Implicit Neural Models for Direct Multi-Shell Diffusion MRI Signal Estimation

Sanna Persson; Fabian Leander Sinzinger; Christoffer Olsson; Rodrigo Moreno

eess.IVOct 16, 2026

Implicit Neural Models for Direct Multi-Shell Diffusion MRI Signal Estimation

Authors: Sanna Persson, Fabian Leander Sinzinger, Christoffer Olsson, Rodrigo Moreno

Organizations: Biomedical Engineering and Health Systems, KTH Royal Institute of Technology, Stockholm, Sweden

Abstract

Diffusion magnetic resonance imaging is a noninvasive technique that is sensitive to the motion of water molecules in human tissue. The image signal is commonly described in q-space; however, a complete coverage of this space is often infeasible due to hardware and acquisition time limitations. In this work, we propose to represent the entire diffusion signal using an implicit neural representation network with wavelet activation functions. The signal space is encoded using Fourier features and spherical harmonics basis functions for a continuous output distribution of the raw diffusion data, enabling both super-resolution and q-space interpolation. Additionally, we introduce a pretraining strategy using a group template, enabling faster per-subject optimisation when combined with nonlinear registrations of subjects into template space. We tested the method with 10 subjects from the Human Connectome Project and 26 subjects from in-house data (multidimensional diffusion MRI from a Parkinson's disease cohort). The results show that the proposed method yields high-fidelity reconstructions with significantly reduced memory requirements compared to raw diffusion-weighted images. The pretraining scheme showed good generalisability properties on data acquired with very different protocols. We further assess the method on downstream metrics relevant to diffusion MRI data from the learned representations.

Figures & tables

Figure 1: Overview of the proposed representation for learning the DWI signal using an implicit neural representation (INR). The multi-shell DWI is represented as pairs of signal-space coordinates and corresponding scalar amplitude values. The spherical q-space signal-space coordinates are encoded via spherical harmonics function evaluations (bottom, centre).

Figure 2: a) First, the DWI signal, defined at multi-shell gradient directions, is transformed into the spherical harmonics domain by performing a linear least squares fit of the harmonic coefficients per shell. b) Then, individual shells in spherical harmonics form are affine-transformed and subsequently nonlinearly warped to MNI152 space. The affine and nonlinear deformations were obtained from the official HCP preprocessing pipeline [ 36 ] . c) Next, the spherical harmonics coefficients are averaged across all 50 subjects. d) Finally, a DWI template in spatial form is reconstructed by sampling the spherical harmonics representation at discrete gradient directions.

Method	MSE $(\times 10^{-4})$ $\downarrow$	MAE $\downarrow$	RE (%) $\downarrow$	PSNR (dB) $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
Leaky	2.52	0.0113	7.05	32.37	0.972	0.0164
SIREN	1.81	0.0100	7.35	33.75	0.978	0.0120
Wire	0.73	0.0063	4.43	37.74	0.991	0.0041

Table 1: Overall reconstruction performance (averaged across all voxels and shells) for the DWI template. Lower is better for MSE/MAE/RE/LPIPS, higher is better for PSNR/SSIM.

Training setup	MSE ( $\times 10^{-4}$ ) $\downarrow$	MAE $\downarrow$	RE (%) $\downarrow$	PSNR (dB) $\uparrow$	SSIM $\uparrow$
Unregistered	$0.82\!\pm\!0.18$	$0.0065\!\pm\!0.00078$	$20.43\!\pm\!0.39$	$29.25\!\pm\!0.64$	$0.989\!\pm\!0.002$
Pretrained Unreg.	$0.92\!\pm\!0.21$	$0.0069\!\pm\!0.00080$	$21.56\!\pm\!0.31$	$27.09\!\pm\!0.58$	$0.844\!\pm\!0.009$
Registered	$0.40\!\pm\!0.05$	$0.0047\!\pm\!0.00029$	$5.30\!\pm\!1.15$	$35.46\!\pm\!0.65$	$0.989\!\pm\!0.001$
Pretrained Reg.	$0.70\!\pm\!0.07$	$0.0059\!\pm\!0.00037$	$6.10\!\pm\!1.26$	$33.49\!\pm\!0.56$	$0.986\!\pm\!0.001$

Table 2: Overall reconstruction performance (mean ± std, averaged across all voxels and shells) on HCP subjects. Training time for the pretrained setups is a factor of 1/4 compared to the not pretrained ones.

Training setup	MSE ( $\times 10^{-4}$ ) $\downarrow$	MAE $\downarrow$	RE (%) $\downarrow$	PSNR (dB) $\uparrow$	SSIM $\uparrow$
Unregistered	$5.80\!\pm\!0.15$	$0.0146\!\pm\!0.0018$	$15.88\!\pm\!3.92$	$26.39\!\pm\!1.17$	$0.972\!\pm\!0.005$
Pretrained Unreg.	$10.42\!\pm\!1.73$	$0.0195\!\pm\!0.0018$	$22.22\!\pm\!6.16$	$24.13\!\pm\!1.46$	$0.950\!\pm\!0.008$
Registered	$1.77\!\pm\!0.26$	$0.0092\!\pm\!0.0007$	$7.52\!\pm\!1.07$	$29.42\!\pm\!0.51$	$0.972\!\pm\!0.003$
Pretrained Reg.	$2.37\!\pm\!0.33$	$0.0103\!\pm\!0.0008$	$7.95\!\pm\!1.23$	$27.97\!\pm\!0.59$	$0.963\!\pm\!0.004$

Table 3: Overall reconstruction performance (mean ± std across subjects, averaged across all voxels and shells) for 26 subjects from the ParkMR Dataset.

Training setup	b-value	Metric
Training setup	b-value	MSE ( $\times 10^{-4}$ ) $\downarrow$	MAE $\downarrow$	RE (%) $\downarrow$	PSNR (dB) $\uparrow$	SSIM $\uparrow$
Unregistered	0	$1.70\!\pm\!0.37$	$0.0095\!\pm\!0.0011$	$17.94\!\pm\!2.01$	$37.28\!\pm\!0.95$	$0.991\!\pm\!0.001$
	1000	$0.64\!\pm\!0.16$	$0.0062\!\pm\!0.00080$	$16.11\!\pm\!0.42$	$28.15\!\pm\!0.85$	$0.987\!\pm\!0.002$
	2000	$0.52\!\pm\!0.14$	$0.0056\!\pm\!0.00074$	$22.09\!\pm\!0.55$	$25.85\!\pm\!0.91$	$0.988\!\pm\!0.002$
	3000	$0.41\!\pm\!0.11$	$0.0049\!\pm\!0.00068$	$25.52\!\pm\!0.38$	$25.71\!\pm\!0.95$	$0.990\!\pm\!0.002$
Pretrained Unreg.	0	$1.89\!\pm\!0.42$	$0.0100\!\pm\!0.0011$	$20.74\!\pm\!2.38$	$26.30\!\pm\!0.96$	$0.919\!\pm\!0.012$

Table 4: Reconstruction quality per b-value for the four training setups (mean ± std across subjects) for the HCP data.

Figure 3: Relative error (mean $\pm$ std) of the Wire model when trained with 10-70% of diffusion directions for the registered HCP data with 288 gradient directions in total.

% of training	MSE ( $\times 10^{-4}$ ) $\downarrow$	MAE $\downarrow$	RE (%) $\downarrow$	PSNR (dB) $\uparrow$	SSIM $\uparrow$
10%	3.63 ± 0.70	0.0125 ± 0.0013	12.6 ± 2.6	26.7 ± 0.5	0.950 ± 0.006
30%	1.17 ± 0.13	0.0074 ± 0.0004	7.7 ± 1.6	31.4 ± 0.6	0.978 ± 0.001
50%	0.74 ± 0.14	0.0062 ± 0.0006	6.5 ± 1.4	33.1 ± 0.7	0.984 ± 0.002
70%	0.60 ± 0.10	0.0056 ± 0.0005	6.0 ± 1.2	33.8 ± 0.5	0.986 ± 0.002

Table 5: Overall reconstruction quality on the HCP subset for the Wire model with varying fractions of gradient directions used for training (mean ± std across subjects)

Figure 4: Visualisation of predicted and ground truth B0, MD and FA for a HCP subject.

Training setup	Parameter	Metric
Training setup	Parameter	MSE	MAE	Mean RE (%)	(25th RE, 75th RE) %
Unregistered	MD	$0.000\pm 0.0000$	$0.000061\pm 0.000005$	$7.35\pm 0.45$	( $2.56,10.31$ )
Unregistered	FA (WM)	$0.027\pm 0.0039$	$0.1371\pm 0.0110$	$34.41\pm 2.82$	( $17.14,50.25$ )
Pretrained Unreg	MD	$0.000\pm 0.0000$	$0.000069\pm 0.000006$	$8.50\pm 0.50$	( $2.92,12.00$ )
Pretrained Unreg	FA (WM)	$0.039\pm 0.0054$	$0.1678\pm 0.0131$	$41.74\pm 3.23$	( $23.63,59.50$ )
Registered	MD	$0.000\pm 0.0000$	$0.000021\pm 0.000006$	$4.37\pm 0.77$	( $1.55,6.07$ )

Table 6: Reconstruction quality for MD and FA (WM) parameters for the four training variants (mean

\pm

std for MSE, MAE, Mean RE; mean for percentile RE) for the HCP subset.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Signal flow diagram of the ParkMR nonlinear registration pipeline, exported from Nipype. The pipeline include the following stages: (1) general DWI preprocessing using MRtrix3; (2.a) nonlinear registration of T1 images to MNI 2 mm space using ANTs with affine and SyN transformations; (2.b) rigid registration of the mean $b_{0}$ volume to the T1 image; (3) conversion of shell-specific DWI amplitudes to spherical harmonics (SH) with $l_{\text{max}}=8$ ; (4) spatial transformation of SH volumes using MRtrix3 with FOD reorientation; and (5) projection of SH signals onto a predefined, shell-specific gradient scheme, including duplication of the $b_{0}$ image. Nodes use MRtrix3, ANTs, and utility functions within Nipype.

Subject Info	Controls	PD Patients
Mean age $\pm$ SD [min–max]	$60\pm 6\,[50\text{\textendash}78]$	$63\pm 9\,[47\text{\textendash}78]$
Gender [F:M]	3:14	3:9
Disease duration [years]	–	$6.4\pm 5.3\,[1\text{\textendash}17]$
LEDD [mg]	–	$479\pm 301\,[72\text{\textendash}958]$
H & Y stage	–	$1.75\pm 0.45\,[1\text{\textendash}2]$
MoCA	–	$28.9\pm 1.2\,[27\text{\textendash}30]$

Appendix

Table 7: Demographic and clinical characteristics of the study cohort for the MD-dMRI data. Values are reported as mean ± SD with the range in square brackets. A dash (–) indicates “not applicable”.

Training setup	b-value	Metric
Training setup	b-value	MSE ( $\times 10^{-4}$ ) $\downarrow$	MAE $\downarrow$	RE (%) $\downarrow$	PSNR (dB) $\uparrow$	SSIM $\uparrow$
Unregistered	0	$19.84\!\pm\!5.30$	$0.0317\!\pm\!0.0041$	$19.13\!\pm\!3.42$	$27.04\!\pm\!1.15$	$0.9546\!\pm\!0.0076$
	100	$5.55\!\pm\!1.52$	$0.0164\!\pm\!0.0022$	$11.69\!\pm\!2.33$	$30.06\!\pm\!0.81$	$0.9778\!\pm\!0.0035$
	700	$1.51\!\pm\!0.39$	$0.0094\!\pm\!0.0012$	$13.38\!\pm\!3.51$	$25.47\!\pm\!1.89$	$0.9733\!\pm\!0.0054$
	1400	$1.14\!\pm\!0.32$	$0.0081\!\pm\!0.0012$	$16.92\!\pm\!5.24$	$24.58\!\pm\!1.98$	$0.9761\!\pm\!0.0060$
	2000	$0.95\!\pm\!0.28$	$0.0073\!\pm\!0.0011$	$18.27\!\pm\!6.13$	$24.78\!\pm\!2.11$	$0.9786\!\pm\!0.0060$

Appendix

Table 8: Quantitative metrics (mean ± std across subjects) for the ParkMR data. Lower is better for MSE/MAE/RE, higher is better for PSNR/SSIM.

Method	b-value	Metric
Method	b-value	MSE $(\times 10^{-4})$ $\downarrow$	MAE $\downarrow$	RE (%) $\downarrow$	PSNR (dB) $\uparrow$	SSIM $\uparrow$
Leaky	0	2.24	0.0108	7.18	36.50	0.980
	1000	4.07	0.0148	6.51	31.57	0.969
	2000	2.22	0.0108	7.08	30.75	0.968
	3000	1.58	0.0088	7.41	30.63	0.970
Siren	0	1.09	0.0079	6.86	39.63	0.990

Appendix

Table 9: Reconstruction quality per b-value for the Leaky , Siren and Wire models on the DWI template. Lower is better for MSE/MAE/RE, higher is better for PSNR/SSIM.

Figure 6: Fibre Orientation Distributions (FODs) from a coronal slice of the group-averaged DWI template. Ground truth FODs (top) and predictions from the Wire model (bottom) are shown.

Figure 7: Native space. Qualitative comparison in native DWI space (no template registration). Four columns correspond to the target shells ( $b=0$ , $1000$ , $2000$ , $3000$ s/mm 2 ); within each column, the top pair shows the INR prediction ( Pred ) and the bottom pair the acquired ground truth ( GT ). Rows 1–2: INR trained from scratch in native space ( Unregistered ). Rows 3–4: template-initialised INR fitted in native space ( Pretrained – Reg ). Template pretraining reduces training time to one-quarter.

Figure 8: Registered space. Same layout as Fig. 7 , but after rigid registration to the $b\!=\!0$ volume. Rows 1–2: INR trained from scratch in registered space ( Registered ). Rows 3–4: template-initialised INR ( Pretrained + Reg ) with 1/4 of training time.

Parameter Category	Hyperparameter	Value
Optimization	Learning Rate ( lr )	varied
	Cosine LR Scheduler ( use_cos_lrsheduler )	True
	Training Batch Size ( train_batch_size )	262,144
	Epochs ( epochs )	varied
Model Architecture	Network Type ( network_name )	varied
	Input Dimensionality ( in_size )	301

Appendix

Table 10: Key hyperparameters for the model training and architecture. Variable names from the configuration are shown in monospace . The parameters that differ across the experiments are indicated by: varied .

Model Type	Parameter	Value
Siren	First Layer $\omega_{0}$	30.0
	Hidden Layer $\omega_{h}$	30.0
Wire	$\omega$	10.0
	$\sigma$	40.0
Shared Encoders	Fourier Encoder Frequencies	128
	Fourier Feature Dim.	256

Appendix

Table 11: Model-specific parameters for Wire , Siren models and feature encoding methods.

Parameter Category	Hyperparameter	Value
Optimization	Learning Rate ( lr )	$0.001$
	Cosine LR Scheduler ( use_cos_lrsheduler )	True
	Training Batch Size ( train_batch_size )	262,144
	Epochs ( epochs )	64
Model Architecture	Network Type ( network_name )	Leaky ReLU, Siren , Wire
	Input Dimensionality ( in_size )	301

Appendix

Table 12: Hyperparameters for training the INR on the group-averaged DWI template.

Parameter Category	Hyperparameter	Value
Optimization	Learning Rate ( lr )	$0.0001$
	Cosine LR Scheduler ( use_cos_lrsheduler )	True
	Training Batch Size ( train_batch_size )	262,144
	Epochs ( epochs )	Baseline: 64, Pretrained: 16
Model Architecture	Network Type ( network_name )	Wire
	Input Dimensionality ( in_size )	301

Appendix

Table 13: Hyperparameters for pretraining and registration experiments.

Parameter Category	Hyperparameter	Value
Optimization	Learning Rate ( lr )	$0.001$
	Cosine LR Scheduler ( use_cos_lrsheduler )	True
	Training Batch Size ( train_batch_size )	262,144
	Epochs ( epochs )	16
Model Architecture	Network Type ( network_name )	Wire (from checkpoint)
	Input Dimensionality ( in_size )	301

Appendix

Table 14: Key hyperparameters for the b-vector interpolation experiments.

Parameter Category	Hyperparameter	Value
Optimization	Learning Rate ( lr )	0.001
	Cosine LR Scheduler ( use_cos_lrsheduler )	True
	Training Batch Size ( train_batch_size )	262,144
	Epochs ( epochs )	32
	L2-regularisation strength	0.0001
	b-value dropout	0.5

Appendix

Table 15: Key hyperparameters for the b-value interpolation experiments.

Figure 9: Relative error (mean ± std) of the SHORE model when fitted to 10–70% of diffusion directions for the registered HCP data with 288 gradient directions in total. Data and setup correspond to the main paper, Figure 3.

Parameter Category	Hyperparameter	Value
Optimization	Learning Rate ( lr )	$0.001$
	Cosine LR Scheduler ( use_cos_lrsheduler )	True
	Training Batch Size ( train_batch_size )	262,144
	Epochs ( epochs )	32
Model Architecture	Network Type ( network_name )	Wire (from checkpoint)
	Input Dimensionality ( in_size )	301

Appendix

Table 16: Hyperparameters used for superresolution experiments.

Upscaling factor	b-value	Metric
Upscaling factor	b-value	MSE ( $\times 10^{-3}$ ) $\downarrow$	MAE $\downarrow$	RE (%) $\downarrow$	PSNR (dB) $\uparrow$	SSIM $\uparrow$
4×	0	$6.67\!\pm\!1.57$	$0.051\!\pm\!0.010$	$28.8\!\pm\!4.0$	$21.9\!\pm\!1.0$	$0.794\!\pm\!0.019$
	1000	$1.30\!\pm\!0.68$	$0.027\!\pm\!0.008$	$28.3\!\pm\!6.0$	$20.8\!\pm\!1.4$	$0.858\!\pm\!0.028$
	2000	$0.829\!\pm\!0.772$	$0.023\!\pm\!0.011$	$28.9\!\pm\!10.0$	$19.6\!\pm\!2.5$	$0.896\!\pm\!0.038$
	3000	$0.731\!\pm\!0.823$	$0.022\!\pm\!0.012$	$30.5\!\pm\!12.3$	$18.9\!\pm\!3.2$	$0.911\!\pm\!0.046$
2×	0	$4.17\!\pm\!1.17$	$0.039\!\pm\!0.010$	$20.4\!\pm\!3.7$	$24.0\!\pm\!1.2$	$0.879\!\pm\!0.009$

Appendix

Table 17: Reconstruction quality per b-value (mean ± std across subjects) for 4× and 2× superresolution on the HCP data.

Upscaling factor	MSE ( $\times 10^{-3}$ ) $\downarrow$	MAE $\downarrow$	RE (%) $\downarrow$	PSNR (dB) $\uparrow$	SSIM $\uparrow$
4×	$2.383\!\pm\!0.785$	$0.031\!\pm\!0.009$	$29.1\!\pm\!6.9$	$20.3\!\pm\!1.7$	$0.865\!\pm\!0.031$
2×	$1.372\!\pm\!0.472$	$0.022\!\pm\!0.007$	$19.9\!\pm\!5.1$	$23.4\!\pm\!1.7$	$0.930\!\pm\!0.014$
No upscaling	$0.40\!\pm\!0.05$	$0.0047\!\pm\!0.00029$	$5.30\!\pm\!1.15$	$35.46\!\pm\!0.65$	$0.989\!\pm\!0.001$

Appendix

Table 18: Overall reconstruction performance (mean ± std, averaged across all voxels and shells) for the two upscaling factors.

Diffusion MRI

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Implicit Neural Representations Medical Image Super-Resolution Transfer Learning Diffusion MRI

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Resolving complex fiber geometries in brain white matter requires high-resolution diffusion MRI at the cost of long acquisition times. This leads many clinical protocols to opt for low-resolution scans, making downstream microstructure estimation and tractography challenging. Implicit neural representations (INRs) can model the diffusion signal continuously, enabling native single-subject super-resolution by querying the network at arbitrary spatial coordinates, yet existing methods often suffer from long training times and lack a mechanism to incorporate anatomical priors to regularize super-resolution by constraining the space of plausible reconstructions. To address these limitations, we propose a novel transfer-learning framework that pre-trains an INR on a high-resolution template and then adapts it to subject-specific scans via registration and fine-tuning. For

4\times

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faster training, outperforming competing INR-based methods across both image quality and domain-specific metrics. Code is available on the project page at https://abdulkaderghandoura.github.io/research/msc-thesis/ .

Abstract

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