pith. sign in

arxiv: 1907.04834 · v1 · pith:JK3TCWJVnew · submitted 2019-07-10 · 💻 cs.CV

Barnes-Hut Approximation for Point SetGeodesic Shooting

Jiancong Wang , Long Xie , Paul Yushkevich , James Gee This is my paper

Pith reviewed 2026-05-24 23:38 UTC · model grok-4.3

classification 💻 cs.CV
keywords geodesic shootingBarnes-Hut approximationpoint set registrationdiffeomorphic registrationmedical image analysiscomputational complexity reduction
0
0 comments X

The pith

Barnes-Hut approximation reduces point set geodesic shooting from O(N squared) to O(N log N) while preserving accuracy.

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

The paper introduces a Barnes-Hut based approximation for geodesic shooting between point sets used in diffeomorphic registration. Exact shooting requires quadratic operations per time step on the number of points, but the grouping of distant points cuts this to O(Nb + N log N). Evaluation on simulated data and real medial temporal lobe thickness measurements shows the approximated results match the exact method closely enough to support the same clinical conclusions, while delivering up to three times the speed.

Core claim

The Barnes-Hut approximation for point set geodesic shooting reduces algorithm complexity to O(Nb + N log N) and demonstrates comparable accuracy to exact point set geodesic shooting in both simulated images and medial temporal lobe thickness analysis.

What carries the argument

The Barnes-Hut tree, which groups distant points into single representative interactions to approximate the velocity field updates during geodesic shooting.

If this is right

  • Larger point sets become feasible for geodesic shooting registration without prohibitive run times.
  • The method supports the same downstream measurements, such as regional thickness, that the exact algorithm produces.
  • Up to threefold reduction in computation time is realized on the tested image sizes.
  • Clinical research studies that rely on point-set diffeomorphic registration can now scale to more subjects or higher-resolution data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same tree-based grouping could be tested on other particle interaction problems that appear in shape analysis beyond brain imaging.
  • If the error bound holds across modalities, the approximation may allow online or interactive registration pipelines that were previously offline only.

Load-bearing premise

The error introduced by grouping distant points remains small enough not to alter the final diffeomorphic registration outcome or downstream clinical measurements.

What would settle it

A direct comparison on the medial temporal lobe dataset in which the thickness maps or registration error from the approximated method differ by more than the variability seen between exact runs on the same data.

Figures

Figures reproduced from arXiv: 1907.04834 by James Gee, Jiancong Wang, Long Xie, Paul Yushkevich.

Figure 1
Figure 1. Figure 1: 2D graphic illustration of the recursive partition and Gaussian calculation. (a) Sample points. (b) Quad-tree subdivision. For a point in action (red), the Gaussian interaction (1) of a single-point node is directly calculated, (2) of a node outside the distance threshold (red circle) is approximated, (3) of a node with multiple points and at least one within the distance threshold is recursively calculate… view at source ↗
Figure 2
Figure 2. Figure 2: Test of proposed algorithm on synthetic shapes. In (b) and (c), blue, red, and green colors are the moving shape, the fixed shape and the deformation respectively. In (a), the blue and green circles are registered to the red and purple circles simultaneously. 3.2 Evaluation on Cortical Thickness Analysis of the Medial Temporal Lobe in Alzheimer’s Disease To demonstrate the utility of the proposed technique… view at source ↗
Figure 3
Figure 3. Figure 3: An illustration of MTL thickness measurement pipeline Template Matching Baseline T1-weighted MRI scans of 665 subjects from Alzheimer’s disease neuroimaging initiative (ADNI) were chosen in this study. An open-source multi-atlas segmentation pipeline proposed by Xie et al. [17] was applied to segment the MTL cortex subregions including the entorhinal cortex (ERC), Brodmann areas 35 and 36 (BA35/36) and par… view at source ↗
read the original abstract

Geodesic shooting has been successfully applied to diffeo-morphic registration of point sets. Exact computation of the geodesicshooting between point sets, however, requiresO(N2) calculations each time step on the number of points in the point set. We proposean approximation approach based on the Barnes-Hut algorithm to speedup point set geodesic shooting. This approximation can reduce the al-gorithm complexity toO(N b+N logN). The evaluation of the proposedmethod in both simulated images and the medial temporal lobe thick-ness analysis demonstrates a comparable accuracy to the exact point set geodesic shooting while offering up to 3-fold speed up. This improvementopens up a range of clinical research studies and practical problems towhich the method can be effectively applied.

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 Barnes-Hut approximation to accelerate exact point-set geodesic shooting for diffeomorphic registration. It claims this reduces per-timestep complexity from O(N²) to O(Nb + N log N), yields registration outcomes and downstream measurements (medial temporal lobe thickness) comparable to the exact method, and delivers up to a 3-fold speedup on simulated images and one clinical dataset.

Significance. If the accuracy claim holds under quantitative scrutiny, the work would remove a practical barrier to applying geodesic shooting on larger point sets, directly enabling new clinical studies that currently cannot afford the quadratic cost. The complexity reduction itself follows standard Barnes-Hut analysis and introduces no circularity.

major comments (2)
  1. [Evaluation] Evaluation section: the central claim of 'comparable accuracy' is supported only by a qualitative statement in the abstract; no error tables, per-landmark or per-subject distance metrics, parameter values for the Barnes-Hut opening-angle threshold, or statistical tests (e.g., paired t-tests or equivalence bounds) are supplied. This directly affects the weakest assumption that approximation error does not propagate to alter diffeomorphic maps or clinical measurements.
  2. [Method] Method / Complexity paragraph: while the O(Nb + N log N) bound is asserted, the manuscript does not show how the Barnes-Hut grouping is integrated into the specific force and velocity computations of geodesic shooting (e.g., which kernel evaluations are approximated and under what regularity conditions on the diffeomorphism). This detail is load-bearing for reproducibility of the claimed speedup.
minor comments (2)
  1. Notation for the Barnes-Hut parameter b (or θ) is introduced without an explicit definition or reference to the standard formulation used in the implementation.
  2. [Evaluation] The abstract states 'up to 3-fold speed up' but the evaluation does not report wall-clock timings, hardware, or N values at which the speedup was measured.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight areas where additional quantitative evidence and implementation details will strengthen the manuscript. We address each major comment below and will incorporate revisions as noted.

read point-by-point responses

  1. Referee: [Evaluation] Evaluation section: the central claim of 'comparable accuracy' is supported only by a qualitative statement in the abstract; no error tables, per-landmark or per-subject distance metrics, parameter values for the Barnes-Hut opening-angle threshold, or statistical tests (e.g., paired t-tests or equivalence bounds) are supplied. This directly affects the weakest assumption that approximation error does not propagate to alter diffeomorphic maps or clinical measurements.

    Authors: We agree that quantitative support for the accuracy claim is essential. The current manuscript relies on qualitative statements and the reported clinical outcome (medial temporal lobe thickness) without accompanying error tables or statistical comparisons. In the revision we will add: (1) per-subject and per-landmark L2 distance tables between exact and approximate velocity fields, (2) the specific opening-angle threshold used in all experiments, and (3) paired statistical tests (t-tests and equivalence bounds) on both simulated and clinical data to demonstrate that approximation error does not meaningfully alter the resulting diffeomorphisms or downstream measurements. revision: yes

  2. Referee: [Method] Method / Complexity paragraph: while the O(Nb + N log N) bound is asserted, the manuscript does not show how the Barnes-Hut grouping is integrated into the specific force and velocity computations of geodesic shooting (e.g., which kernel evaluations are approximated and under what regularity conditions on the diffeomorphism). This detail is load-bearing for reproducibility of the claimed speedup.

    Authors: The referee correctly identifies a missing implementation detail. The manuscript states the complexity reduction but does not explicitly describe which kernel summations are replaced by Barnes-Hut tree traversals or the smoothness assumptions required for the approximation to remain valid inside the geodesic-shooting ODE. In the revised version we will insert a dedicated subsection that (a) identifies the exact force and velocity kernel evaluations that are approximated, (b) specifies the tree-construction and traversal procedure, and (c) states the regularity conditions (C^2 diffeomorphism with compactly supported kernel) under which the O(Nb + N log N) bound holds. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper applies the established Barnes-Hut tree approximation to the O(N²) interaction terms in point-set geodesic shooting. The complexity reduction to O(Nb + N log N) follows directly from the standard Barnes-Hut analysis without any redefinition, fitting, or self-referential construction inside the paper. Accuracy is asserted via direct empirical comparison to the exact method on simulated images and clinical thickness data; no parameter is fitted to a subset and then relabeled as a prediction, and no load-bearing self-citation or uniqueness theorem is invoked to close the argument. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work rests on the standard mathematical properties of the Barnes-Hut algorithm and the existing diffeomorphic registration framework; no new free parameters, axioms, or invented entities are introduced in the abstract.

pith-pipeline@v0.9.0 · 5654 in / 1037 out tokens · 17771 ms · 2026-05-24T23:38:13.233450+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

17 extracted references · 17 canonical work pages

  1. [1]

    In: International Workshop on Energy Minimiza- tion Methods in Computer Vision and Pattern Recognition

    Allassonni` ere, S., Trouv´ e, A., Younes, L.: Geodesic shooting and diffeomorphic matching via textured meshes. In: International Workshop on Energy Minimiza- tion Methods in Computer Vision and Pattern Recognition. pp. 365–381. Springer (2005)

    work page 2005

  2. [2]

    Journal de math´ ematiques pures et appliqu´ ees 104(1), 139–178 (2015)

    Arguillere, S., Tr´ elat, E., Trouv´ e, A., Younes, L.: Shape deformation analysis from the optimal control viewpoint. Journal de math´ ematiques pures et appliqu´ ees 104(1), 139–178 (2015)

    work page 2015

  3. [3]

    SIAM Journal on Imaging Sciences 10(1), 303–334 (2017)

    Camassa, R., Kuang, D., Lee, L.: A geodesic landmark shooting algorithm for template matching and its applications. SIAM Journal on Imaging Sciences 10(1), 303–334 (2017)

    work page 2017

  4. [4]

    IEEE Transactions on Visualization and Computer Graphics 18(6), 914–924 (2012)

    Corsini, M., Cignoni, P., Scopigno, R.: Efficient and Flexible Sampling with Blue Noise Properties of Triangular Meshes. IEEE Transactions on Visualization and Computer Graphics 18(6), 914–924 (2012)

    work page 2012

  5. [5]

    Second Edition

    Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis, with Applications in {R}. Second Edition. John Wiley and Sons, Chichester (2016)

    work page 2016

  6. [6]

    In: 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003

    Fletcher, P.T., Lu, C., Joshi, S.: Statistics of shape via principal geodesic analysis on lie groups. In: 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings. vol. 1, pp. I–I. IEEE (2003)

    work page 2003

  7. [7]

    International journal of computer vision 80(3), 317 (2008)

    Glaun` es, J., Qiu, A., Miller, M.I., Younes, L.: Large deformation diffeomorphic metric curve mapping. International journal of computer vision 80(3), 317 (2008)

    work page 2008

  8. [8]

    In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition

    Golyanik, V., Aziz Ali, S., Stricker, D.: Gravitational approach for point set regis- tration. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 5802–5810 (2016)

    work page 2016

  9. [9]

    Kitware, Inc

    Johnson, H.J., McCormick, M.M., Ibanez, L.: The ITK Software Guide Book 1: Introduction and Development Guidelines-Volume 1. Kitware, Inc. (2015)

    work page 2015

  10. [10]

    IEEE transactions on image processing 9(8), 1357–1370 (2000)

    Joshi, S.C., Miller, M.I.: Landmark matching via large deformation diffeomor- phisms. IEEE transactions on image processing 9(8), 1357–1370 (2000)

    work page 2000

  11. [11]

    Mathematical programming 45(1-3), 503–528 (1989)

    Liu, D.C., Nocedal, J.: On the limited memory bfgs method for large scale opti- mization. Mathematical programming 45(1-3), 503–528 (1989)

    work page 1989

  12. [12]

    In: International Conference on Information Processing in Medical Imaging

    Lombaert, H., Sporring, J., Siddiqi, K.: Diffeomorphic spectral matching of cor- tical surfaces. In: International Conference on Information Processing in Medical Imaging. pp. 376–389. Springer (2013)

    work page 2013

  13. [13]

    Pattern Recognition 28(3), 343–359 (1995) Barnes-Hut Approximation for Point Set Geodesic Shooting 9

    Ogniewicz, R., K¨ ubler, O.: Hierarchic Voronoi skeletons. Pattern Recognition 28(3), 343–359 (1995) Barnes-Hut Approximation for Point Set Geodesic Shooting 9

    work page 1995

  14. [14]

    IEEE transactions on medical imaging 35(6), 1369–1380 (2016)

    Pai, A., Sommer, S., Sørensen, L., Darkner, S., Sporring, J., Nielsen, M.: Ker- nel bundle diffeomorphic image registration using stationary velocity fields and wendland basis functions. IEEE transactions on medical imaging 35(6), 1369–1380 (2016)

    work page 2016

  15. [15]

    In: Biennial International Conference on Information Processing in Medical Imaging

    Vaillant, M., Glaun` es, J.: Surface matching via currents. In: Biennial International Conference on Information Processing in Medical Imaging. pp. 381–392. Springer (2005)

    work page 2005

  16. [16]

    Xie, L., Wisse, L.E.M., Das, S.R., Ittyerah, R., Wang, J., Wolk, D.A., Yushkevich, P.A., Initiative, f.t.A.D.N.: Characterizing Anatomical Variability and Alzheimer’s Disease Related Cortical Thinning in the Medial Temporal Lobe Using Graph- Based Groupwise Registration and Point Set Geodesic Shooting. pp. 28–37. Springer, Cham (2018)

    work page 2018

  17. [17]

    In: MICCAI

    Xie, L., Wisse, L.E., Das, S.R., Wang, H., Wolk, D.A., Manj´ on, J.V., Yushke- vich, P.A.: Accounting for the confound of meninges in segmenting entorhinal and perirhinal cortices in t1-weighted mri. In: MICCAI. pp. 564–571. Springer (2016)

    work page 2016