pith. sign in

arxiv: 2603.11806 · v2 · submitted 2026-03-12 · 🧮 math.GR · cs.CV

A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration

Lili Bao , Bin Xiao , Shihui Ying , Stefan Sommer This is my paper

Pith reviewed 2026-05-15 11:53 UTC · model grok-4.3

classification 🧮 math.GR cs.CV
keywords diffeomorphism groupoidLie algebroidimage registrationdiscontinuous deformationsliding motionEuler-Arnold equationsLDDMMpiecewise diffeomorphic
0
0 comments X

The pith

Diffeomorphism groupoids allow modeling of discontinuous sliding motions in image registration while keeping diffeomorphisms within regions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends traditional diffeomorphism Lie groups used in LDDMM to groupoids to handle discontinuities along sliding boundaries in image registration. This framework maintains smoothness inside homogeneous regions but permits jumps across boundaries. By analyzing the associated Lie algebroids and their duals, the authors derive Euler-Arnold equations that govern the optimal flows for such discontinuous deformations. A sympathetic reader would care because many real-world images involve sliding motions that standard continuous methods cannot capture accurately.

Core claim

We extend the diffeomorphism Lie groups to a framework of discontinuous diffeomorphism Lie groupoids, allowing for discontinuities along sliding boundaries while maintaining diffeomorphism within homogeneous regions. We provide a rigorous analysis of the associated mathematical structures, including Lie algebroids and their duals, and derive specific Euler-Arnold equations to govern optimal flows for discontinuous deformations.

What carries the argument

The discontinuous diffeomorphism Lie groupoid, which structures piecewise diffeomorphic transformations with discontinuities isolated to boundaries between regions.

Load-bearing premise

Sliding discontinuities can be cleanly isolated to boundaries between homogeneous regions and the groupoid structure produces valid optimal flows without instabilities.

What would settle it

If numerical solutions of the derived Euler-Arnold equations for test cases with known sliding boundaries fail to recover accurate deformations or produce unstable flows.

Figures

Figures reproduced from arXiv: 2603.11806 by Bin Xiao, Lili Bao, Shihui Ying, Stefan Sommer.

Figure 1
Figure 1. Figure 1: Illustration of the discontinuous image registration based on Lie [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of image registration. achieving registration between the two images. To ensure that the deformation field is as simple and smooth as possible, and close to the identity deformation, the regularization term ER(ϕ) can be defined using the distance given as follows [33, 37]: ER(ϕ) = d 2 V (Id, ϕ) = min v(t)∈V,ϕv 01=ϕ Z 1 0 ∥v(t)∥ 2 V dt. (2.3) The velocity v lies in the admissible Hilbert spa… view at source ↗
Figure 3
Figure 3. Figure 3: (left) Illustration of the geodesic flow on groups Diff(Ω) of diffeomor [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The structure of a groupoid G ⇒ B. (d) Inverse: For each g ∈ G, there exists an element g −1 ∈ G such that src(g −1 ) = trg(g), trg(g −1 ) = src(g), g −1 g = idsrc(g) , and gg−1 = idtrg(g) . The structure of a groupoid, along with its inverses, identities, and multi￾plication operations, is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the composition rule of two groupoid elements [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The structure of the Lie groupoid DDiff( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Registration results on synthetic rectangle images with sliding motion. [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Registration results on synthetic wheel images with sliding motion. [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Registration results on real lung images with sliding motion. Rows [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
read the original abstract

In this paper, we propose a novel mathematical framework for piecewise diffeomorphic image registration that involves discontinuous sliding motion using a diffeomorphism groupoid and algebroid approach. The traditional Large Deformation Diffeomorphic Metric Mapping (LDDMM) registration method builds on Lie groups, which assume continuity and smoothness in velocity fields, limiting its applicability in handling discontinuous sliding motion. To overcome this limitation, we extend the diffeomorphism Lie groups to a framework of discontinuous diffeomorphism Lie groupoids, allowing for discontinuities along sliding boundaries while maintaining diffeomorphism within homogeneous regions. We provide a rigorous analysis of the associated mathematical structures, including Lie algebroids and their duals, and derive specific Euler-Arnold equations to govern optimal flows for discontinuous deformations. Numerical tests are performed to validate the efficiency of the proposed approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a novel framework extending Large Deformation Diffeomorphic Metric Mapping (LDDMM) from Lie groups to discontinuous diffeomorphism Lie groupoids and associated algebroids. This allows piecewise diffeomorphic registrations with sliding discontinuities isolated to boundaries between homogeneous regions, while deriving modified Euler-Arnold equations on the algebroid dual to govern optimal flows and providing numerical validation of the approach.

Significance. If the central derivations hold, the work offers a structured extension of diffeomorphic registration methods to handle sliding motions without ad-hoc constraints, which could improve modeling in applications such as medical imaging of sliding organs or tissues. The groupoid/algebroid formalism provides a potential route to parameter-free optimal flows within regions, building on existing Lie-group techniques.

major comments (2)
  1. [Derivation of Euler-Arnold equations on algebroid dual] The derivation of the Euler-Arnold equations (likely in the section following the Lie algebroid dual construction) must explicitly show how the momentum map and coadjoint action are modified by the groupoid structure to accommodate discontinuities; the abstract and high-level description suggest the standard form is retained, but this requires verification that no additional instability terms arise at the sliding boundaries.
  2. [Numerical validation] The numerical tests section reports validation but lacks quantitative metrics (e.g., Dice scores, target registration error, or comparison against standard LDDMM on synthetic sliding datasets); without these, the claim that the framework produces valid optimal flows cannot be assessed for practical improvement.
minor comments (2)
  1. [Mathematical structures] Define the groupoid multiplication operation explicitly with an equation or diagram early in the mathematical framework section to clarify how it differs from standard Lie group multiplication.
  2. [Introduction] Add a short remark on the smoothness requirements within homogeneous regions to ensure the diffeomorphism property is preserved away from boundaries.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment point by point below, indicating the revisions we will incorporate.

read point-by-point responses

  1. Referee: The derivation of the Euler-Arnold equations (likely in the section following the Lie algebroid dual construction) must explicitly show how the momentum map and coadjoint action are modified by the groupoid structure to accommodate discontinuities; the abstract and high-level description suggest the standard form is retained, but this requires verification that no additional instability terms arise at the sliding boundaries.

    Authors: The derivation in the manuscript already incorporates the groupoid structure by replacing the standard Lie group coadjoint action with the algebroid dual version that respects the piecewise diffeomorphisms and boundary discontinuities. This yields modified Euler-Arnold equations without additional instability terms, as the velocity fields remain smooth within each homogeneous region and the boundary conditions enforce continuity of the momentum across sliding interfaces. To address the request for explicit verification, we will expand the derivation section with a step-by-step breakdown of the modified momentum map and coadjoint action, including a short lemma confirming the absence of instability terms at the boundaries. revision: partial

  2. Referee: The numerical tests section reports validation but lacks quantitative metrics (e.g., Dice scores, target registration error, or comparison against standard LDDMM on synthetic sliding datasets); without these, the claim that the framework produces valid optimal flows cannot be assessed for practical improvement.

    Authors: We agree that quantitative metrics are needed to strengthen the validation claims. The current numerical examples illustrate the qualitative behavior of the discontinuous flows, but we will revise the numerical tests section to include Dice scores, target registration errors, and direct comparisons against standard LDDMM on synthetic datasets with known sliding motions, thereby providing measurable evidence of improvement. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a direct extension of the LDDMM Lie-group framework to diffeomorphism groupoids and algebroids, deriving Euler-Arnold equations on the dual algebroid from the new groupoid structure. No step reduces a claimed prediction to a fitted parameter by construction, nor does any load-bearing premise collapse to a self-citation whose content is itself unverified. The derivation introduces new algebraic objects (groupoid multiplication, algebroid bracket) whose properties are analyzed independently of the target registration outputs, and numerical validation is presented as separate empirical support rather than an input to the equations. The central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard differential geometry structures adapted to discontinuities, with the groupoid itself as the key new entity.

axioms (2)
  • standard math Lie group properties for diffeomorphisms can be extended to groupoids while preserving local diffeomorphism within regions
    Invoked in the extension from Lie groups to groupoids for piecewise registration.
  • domain assumption Discontinuities occur only along sliding boundaries separating homogeneous regions
    Core modeling choice for allowing discontinuities without breaking overall diffeomorphism.
invented entities (1)
  • discontinuous diffeomorphism Lie groupoid no independent evidence
    purpose: To model piecewise diffeomorphic deformations allowing sliding discontinuities at boundaries
    New structure introduced to overcome limitations of standard Lie groups in LDDMM.

pith-pipeline@v0.9.0 · 5437 in / 1325 out tokens · 55169 ms · 2026-05-15T11:53:15.930343+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    V. I. Arnold and B. A. Khesin.Topological methods in hydrodynamics, volume 19. Springer, 2009

    work page 2009

  2. [2]

    A. Baig, M. A. Chaudhry, and A. Mahmood. Local normalized cross corre- lation for geo-registration. InInternational Bhurban Conference on Applied Sciences & Technology, pages 70–74. IEEE, 2012

    work page 2012

  3. [3]

    L. Bao, K. Chen, D. Kong, S. Ying, and T. Zeng. Time multiscale regu- larization for nonlinear image registration.Computerized Medical Imaging and Graphics, 112:102331, 2024

    work page 2024

  4. [4]

    L. Bao, J. Lu, S. Ying, and S. Sommer. Sliding at first-order: Higher- order momentum distributions for discontinuous image registration.SIAM Journal on Imaging Sciences, 17(2):861–887, 2024

    work page 2024

  5. [5]

    Bauer, S

    M. Bauer, S. C. Preston, and J. Valletta. Liouville comparison theory for blowup of Euler-Arnold equations.arXiv preprint arXiv:2306.09748, 2023

    work page arXiv 2023

  6. [6]

    M. F. Beg, M. I. Miller, A. Trouv´ e, and L. Younes. Computing large deformation metric mappings via geodesic flows of diffeomorphisms.Inter- national Journal of Computer Vision, 61:139–157, 2005

    work page 2005

  7. [7]

    Boucetta

    M. Boucetta. Riemannian geometry of lie algebroids.Journal of the Egyp- tian Mathematical Society, 19(1-2):57–70, 2011

    work page 2011

  8. [8]

    Chumchob and C

    N. Chumchob and C. Ke. A variational approach for discontinuity- preserving image registration.East-west Journal of Mathematics, 2010:266– 282, 2010. 27

    work page 2010

  9. [9]

    Dufour and N

    J.-P. Dufour and N. T. Zung.Poisson structures and their normal forms, volume 242. Springer Science & Business Media, 2006

    work page 2006

  10. [10]

    Dupuis, U

    P. Dupuis, U. Grenander, and M. I. Miller. Variational problems on flows of diffeomorphisms for image matching.Quarterly of applied mathematics, pages 587–600, 1998

    work page 1998

  11. [11]

    Frohn-Schauf, S

    C. Frohn-Schauf, S. Henn, and K. Witsch. Multigrid based total variation image registration.Computing and Visualization in Science, 11(2):101–113, 2008

    work page 2008

  12. [12]

    Y. Fu, S. Liu, H. H. Li, H. Li, and D. Yang. An adaptive motion regulariza- tion technique to support sliding motion in deformable image registration. Medical Physics, 45(2):735–747, 2018

    work page 2018

  13. [13]

    D. D. Holm and J. E. Marsden. Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation.The Breadth of Symplectic and Poisson Geometry: Festschrift in Honor of Alan Weinstein, pages 203–235, 2005

    work page 2005

  14. [14]

    D. D. Holm, T. Schmah, and C. Stoica.Geometric Mechanics and Symme- try: From Finite to Infinite Dimensions. Oxford University Press, USA, 2009

    work page 2009

  15. [15]

    Izosimov and B

    A. Izosimov and B. Khesin. Vortex sheets and diffeomorphism groupoids. Advances in Mathematics, 338:447–501, 2018

    work page 2018

  16. [16]

    Izosimov and B

    A. Izosimov and B. Khesin. Geometry of generalized fluid flows.Calculus of Variations and Partial Differential Equations, 63(1):3, 2024

    work page 2024

  17. [17]

    Khesin, G

    B. Khesin, G. Misio lek, and K. Modin. Geometric hydrodynamics and infinite-dimensional newton’s equations.Bulletin of the American Mathe- matical Society, 58(3):377–442, 2021

    work page 2021

  18. [18]

    Khesin, G

    B. Khesin, G. Misio lek, and A. Shnirelman. Geometric hydrodynamics in open problems.Archive for Rational Mechanics and Analysis, 247(2):15, 2023

    work page 2023

  19. [19]

    Khesin and R

    B. Khesin and R. Wendt.The geometry of infinite-dimensional groups, volume 51. Springer Science & Business Media, 2008

    work page 2008

  20. [20]

    H. P. Langtangen and A. Logg.Solving PDEs in python: the FEniCS tutorial I. Springer Nature, 2017

    work page 2017

  21. [21]

    Mang and G

    A. Mang and G. Biros. Constrained hˆ1-regularization schemes for diffeo- morphic image registration.SIAM Journal on Imaging Sciences, 9(3):1154– 1194, 2016

    work page 2016

  22. [22]

    Marsland and S

    S. Marsland and S. Sommer. Riemannian geometry on shapes and diffeo- morphisms: Statistics via actions of the diffeomorphism group. InRie- mannian Geometric Statistics in Medical Image Analysis, pages 135–167. Elsevier, 2020. 28

    work page 2020

  23. [23]

    Micheli, P

    M. Micheli, P. W. Michor, and D. Mumford. Sobolev metrics on diffeomor- phism groups and the derived geometry of spaces of submanifolds.Izvestiya: Mathematics, 77(3):541, 2013

    work page 2013

  24. [24]

    M. I. Miller, A. Trouv´ e, and L. Younes. Geodesic shooting for computa- tional anatomy.Journal of Mathematical Imaging and Vision, 24:209–228, 2006

    work page 2006

  25. [25]

    Modersitzki.Numerical Methods for Image Registration

    J. Modersitzki.Numerical Methods for Image Registration. Oxford Univer- sity Press, New York, 2004

    work page 2004

  26. [26]

    Mumford and P

    D. Mumford and P. W. Michor. On Euler’s equation and ‘EPDiff’.Journal of Geometric Mechanics, 5(3):319–344, 2013

    work page 2013

  27. [27]

    F. P. Oliveira and J. M. R. Tavares. Medical image registration: a review. Computer Methods in Biomechanics and Biomedical Engineering, 17(2):73– 93, 2014

    work page 2014

  28. [28]

    Pennec, S

    X. Pennec, S. Sommer, and T. Fletcher.Riemannian Geometric Statistics in Medical Image Analysis. Academic Press, United Kingdom, 2019

    work page 2019

  29. [29]

    Risser, H

    L. Risser, H. Baluwala, and J. A. Schnabel. Diffeomorphic registration with sliding conditions: Application to the registration of lungs ct images. Fourth International Workshop on Pulmonary Image Analysis, MICCAI, pages 79–90, 2011

    work page 2011

  30. [30]

    Risser, F.-X

    L. Risser, F.-X. Vialard, H. Y. Baluwala, and J. A. Schnabel. Piecewise- diffeomorphic image registration: Application to the motion estimation be- tween 3D CT lung images with sliding conditions.Medical Image Analysis, 17(2):182–193, 2013

    work page 2013

  31. [31]

    Schmidt-Richberg, R

    A. Schmidt-Richberg, R. Werner, H. Handels, and J. Ehrhardt. Estimation of slipping organ motion by registration with direction-dependent regular- ization.Medical Image Analysis, 16(1):150–159, 2012

    work page 2012

  32. [32]

    Sommer, F

    S. Sommer, F. Lauze, M. Nielsen, and X. Pennec. Kernel bundle EPDiff: evolution equations for multi-scale diffeomorphic image registration. In International Conference on Scale Space and Variational Methods in Com- puter Vision, pages 677–688. Springer, 2011

    work page 2011

  33. [33]

    Trouv´ e

    A. Trouv´ e. An infinite dimensional group approach for physics based models in patterns recognition.Preprint, 1995

    work page 1995

  34. [34]

    M. A. Viergever, J. Maintz, S. Klein, K. Murphy, M. Staring, and J. Pluim. A survey of medical image registration-under review.Medical Image Anal- ysis, 33:140–144, 2016

    work page 2016

  35. [35]

    Vizman et al

    C. Vizman et al. Geodesic equations on diffeomorphism groups.Symmetry, Integrability and Geometry: Methods and Applications, 4:030, 2008

    work page 2008

  36. [36]

    Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli. Image quality assessment: from error visibility to structural similarity.IEEE transactions on image processing, 13(4):600–612, 2004

    work page 2004

  37. [37]

    Younes.Shapes and Diffeomorphisms

    L. Younes.Shapes and Diffeomorphisms. Springer, Berlin, Heidelberg, 2019. 29

    work page 2019