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Understanding Trade-Offs in Continuous Neural Representations for Diffeomorphic Image Registration: A Comparative Study of Implicit Neural Representations and Neural Ordinary Differential Equations.

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The authors' code

Python · 107 lines · 3.8 KB · no license

  1. import torch
  2. import torch.nn.functional as F
  4. class NCC(torch.nn.Module):
  5. """
  6. NCC with cumulative sum implementation for acceleration. local (over window) normalized cross correlation.
  7. """
  9. def __init__(self, win=21, eps=1e-5):
  10. super(NCC, self).__init__()
  11. self.eps = eps
  12. self.win = win
  13. self.win_raw = win
  15. def window_sum_cs3D(self, I, win_size):
  16. half_win = int(win_size / 2)
  17. pad = [half_win + 1, half_win] * 3
  19. I_padded = F.pad(I, pad=pad, mode='constant', value=0) # [x+pad, y+pad, z+pad]
  21. # Run the cumulative sum across all 3 dimensions
  22. I_cs_x = torch.cumsum(I_padded, dim=2)
  23. I_cs_xy = torch.cumsum(I_cs_x, dim=3)
  24. I_cs_xyz = torch.cumsum(I_cs_xy, dim=4)
  26. x, y, z = I.shape[2:]
  28. # Use subtraction to calculate the window sum
  29. I_win = I_cs_xyz[:, :, win_size:, win_size:, win_size:] \
  30. - I_cs_xyz[:, :, win_size:, win_size:, :z] \
  31. - I_cs_xyz[:, :, win_size:, :y, win_size:] \
  32. - I_cs_xyz[:, :, :x, win_size:, win_size:] \
  33. + I_cs_xyz[:, :, win_size:, :y, :z] \
  34. + I_cs_xyz[:, :, :x, win_size:, :z] \
  35. + I_cs_xyz[:, :, :x, :y, win_size:] \
  36. - I_cs_xyz[:, :, :x, :y, :z]
  38. return I_win
  40. def forward(self, I, J):
  41. # compute CC squares
  42. I = I.double()
  43. J = J.double()
  45. I2 = I * I
  46. J2 = J * J
  47. IJ = I * J
  49. # compute local sums via cumsum trick
  50. I_sum_cs = self.window_sum_cs3D(I, self.win)
  51. J_sum_cs = self.window_sum_cs3D(J, self.win)
  52. I2_sum_cs = self.window_sum_cs3D(I2, self.win)
  53. J2_sum_cs = self.window_sum_cs3D(J2, self.win)
  54. IJ_sum_cs = self.window_sum_cs3D(IJ, self.win)
  56. win_size_cs = (self.win * 1.) ** 3
  58. u_I_cs = I_sum_cs / win_size_cs
  59. u_J_cs = J_sum_cs / win_size_cs
  61. cross_cs = IJ_sum_cs - u_J_cs * I_sum_cs - u_I_cs * J_sum_cs + u_I_cs * u_J_cs * win_size_cs
  62. I_var_cs = I2_sum_cs - 2 * u_I_cs * I_sum_cs + u_I_cs * u_I_cs * win_size_cs
  63. J_var_cs = J2_sum_cs - 2 * u_J_cs * J_sum_cs + u_J_cs * u_J_cs * win_size_cs
  65. cc_cs = cross_cs * cross_cs / (I_var_cs * J_var_cs + self.eps)
  66. cc2 = cc_cs # cross correlation squared
  68. # return negative cc.
  69. return 1. - torch.mean(cc2).float()
  71. def JacboianDet(J):
  72. if J.size(-1) != 3:
  73. J = J.permute(0, 2, 3, 4, 1)
  74. J = J + 1
  75. J = J / 2.
  76. scale_factor = torch.tensor([J.size(1), J.size(2), J.size(3)]).to(J).view(1, 1, 1, 1, 3) * 1.
  77. J = J * scale_factor
  79. dy = J[:, 1:, :-1, :-1, :] - J[:, :-1, :-1, :-1, :]
  80. dx = J[:, :-1, 1:, :-1, :] - J[:, :-1, :-1, :-1, :]
  81. dz = J[:, :-1, :-1, 1:, :] - J[:, :-1, :-1, :-1, :]
  83. Jdet0 = dx[:, :, :, :, 0] * (dy[:, :, :, :, 1] * dz[:, :, :, :, 2] - dy[:, :, :, :, 2] * dz[:, :, :, :, 1])
  84. Jdet1 = dx[:, :, :, :, 1] * (dy[:, :, :, :, 0] * dz[:, :, :, :, 2] - dy[:, :, :, :, 2] * dz[:, :, :, :, 0])
  85. Jdet2 = dx[:, :, :, :, 2] * (dy[:, :, :, :, 0] * dz[:, :, :, :, 1] - dy[:, :, :, :, 1] * dz[:, :, :, :, 0])
  87. Jdet = Jdet0 - Jdet1 + Jdet2
  88. return Jdet
  90. def neg_Jdet_loss(J):
  91. Jdet = JacboianDet(J)
  92. neg_Jdet = -1.0 * (Jdet - 0.5)
  93. selected_neg_Jdet = F.relu(neg_Jdet)
  94. return torch.mean(selected_neg_Jdet ** 2)
  96. def smoothloss_loss(df):
  97. return (((df[:, :, 1:, :, :] - df[:, :, :-1, :, :]) ** 2).mean() + \
  98. ((df[:, :, :, 1:, :] - df[:, :, :, :-1, :]) ** 2).mean() + \
  99. ((df[:, :, :, :, 1:] - df[:, :, :, :, :-1]) ** 2).mean())
  101. def magnitude_loss(all_v):
  102. all_v_x_2 = all_v[:, :, 0, :, :, :] * all_v[:, :, 0, :, :, :]
  103. all_v_y_2 = all_v[:, :, 1, :, :, :] * all_v[:, :, 1, :, :, :]
  104. all_v_z_2 = all_v[:, :, 2, :, :, :] * all_v[:, :, 2, :, :, :]
  105. all_v_magnitude = torch.mean(all_v_x_2 + all_v_y_2 + all_v_z_2)
  106. return all_v_magnitude

Loss.py at commit fb01066, no license · at the source

Overview

Authors: Salvador Rodriguez-Sanz1,2, Carlos Paesa-Lia1,2, Monica Hernandez1,2
ORCID iDs: Carlos Paesa-Lia, Monica Hernandez
  1. Aragon Institute of Engineering Research (I3A), University of Zaragoza (UZ), 50018 Zaragoza, Spain; (S.R.-S.); (C.P.-L.)
  2. Computer Science and Systems Engineering Department, University of Zaragoza (UZ), 50018 Zaragoza, Spain
Institutions: Universidad de Zaragoza (Spain); Instituto de Investigación en Ingeniería de Aragón (Spain)
Journal: Journal of imaging, volume 12, issue 8, article 384
Dates: received 1 July 2026; accepted 4 August 2026; published online 14 August 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.3390/jimaging12080384 · PMID 42645983 · PMCID PMC13514420 · OpenAlex W7203478090
Open access: gold, a free copy (OpenAlex)
Status: code verified
Categories: systems (subfield)
Methods: Connectivity, Statistics
Keywords: Large Deformation Diffeomorphic Metric Mapping (LDDMM), diffeomorphic image registration, implicit neural representations (INRs), neural ordinary differential equations (NODEs), neural representations, continuous transformation modeling
Topic: Medical Image Segmentation Techniques (Computer Vision and Pattern Recognition, Computer Science), according to OpenAlex
Funding: Gobierno de Aragón (T64_23R, ECU/1871/2023, PROY_B50_24); European Union (CLASiK - PCI2025-167187-2); Instituto de Salud Carlos III (RD24/0007/0022); European Union Satellite Centre (HORIZON-MSCA-2024-SE-01 G.A. 10123661); Ministerio de Ciencia Innovación y Universidades (PID2023-148219OB-C22, PID2022-138703OB-I00)
Citations: not cited yet (Europe PMC); 42 references in the paper

Abstract

Non-rigid image registration is a fundamental problem in medical imaging and a representative example of continuous transformation modeling in image processing. Diffeomorphic registration methods, such as Large Deformation Diffeomorphic Metric Mapping (LDDMM) and its PDE-constrained variants (PDE-LDDMM), provide mathematically grounded formulations with strong geometric guarantees for transformation quality. However, existing approaches face persistent trade-offs between numerical stability, accuracy, and computational efficiency. Recent work has explored implicit neural representations (INRs) and neural ordinary differential equations (NODEs) as flexible neural representations for modeling continuous transformations. Despite their increasing adoption, their practical behavior and limitations in diffeomorphic registration remain insufficiently understood. In this paper, we present a unified formulation of INR- and NODE-based registration methods within LDDMM and PDE-LDDMM, enabling a systematic and controlled comparison across architectures, sampling strategies, and numerical solvers. Our analysis reveals fundamental trade-offs between these approaches. In particular, we show that MLP-based INR formulations introduce significant computational overhead and rely on sampling strategies that can degrade smoothness and lead to the increased occurrence of non-diffeomorphic transformations at higher resolutions. Moreover, these approximations do not fully alleviate the computational cost, with some variants exceeding the costs of expensive classical optimization-based methods. In contrast, NODE-based formulations and downsampling strategies consistently provide transformations with more controlled Jacobian extrema while maintaining competitive computational performance. Among the evaluated methods, the original NODE-LDDMM and NODE-PDE-LDDMM formulations achieve the most favorable trade-offs between registration accuracy, geometric consistency, and computational efficiency. These findings provide clear insights into the design of neural representations for continuous transformation modeling, with practical implications for diffeomorphic registration and computational anatomy applications.

Reproduced under the paper's license (CC BY), from the paper cited above.

Repositories

Its files are read in the Code ↔ Paper reader above.

yifannnwu/NODEO-DIR

License: none: the authors keep all their rights
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Commit: fb010664d02ca6c6bca01c0f50f2de17f61d23ed, 4 February 2024
Languages: Python (5)
Size: 10 files, 5 scripts
Software Heritage: not archived
Found in: the text, “5.3. Implementation Details and Parameters”
Holds: README
Not found: license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: PyTorch (4 files), NumPy (3 files), NiBabel (1 file), SciPy (1 file)
Availability: 1 check, the latest on 27 September 2026: the link answers
  • 27 September 2026: the link answers
6 files
At the source: github.com/yifannnwu/NODEO-DIR

MIAGroupUT/IDIR

License: MIT
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Commit: 9ab3ccfbbf09dc8fc65fa848332e7d9a0b3b81b3, 23 March 2022
Languages: Python (12)
Size: 21 files, 12 scripts
Software Heritage: not archived
Found in: the text, “5.3. Implementation Details and Parameters”
Holds: README, license file, environment (environment.yml), continuous integration
Not found: CITATION.cff, tests, documentation
Tools: PyTorch (5 files), NumPy (2 files), SimpleITK (1 file)
Availability: 1 check, the latest on 27 September 2026: the link answers
  • 27 September 2026: the link answers
14 files
At the source: github.com/MIAGroupUT/IDIR

BrainImageAnalysis/INRsRegExp

License: MIT
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Commit: 74036e2b761b252375dfda40dca3fd6bfa19deaa, 5 December 2023
Languages: Python (5), Jupyter (2)
Size: 16 files, 7 scripts
Software Heritage: not archived
Found in: the text, “5.3. Implementation Details and Parameters”
Holds: README, license file, 2 notebooks
Not found: CITATION.cff, environment file, tests, continuous integration, documentation
Tools: NumPy (6 files), PyTorch (6 files), Matplotlib (3 files), NiBabel (1 file), SciPy (1 file)
Availability: 1 check, the latest on 27 September 2026: the link answers
  • 27 September 2026: the link answers
9 files
At the source: github.com/BrainImageAnalysis/INRsRegExp

mhglddmm/INRvsNODE_LDDMM

License: GPL-3.0
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Commit: cc9b0536f74f33bc05dab0d46abe76c994ed6e11, 28 July 2026
Size: 1 file, 0 scripts
Software Heritage: not archived
Found in: “Data Availability Statement”
Holds: license file
Not found: README, CITATION.cff, environment file, tests, continuous integration, documentation
Availability: 1 check, the latest on 27 September 2026: the link answers
  • 27 September 2026: the link answers
1 file
At the source: github.com/mhglddmm/INRvsNODE_LDDMM

The paper's code and data availability statement is in the Data section.

Tracing map

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  • 4 repositories of the authors' code, each at its verified commit, with its license and how the link was found in the paper;
  • 24 scripts, each with its path and the digest of its content;
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Data

Datasets cited

Data Availability Statement

The NIREP dataset is publicly available through the repository maintained by Andreas Mang https://github.com/andreasmang/nirep (accessed on 3 August 2026). Additional information regarding the Non-rigid Image Registration Evaluation Project (NIREP) can be found at https://www.nitrc.org/projects/nirep/ (accessed on 3 August 2026). The OASIS Learn2Reg dataset is publicly available through the Learn2Reg challenge website https://learn2reg.grand-challenge.org/Learn2Reg2021 (accessed on 3 August 2026). The main experimental results generated in this work will be made publicly available through a public data repository upon acceptance of the manuscript. The source code used in this study, together with the documentation required to reproduce the main experiments, will be made publicly available upon acceptance of the manuscript at https://github.com/mhglddmm/INRvsNODE_LDDMM (accessed on 3 August 2026).

Reproduced under the paper's license (CC BY), from the paper cited above.

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Version 1, 27 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 3 authors, 6 keywords, 5 funders, 38 references.

Cite

This paper

Rodriguez-Sanz, S., Paesa-Lia, C., & Hernandez, M. (2026). Understanding Trade-Offs in Continuous Neural Representations for Diffeomorphic Image Registration: A Comparative Study of Implicit Neural Representations and Neural Ordinary Differential Equations. Journal of imaging, 12(8), 384. https://doi.org/10.3390/jimaging12080384

BibTeX

@article{rodriguezsanz2026understanding,
author = {Rodriguez-Sanz, Salvador and Paesa-Lia, Carlos and Hernandez, Monica},
title = {{Understanding Trade-Offs in Continuous Neural Representations for Diffeomorphic Image Registration: A Comparative Study of Implicit Neural Representations and Neural Ordinary Differential Equations}},
journal = {Journal of imaging},
year = {2026},
month = aug,
volume = {12},
number = {8},
pages = {384},
publisher = {Multidisciplinary Digital Publishing Institute (MDPI)},
issn = {2313-433X},
doi = {10.3390/jimaging12080384},
url = {https://doi.org/10.3390/jimaging12080384},
pmid = {42645983},
pmcid = {PMC13514420}
}

RIS

TY - JOUR
AU - Rodriguez-Sanz, Salvador
AU - Paesa-Lia, Carlos
AU - Hernandez, Monica
TI - Understanding Trade-Offs in Continuous Neural Representations for Diffeomorphic Image Registration: A Comparative Study of Implicit Neural Representations and Neural Ordinary Differential Equations
T2 - Journal of imaging
J2 - J Imaging
PY - 2026
DA - 2026/08/14
VL - 12
IS - 8
SP - 384
SN - 2313-433X
PB - Multidisciplinary Digital Publishing Institute (MDPI)
DO - 10.3390/jimaging12080384
UR - https://doi.org/10.3390/jimaging12080384
LA - en
ER -

CSL-JSON

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"PMCID": "PMC13514420",
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