31]. The weights for this model were trained on noisy HCP data and were obtained from the original authors of the work. The other models presented in this section were trained exactly as outlined previously, with the only difference being that the data were not denoised prior to training.
Figure 6 shows an axial slice of dMRI mean absolute error (MAE) in three models when inferring multi-shell data. Whilst this is only a qualitative example, the Q-Space CGAN has significantly higher MAE across the slice as compared to the other two models. This is likely due to lack of diffusion weighted context provided to the Q-Space CGAN as input.
Table 13 presents results from the multi-shell AFD experiment, as shown in Section 4.2.1 using models trained with non-denoised data. These results follow similar trends as present in Table 4, whereby the FOD-Net generally performs best in AFD AE whilst the PCCNN variants perform best within FOD ACC across sub-sampling schemes.
Figure 6: Axial slice of mean absolute error (MAE) within one subject across different models. RCNN and PCCNN-Bv-Sp models use noisy single-shell data ( $b = 1000\mathrm{s} / \mathrm{mm}^2$ ) with angular dimension size  $q_{\mathrm{in}} = 6$  as input, and produce  $84b = 1000\mathrm{s} / \mathrm{mm}^2$ ,  $90b = 2000\mathrm{s} / \mathrm{mm}^2$ , and  $90b = 3000\mathrm{s} / \mathrm{mm}^2$  inferred volumes. Q-Space CGAN uses b0, T1w, and T2w volumes to infer the same output.
Table 13: Comparison of metrics in fixed-based analysis for eight subjects with metrics derived from both noisy input data and inferred data using various models trained on noisy data. Models use single-shell data ( $b = 1000\mathrm{s} / \mathrm{mm}^2$ ) with angular dimension size  $q_{\mathrm{in}}$  as input, and produce 80  $b = 1000\mathrm{s} / \mathrm{mm}^2$ , 90  $b = 2000\mathrm{s} / \mathrm{mm}^2$ , and 90  $b = 3000\mathrm{s} / \mathrm{mm}^2$  inferred volumes. Lowres denotes metrics derived from single-shell input only. Fibre orientation distribution (FOD) angular correlation coefficient (ACC) is defined by Equation (5), whilst apparent fibre density (AFD) absolute error (AE) is calculated using the magnitudes of all pixels within a voxel.  $\uparrow$  denotes that a higher value is better.
qin=6	qin=10	qin=20
FOD ACC ↑	AFD AE ↓	FOD ACC ↑	AFD AE ↓	FOD ACC ↑	AFD AE ↓
Lowres	0.509 ± 0.006	0.187 ± 0.015	0.547 ± 0.004	0.145 ± 0.011	0.590 ± 0.006	0.122 ± 0.009
FOD-Net	0.645 ± 0.007	0.097 ± 0.004	0.654 ± 0.005	0.084 ± 0.003	0.669 ± 0.007	0.076 ± 0.004
Q-Space CGAN	0.486 ± 0.014	0.160 ± 0.013	0.496 ± 0.014	0.160 ± 0.013	0.518 ± 0.013	0.156 ± 0.012
RCNN	0.641 ± 0.010	0.108 ± 0.005	0.707 ± 0.017	0.091 ± 0.004	0.751 ± 0.017	0.085 ± 0.001
PCCNN-Bv-Sp	0.648 ± 0.008	0.098 ± 0.003	0.723 ± 0.010	0.084 ± 0.002	0.762 ± 0.011	0.078 ± 0.001