SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 1: Total number of parameters, in millions, of different deep learning approaches.
Model	Parameters
SR-q-DL [41]	0.52
PCCNN	0.77
PCCNN-Bv	0.79
PCCNN-Sp	0.83
PCCNN-Bv-Sp	0.85
RCNN [26]	6.82
FOD-Net [44]	48.17
compared against low angular resolution data. All error metrics were calculated with respect to the full angular resolution HCP dataset.
The reported error values consist of the mean and standard deviation, with distribution statistics generated from the mean performance per test subject. Only voxels within the brain are included
Table 2: Absolute error of dMRI intensity in eight subjects with data derived from various models. Models use single-shell data with angular dimension size  $q_{\mathrm{in}}$  as input, and produce inferred data with angular dimension size  $90 - q_{\mathrm{in}}$  from the same shell. b-values are quoted in units of  $\mathrm{s} / \mathrm{mm}^2$ .
qin=6	qin=10	qin=20
b=1000	b=3000	b=1000	b=3000	b=1000	b=3000
SH Interpolation	84.85 ± 5.47	106.40 ± 4.29	47.25 ± 2.85	46.14 ± 2.02	45.07 ± 3.10	42.27 ± 1.86
RCNN	74.01 ± 4.03	52.11 ± 1.97	56.01 ± 3.50	38.68 ± 1.45	53.89 ± 3.92	34.31 ± 1.39
PCCNN	74.55 ± 3.69	57.99 ± 2.60	48.41 ± 2.98	34.99 ± 1.63	44.03 ± 3.09	24.86 ± 1.23
PCCNN-Bv	73.55 ± 3.63	56.71 ± 2.56	48.40 ± 2.89	34.18 ± 1.65	45.32 ± 3.13	24.77 ± 1.20
PCCNN-Sp	75.01 ± 3.64	57.41 ± 2.57	47.79 ± 2.99	34.16 ± 1.49	41.88 ± 2.76	23.88 ± 0.98
PCCNN-Bv-Sp	74.87 ± 3.51	56.78 ± 2.48	46.92 ± 2.53	33.53 ± 1.47	42.52 ± 2.57	23.70 ± 0.98
in the calculations, as determined by a brain mask segmentation. The best mean value in a given category is denoted in bold.
4.1 Single-shell dMRI
Within single-shell inference, we evaluated performance across two analyses. Firstly, on raw dMRI data, of which any downstream analysis task would be derived from. We compared performance across all three shells available within the HCP dataset, as well as in three q-space sub-sampling regimes, as outlined in Table 2. Secondly, we compared performance within FBA, where metrics were derived from the inferred dMRI data. For this, we considered the  $b = 1000$  s/mm² shell at three sub-sampling regimes.
4.1.1 dMRI Analysis
Table 2 presents the absolute error (AE) of the inferred dMRI data as compared to the high resolution ground truth. For the  $b = 2000\mathrm{s} / \mathrm{mm}^2$  results, see Table 8 within the Appendix. The PCCNN-Bv-Sp model had the lowest AE in five out of the nine shell and sub-sampling combinations and had the overall best performance when averaged across all data. Despite having approximately a tenth of the number of parameters of the RCNN [26] model, the PCCNN models had lower mean errors across all data regimes except  $q_{\mathrm{in}} = 6$  in shells  $b = \{2000,3000\} \mathrm{s} / \mathrm{mm}^2$ .
4.1.2 Fixed Analysis
FBA is an analysis method used to estimate the intra-voxel fibre populations. To generate fixed data [10], response functions were first estimated in an unsupervised manner [9], and Single-Shell 3-Tissue (SS3T) constrained spherical deconvolution (CSD) [8] was performed to obtain white matter (WM) FODs. WM FODs were then normalised [11], and finally segmented into fixels. This workflow used the MRtrix3 [37] package. FOD data is stored as a finite SH expansion, and as such, was compared using the angular correlation coefficient (ACC) [2], given by Equation (5). To compare FBA data, the AE in the apparent fibre density (AFD) within each voxel was calculated by taking the absolute difference between the magnitudes of each inferred fibre population with respect to the ground truth values.
Table 3 summarises the performance in FBA in the low resolution baseline and model based reconstructions compared to the high resolution ground truth, whilst Figure 3 visualises a subset of reconstructed FODs within an axial slice. Here, all PCCNN models performed better than the low resolution baseline in all three sampling schemes in FOD ACC, and had lower AFD AE in the two lowest sampling schemes. The PCCNN models performed competitively against the RCNN. However, in the smallest sub-sampling scheme  $q_{\mathrm{in}} = 6$ , or equivalently  $6.6\%$  of the original number of voxels, the RCNN performed slightly better. Whilst the low resolution data had lower AFD AE at  $q_{\mathrm{in}} = 20$ , the PCCNN models had, in the worse case, a relative increased mean error of approximately  $6\%$ , compared to  $35\%$  increase in the RCNN model. Additionally, the PCCNN models had greater ACC compared to the low resolution baseline across all sub-sampling schemes, suggesting that their use in probabilistic tractography [33] and connectomics [4] would be beneficial.
Table 3: Comparison of metrics in fixel-based analysis for eight subjects with metrics derived from both input data and inferred data using various models. Models use single-shell data ( $b = 1000\mathrm{s} / \mathrm{mm}^2$ ) with angular dimension size  $q_{\mathrm{in}}$  as input, and produce  $b = 1000\mathrm{s} / \mathrm{mm}^2$  inferred data with angular dimension size  $90 - q_{\mathrm{in}}$ . 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 fixels 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	0.699 ± 0.007	0.133 ± 0.011	0.793 ± 0.009	0.085 ± 0.006	0.850 ± 0.007	0.048 ± 0.003
SH Interpolation	0.702 ± 0.010	0.180 ± 0.014	0.792 ± 0.008	0.090 ± 0.005	0.833 ± 0.006	0.078 ± 0.004
RCNN	0.708 ± 0.005	0.081 ± 0.006	0.784 ± 0.007	0.071 ± 0.005	0.823 ± 0.005	0.065 ± 0.004
PCCNN	0.705 ± 0.007	0.082 ± 0.006	0.806 ± 0.008	0.063 ± 0.004	0.856 ± 0.006	0.049 ± 0.003
PCCNN-Bv	0.708 ± 0.006	0.079 ± 0.005	0.802 ± 0.006	0.063 ± 0.004	0.853 ± 0.006	0.051 ± 0.003
PCCNN-Sp	0.699 ± 0.007	0.082 ± 0.005	0.803 ± 0.008	0.062 ± 0.004	0.855 ± 0.006	0.048 ± 0.003
PCCNN-Bv-Sp	0.704 ± 0.007	0.081 ± 0.005	0.802 ± 0.007	0.062 ± 0.004	0.852 ± 0.006	0.049 ± 0.003
(a) ROI
Figure 3: Fibre orientation distributions (FOD)s in one ROI slice for a test subject with metrics derived from both input data and inferred data using various models. Models use single-shell data  $(b = 1000\mathrm{s} / \mathrm{mm}^2)$  with angular dimension size  $q_{\mathrm{in}} = 10$  as input, and produce  $b = 1000\mathrm{s} / \mathrm{mm}^2$  inferred data with angular dimension size 80. Low Resolution denotes FODs derived from single-shell input only.
(b) High Resolution
(c) Low Resolution
(d) RCNN
(e) PCCNN-Bv-Sp
4.2 Multi-shell dMRI
To evaluate the performance of multi-shell inference, we used a varying  $q_{\mathrm{in}}$  number of  $b = 1000\mathrm{s} / \mathrm{mm}^2$  volumes as input, and inferred the remaining known volumes from all three shells as output. The output volumes set contained the output volumes from single-shell inference, in addition to the other two shells. The  $q_{\mathrm{in}} = 6$ , 10, and 20 sub-sampling schemes correspond to an under-sampling rate of  $2.2\%$ ,  $3.7\%$ , and  $7.4\%$  respectively. Outlined in the sections below are the performances of the models evaluated within FBA and NODDI metrics.
4.2.1 Fixed Analysis
Fixed data were generated in the same manner as outlined in Section 4.1.2, except where the MultiShell Multi-Tissue (MSMT) algorithm [18] was used in place of SS3T. Table 4 outlines the performance within multi-shell inference. In contrast to single-shell inference, here FOD ACC and AFD AE for PCCNN models were lower than the baseline in all sub-sampling schemes. The minimum difference in AFD AE, as compared to baseline, was  $13.8\%$  at  $q_{\mathrm{in}} = 20$ , whilst the maximum difference was  $43.04\%$  at  $q_{\mathrm{in}} = 6$ . The FOD-Net model had the best performance overall in this experiment, obtaining the best error rate in three out of the six metrics. Despite having approximately 50-fold less parameters, the PCCNN models still perform competitively, and boasts the highest FOD ACC in  $q_{\mathrm{in}} = \{10, 20\}$ .
4.2.2 NODDI Analysis
NODDI metrics were generated from input and inferred output volumes using the cuDIMOT [17] package. The full angular resolution derived NODDI metrics were used as ground truth. Data for  $q_{\mathrm{in}} = \{6,20\}$  are presented in Table 5, whilst  $q_{\mathrm{in}} = 10$  can be found in Table 11 within the Appendix.
Table 4: Comparison of metrics in fixed-based analysis for eight subjects with metrics derived from both input data and inferred data using various models. Models use single-shell data ( $b = 1000\mathrm{s/mm}^2$ ) with angular dimension size  $q_{\mathrm{in}}$  as input, and produce  $90 - q_{\mathrm{in}}$ $b = 1000\mathrm{s/mm}^2$ ,  $90$ $b = 2000\mathrm{s/mm}^2$ , and  $90$ $b = 3000\mathrm{s/mm}^2$  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 fixes 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	0.653 ± 0.008	0.157 ± 0.014	0.724 ± 0.008	0.119 ± 0.012	0.757 ± 0.010	0.086 ± 0.011
FOD-Net	0.743 ± 0.008	0.087 ± 0.005	0.767 ± 0.006	0.072 ± 0.004	0.776 ± 0.008	0.062 ± 0.004
RCNN	0.685 ± 0.010	0.087 ± 0.006	0.749 ± 0.009	0.080 ± 0.006	0.765 ± 0.010	0.079 ± 0.006
PCCNN	0.658 ± 0.009	0.090 ± 0.005	0.753 ± 0.008	0.077 ± 0.005	0.792 ± 0.009	0.068 ± 0.006
PCCNN-Bv	0.681 ± 0.010	0.091 ± 0.005	0.770 ± 0.011	0.080 ± 0.006	0.807 ± 0.011	0.074 ± 0.006
PCCNN-Sp	0.670 ± 0.013	0.092 ± 0.006	0.761 ± 0.013	0.075 ± 0.006	0.791 ± 0.014	0.070 ± 0.007
PCCNN-Bv-Sp	0.675 ± 0.013	0.089 ± 0.005	0.766 ± 0.014	0.075 ± 0.006	0.798 ± 0.015	0.067 ± 0.006
Table 5: Absolute error (AE) of orientation dispersion index (OD),  $f_{\mathrm{intra}}$ , and  $f_{\mathrm{iso}}$  derived from neurite orientation dispersion and density imaging using input data and inferred data from various models. Models use single-shell data ( $b = 1000\mathrm{s} / \mathrm{mm}^2$ ) with angular dimension size  $q_{\mathrm{in}}$  as input, and produce  $90 - q_{\mathrm{in}}b = 1000\mathrm{s} / \mathrm{mm}^2$ ,  $90b = 2000\mathrm{s} / \mathrm{mm}^2$ , and  $90b = 3000\mathrm{s} / \mathrm{mm}^2$  inferred volumes.
qin=6	qin=20
OD	fiso	fintra	OD	fiso	fintra
SR-q-DL	0.095 ± 0.004	0.162 ± 0.015	0.132 ± 0.010	0.089 ± 0.003	0.162 ± 0.015	0.129 ± 0.010
RCNN	0.074 ± 0.007	0.030 ± 0.003	0.052 ± 0.005	0.045 ± 0.005	0.025 ± 0.003	0.048 ± 0.006
PCCNN	0.084 ± 0.006	0.030 ± 0.003	0.054 ± 0.006	0.039 ± 0.006	0.024 ± 0.003	0.045 ± 0.004
PCCNN-Bv	0.075 ± 0.006	0.026 ± 0.003	0.050 ± 0.006	0.036 ± 0.004	0.023 ± 0.003	0.045 ± 0.006
PCCNN-Sp	0.080 ± 0.008	0.026 ± 0.004	0.050 ± 0.010	0.038 ± 0.006	0.025 ± 0.004	0.051 ± 0.012
PCCNN-Bv-Sp	0.083 ± 0.008	0.025 ± 0.003	0.050 ± 0.012	0.040 ± 0.007	0.023 ± 0.004	0.047 ± 0.012
Input, or low resolution, data were not included in this analysis as multi-shell data is required to reconstruct NODDI parameters. The parameters  $f_{\mathrm{intra}}$  and  $f_{\mathrm{iso}}$  refer to the estimated volume fraction for intra-cellular and cerebrospinal fluid respectively, whilst orientation dispersion index (OD) is an estimation of the dispersion of white matter fibre bundles.
In this experiment, the PCCNN models perform competitively, yielding the lowest error in  $f_{\mathrm{iso}}$  and  $f_{\mathrm{intra}}$  across all three sub-sampling rates. The relative error in OD across different models mirrors that of the FOD ACC found in Table 4, suggesting that the trained models are able to infer data that generalise across different downstream analyses. The worst performing model across all sub-sampling schemes was the SR-q-DL model, as is demonstrated qualitatively by Figure 4. Whilst this model could benefit from the more constrained task of inferring NODDI data directly, the lack of geometric prior information, such as b-vector coordinates, suggests that these additions within the PCCNN models were important to its relatively high performance in this task.
The results of the experiments show that the proposed PCConv operation can be used to build a flexible low parameter network to effectively infer high angular resolution raw dMRI data. It does this through appropriate incorporation of task specific geometric properties, as well as suitable parameter sharing via a hypernetwork. We show that inference of raw dMRI data can be used in downstream analysis with moderately high fidelity, and therefore demonstrates the generalisability of the model.
All three of the PCCNN models with additional modifications (PCCNN-Bv, PCCNN-Sp, PCCNN-Bv-Sp) performed greater or equal to the standard PCCNN in approximately  $47\%$  of the experiments presented in this work. Additionally, at least one of the modified PCCNNs outperformed the standard PCCNN in approximately  $87\%$  of the experiments. Overall, this suggests that the inclusion of additional domain specific context through the coordinate embedding was beneficial to the models performance.
(a) High Resolution
(b) SR-q-DL
(c) RCNN
(d) PCCNN-Bv
Figure 4: Axial slice of orientation dispersion index (OD) within one subject across different models. Models use single-shell data  $(b = 1000\mathrm{s} / \mathrm{mm}^2)$  with angular dimension size  $q_{\mathrm{in}} = 10$  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.
The application of super-resolution in a medical imaging context presents an inherent risk, as deep learning models can be susceptible to hallucinations, or high error rates, when making predictions on out-of-distribution data. Given that this work only includes data from relatively young healthy adults, future work will be needed to validate these methods in a diverse set of diagnostic settings such as datasets with pathologies, wide age ranges, and varying acquisition parameters.
Modern deep learning frameworks rely on highly optimised subroutines to perform standard discrete convolutions. Subsequently, despite having a lower number of parameters, the PCCNNs inference and train times were longer than the other methods presented in this work. This is because the PCConv uses a bespoke implementation that does not benefit from the aforementioned subroutines. This highlights the need for future research into more efficient implementations of the PCConv operation.
In summary, our work further explores the integration of geometric priors into the distinct geometry of dMRI data, and in doing so provides a flexible framework to incorporate arbitrary geometries in different domains and problem formulations.
This work was funded by the Engineering and Physical Sciences Research Council (EPSRC) Doctoral Training Partnership (DTP) Scholarship. Data were provided (in part) by the HCP, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.
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A.1 Model Hyperparameters
Figure 5: PCCNN model architecture. Angular kernel size is  $k_{q} = q_{\mathrm{in}} = 20$ .  $C_1$  and  $C_3$  denote the number of output channels for a  $(1\times 1\times 1\times 20)$  and  $(3\times 3\times 3\times 20)$  PCConv kernel respectively.  $W_{h}$  denotes the size of the hidden units within each PCConv's hypernetwork.  $L$  is a hyperparameter of the coordinate embedding  $\mathbf{e}_s$  as shown in Equation (3).
A.2 RCNN Data and Training
Training parameters, the hardware used, and the data sampling scheme used to train the RCNN were the same as outlined within Section 3.6. Training took approximately 3 days to complete.
A.3 FOD-Net Data and Training
In this work we modify the training scheme used within Zeng et al. [44] to more closely match the PCCNN training methodology. Specifically, each training example input was a low angular resolution FOD volume generated from single-shell dMRI data with b-value  $b_{\mathrm{in}}$  and angular dimension  $q_{\mathrm{in}}$  determined via
$$
q _ {\mathrm {i n}} \sim U (q _ {\mathrm {s a m p l e}}), q _ {\mathrm {s a m p l e}} = \{6, 1 0, 2 0 \}, b _ {\mathrm {i n}} \sim U (b _ {\mathrm {s a m p l e}}), b _ {\mathrm {s a m p l e}} = \{1 0 0 0, 2 0 0 0, 3 0 0 0 \}.
$$
Table 6: Peak signal-to-noise ratio (PSNR) of dMRI intensity in eight subjects with data derived from various models. Models use single-shell data with angular dimension size  $q_{\mathrm{in}}$  as input, and produce inferred data with angular dimension size  $90 - q_{\mathrm{in}}$  from the same shell. b-values are quoted in units of  $\mathrm{s} / \mathrm{mm}^2$ .
qin=6	qin=10	qin=20
b=1000	b=3000	b=1000	b=3000	b=1000	b=3000
SH Interpolation	33.83 ± 1.48	25.23 ± 1.15	39.62 ± 1.67	33.54 ± 1.24	40.04 ± 1.65	34.50 ± 1.24
RCNN	35.64 ± 1.49	32.61 ± 1.12	38.37 ± 1.51	35.80 ± 1.15	38.72 ± 1.45	37.04 ± 1.16
PCCNN	35.18 ± 1.50	31.13 ± 1.15	39.52 ± 1.54	36.43 ± 1.26	40.29 ± 1.49	39.93 ± 1.40
PCCNN-Bv	35.41 ± 1.58	31.45 ± 1.13	39.44 ± 1.76	36.77 ± 1.27	39.91 ± 1.76	39.96 ± 1.37
PCCNN-Sp	35.11 ± 1.59	31.25 ± 1.15	39.60 ± 1.56	36.70 ± 1.24	40.62 ± 1.58	40.28 ± 1.33
PCCNN-Bv-Sp	35.04 ± 1.38	31.35 ± 1.18	39.64 ± 1.70	36.93 ± 1.23	40.27 ± 1.66	40.35 ± 1.34
Table 7: Mean structural similarity index (MSSIM) of dMRI intensity in eight subjects with data derived from various models. Models use single-shell data with angular dimension size  $q_{\mathrm{in}}$  as input, and produce inferred data with angular dimension size  $90 - q_{\mathrm{in}}$  from the same shell. b-values are quoted in units of  $\mathrm{s} / \mathrm{mm}^2$ .
qin=6	qin=10	qin=20
b=1000	b=3000	b=1000	b=3000	b=1000	b=3000
SH Interpolation	0.957 ± 0.004	0.788 ± 0.010	0.989 ± 0.001	0.945 ± 0.004	0.990 ± 0.001	0.954 ± 0.004
RCNN	0.973 ± 0.002	0.932 ± 0.005	0.986 ± 0.001	0.964 ± 0.003	0.988 ± 0.001	0.971 ± 0.002
PCCNN	0.968 ± 0.003	0.910 ± 0.006	0.989 ± 0.001	0.970 ± 0.003	0.992 ± 0.001	0.985 ± 0.002
PCCNN-Bv	0.970 ± 0.003	0.916 ± 0.006	0.989 ± 0.001	0.971 ± 0.003	0.991 ± 0.001	0.986 ± 0.002
PCCNN-Sp	0.968 ± 0.003	0.911 ± 0.006	0.989 ± 0.001	0.970 ± 0.003	0.992 ± 0.001	0.986 ± 0.001
PCCNN-Bv-Sp	0.968 ± 0.003	0.913 ± 0.007	0.989 ± 0.001	0.972 ± 0.002	0.992 ± 0.001	0.986 ± 0.001
Training parameters such as number of iterations, batch size, learning rate, hardware used, and optimisation method remain consistent with those detailed in Section 3.6. Model training took approximately 6 hours to complete.
A.4 SR-q-DL Data and Training
The data sampling used to train the SR-q-DL for this work differs from that outlined in Ye et al. [41]. As in Section A.3, this was done to match the PCCNN training scheme. Here, each training example input was a low angular resolution single-shell dMRI dataset with b-value  $b_{\mathrm{in}}$  and angular dimension  $q_{\mathrm{in}}$  determined via
$$
q _ {\mathrm {i n}} \sim U (q _ {\mathrm {s a m p l e}}), q _ {\mathrm {s a m p l e}} = \{6, 7, \dots , 1 9, 2 0 \}, b _ {\mathrm {i n}} \sim U (b _ {\mathrm {s a m p l e}}), b _ {\mathrm {s a m p l e}} = \{1 0 0 0, 2 0 0 0, 3 0 0 0 \}.
$$
Training parameters such as number of iterations, batch size, learning rate, hardware used, and optimisation method remain consistent with those detailed in Section 3.6. Model training took approximately 30 minutes to complete.
A.5 Spherical Harmonic Interpolation
SH interpolation of single-shell dMRI data followed the same procedure as outlined in Lyon et al. [26]. Briefly, SH coefficients for each spatial voxel within the low angular resolution dataset were fit using the pseudo-inverse least squares method using a spherical harmonic order of 2. These derived SH coefficients were then used to calculate the interpolated spatial voxels for the target b-vectors.
A.6 Single-shell dMRI PSNR and MSSIM
Within Table 6 we present the peak signal-to-noise ratio (PSNR) of dMRI intensity within the single-shell experiment. Similarly, Table 7 outlines the mean structural similarity index (MSSIM) of dMRI intensity within the same experiment.
A.7 Additional Single-shell dMRI Analysis
This section provides performance metrics of models in single-shell inference using  $b = 2000\mathrm{s} / \mathrm{mm}^2$  data. Specifically, Table 8 outlines the AE of dMRI intensity, whilst Table 9 and Table 10 outline the PSNR and MSSIM of dMRI intensity respectively.
Table 8: Absolute error (AE) 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  ${q}_{\text{in }}$  as input,and produce inferred  $b = {2000}\mathrm{\;s}/{\mathrm{{mm}}}^{2}$  data with angular dimension size  ${90} - {q}_{\text{in }}$  .
qin = 6	qin = 10	qin = 20
SH Interpolation	103.53 ± 4.64	45.63 ± 2.21	43.67 ± 2.25
RCNN	61.60 ± 2.43	42.79 ± 1.75	39.14 ± 1.93
PCCNN	65.72 ± 2.83	37.93 ± 1.93	30.21 ± 1.81
PCCNN-Bv	64.63 ± 2.74	36.98 ± 1.94	30.84 ± 1.93
PCCNN-Sp	65.74 ± 2.78	36.82 ± 1.77	28.39 ± 1.49
PCCNN-Bv-Sp	65.33 ± 2.71	36.18 ± 1.66	28.32 ± 1.48
A.8 Angular Correlation Coefficient
The angular correlation coefficient (ACC) is given by
$$
\operatorname {A C C} (\alpha , \beta) = \frac {\sum_ {l = 0} ^ {8} \sum_ {m = - l} ^ {l} \alpha_ {l , m} \beta_ {l , m} ^ {*}}{\sqrt {\sum_ {l = 0} ^ {8} \sum_ {m = - l} ^ {l} \alpha_ {l , m} ^ {2}} \sqrt {\sum_ {l = 0} ^ {8} \sum_ {m = - l} ^ {l} \beta_ {l , m} ^ {2}}}, \tag {5}
$$