pith. sign in

arxiv: 2605.28619 · v1 · pith:JEAM6QSAnew · submitted 2026-05-27 · 💻 cs.CV · nlin.AO

A Multiscale Kinetic Framework for Image Segmentation: From Particle Systems to Continuum Models

Horacio Tettamanti , Giulia Guicciardi , Mattia Zanella This is my paper

Pith reviewed 2026-06-29 13:09 UTC · model grok-4.3

classification 💻 cs.CV nlin.AO
keywords image segmentationkinetic modelsparticle systemsmultiscale modelingconsensus-based segmentationmacroscopic limitnoise robustness
0
0 comments X

The pith

Particles with positions and color features interact under a coupled scheme that scales to a macroscopic model for image segmentation.

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

The paper treats each pixel as a particle carrying both a spatial location and a color feature. A coupled interaction rule drives aggregation in both position and feature space. From the resulting particle system the authors derive a kinetic density equation that incorporates transport, aggregation and diffusion. A further scaling limit reduces this to a first-order macroscopic equation that tracks the fraction of pixels sharing a given feature. This continuum description then supplies the objective for a particle-based optimization procedure that produces segmentations shown to remain effective under added noise.

Core claim

Interpreting an image as a system of interacting particles, each characterised by spatial position and an internal feature encoding color, a coupled interaction scheme is introduced that governs evolution in both spaces. The scheme produces a kinetic formulation for the particle density combining transport, aggregation and diffusion. Through a suitable scaling this yields a first-order macroscopic model describing the evolution of the fraction of pixels carrying a certain feature, which underpins a data-oriented segmentation method based on particle optimisation.

What carries the argument

Coupled interaction scheme in position and feature spaces, scaled to a first-order macroscopic model for the fraction of pixels sharing a given feature.

If this is right

  • The reduced macroscopic model enables a data-oriented segmentation procedure that relies on particle-based optimisation.
  • Numerical tests confirm the resulting segmentations remain accurate under different noise conditions.
  • The multiscale construction links microscopic particle rules directly to a continuum description usable for image tasks.
  • The framework supplies a concrete route from kinetic density equations to practical optimisation objectives for pixel classification.

Where Pith is reading between the lines

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

  • The same interaction structure could be applied to other pixel attributes such as texture or depth by extending the feature space.
  • Because the macroscopic equation is first-order, it may admit faster numerical solvers than the full kinetic or particle descriptions for large images.
  • The approach suggests a template for deriving continuum limits in related tasks such as clustering or denoising where both location and attribute information matter.

Load-bearing premise

The chosen scaling makes the macroscopic solutions remain faithful to the original particle dynamics for the purpose of segmentation.

What would settle it

Direct numerical comparison on the same noisy test images where the macroscopic model produces segmentations that differ substantially from those obtained by evolving the full particle system would show the scaling fails to preserve the required dynamics.

Figures

Figures reproduced from arXiv: 2605.28619 by Giulia Guicciardi, Horacio Tettamanti, Mattia Zanella.

Figure 1
Figure 1. Figure 1: Test 1. Time evolution of F(c, t) = R R2 xf(x, c, t) dx with ε = 10−2 , 10−1 , 5 × 10−1 . It can be observed that as, for small values of ε > 0, the quantity F(c, t) approaches to a constant value in time as expected given that it is a conserved quantity of the operator Qˆ S. −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 x 0.00 0.02 0.04 0.06 0.08 0.10 R f(x, y, c) dy dc ε = 5 × 10−1 ε = 10−1 ε = 10−2 p… view at source ↗
Figure 2
Figure 2. Figure 2: Test 1. Comparison between the marginal distributions of the quasi-equilibrium distri￾bution f∞ F,ρ(x, c) given by Equation 28 and f∞ ϵ,N (x, c) obtained with the DSMC method with a final time T = 50 and N = 105 particles for different values of c with ε = 5 × 10−1 , 10−2 . We observe how the marginal distribution of the particle system converges to the marginal distribution of the quasi-equilibrium distri… view at source ↗
Figure 3
Figure 3. Figure 3: Test 2. We compute the first order moment of the distribution ρ(c, t)m(c, t) = R R2 xf(x, c, t) dx with ε = 10−2 as computed from the particle distribution f∞ ϵ,N (x, c) and the quasi-equilibrium distribution f∞ F,ρ(x, c) shown in Equation 28. We recall that this quantity is not a conserved quantity of the system and we observed a good agreement between both quantities for ε = 10−2 and final time T = 50. I… view at source ↗
Figure 4
Figure 4. Figure 4: Test 2. We show the large-time distribution ρ∞(c) obtained with the Boltzmann-type equation for τ = 1 and τ = 10−3 compared to the asymptotic distribution of the macroscopic model obtained evolving Equation 32. We observe that as τ → 0 + the solution of the Boltzmann￾type equation approaches to the solution of the macroscopic model. where the quantity A(c, t) is defined in Equation 31. The system is solved… view at source ↗
Figure 5
Figure 5. Figure 5: Test 3.1. Top row: Gaussian noise, centered square. Second row: Uniform noise, centered circle. Third row: Speckle noise, centered triangle. Fourth row: Poisson noise, centered rhombus. In each case, from left to right, we show the noisy image, the Ground Truth Segmentation Mask (GTSM), and the segmentation obtained with the optimized parameters. feature density to match the expected distribution mask with… view at source ↗
Figure 6
Figure 6. Figure 6: Test 3.1. Feature density distributions for the different geometric shapes with varying noise configurations. In each case, we report the initial distribution ρ(c, 0), the Ground Truth Segmentation Mask (GTSM) ρGTSM(c) and the large time distribution obtained from the macro￾scopic model ρ(c, T) for T = 20 obtained with the optimized parameters. Top Left: Gaussian noise, centered square. Top Right: Uniform … view at source ↗
Figure 7
Figure 7. Figure 7: Test 3.2. We report the evolution of the density ρ(c, t) for four square images with pro￾gressively increasing Gaussian noise levels (σ 2 = 2, 5, 10, 15). The optimized parameters effectively capture the expected behaviour of the GTSM, redistributing the density such that it concentrates near c = 0, 1. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Test 3.2. We report the loss values obtained for the different combination of the model parameters for the case of the Speckle noise with σ 2 = 0.04. We notice the difficulty this optimization problem poses given that the loss landscape is non-convex and flat on large regions of the parameter space which makes it very likely to get stuck in local minima. The red point represents the optimal combination of … view at source ↗
Figure 9
Figure 9. Figure 9: Test 4. Segmentation of a 28 × 28 color image with Gaussian noise in all channels with intensity σ 2 R = σ 2 G = σ 2 B = 0.2. Top: We show the original image, the Ground Truth Segmentation Mask (GTSM), and the segmentation obtained with the optimal parameters. Bottom: We report the marginal distributions ρR(cR), ρG(cG), and ρB(cB) obtained with the optimal parameters compared to the marginals of ρGTSM. We … view at source ↗
read the original abstract

In this work, we present a multiscale kinetic framework for consensus-based image segmentation. By interpreting an image as a system of interacting particles, each pixel is characterised by its spatial position and an internal feature encoding color information. We introduce a coupled interaction scheme governing the evolution of particles in both position and feature spaces, from which we derive a kinetic formulation for the particle density in the space-feature domain combining transport, aggregation, and diffusion effects. Furthermore, through a suitable scaling, we obtain a first-order macroscopic model describing the evolution of the fraction of pixels carrying information on the fraction of pixels having a certain feature. Based on this reduced-complexity model, we present a data-oriented approach where we make use of particle-based optimisation techniques for the accurate segmentation of images. Numerical tests show the effectiveness of the proposed framework and its robustness under different noise conditions.

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

1 major / 0 minor

Summary. The paper develops a multiscale kinetic framework for consensus-based image segmentation. An image is interpreted as a system of interacting particles, each with spatial position and an internal color feature. A coupled interaction scheme in position and feature spaces yields a kinetic formulation for the particle density combining transport, aggregation, and diffusion. A suitable scaling produces a first-order macroscopic model for the evolution of the fraction of pixels with a given feature. Particle-based optimization is then applied to this reduced model for segmentation, with numerical tests claimed to demonstrate effectiveness and robustness under varying noise conditions.

Significance. If the scaling limit is rigorously justified and the macroscopic model remains faithful to the underlying particle dynamics, the work could provide a principled kinetic-to-continuum bridge for segmentation algorithms, potentially improving theoretical grounding and noise robustness in data-oriented computer vision methods. The explicit use of particle optimization on the reduced model is a concrete strength if fidelity holds.

major comments (1)
  1. Abstract (scaling paragraph): the claim that 'a suitable scaling' produces a first-order macroscopic model whose solutions remain faithful to the original particle dynamics for segmentation purposes is load-bearing for the central contribution, yet no scaling parameter, limit procedure, or error estimate is supplied; without these the reduced-complexity claim cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the constructive comment on the scaling claim. We address the point below.

read point-by-point responses

  1. Referee: [—] Abstract (scaling paragraph): the claim that 'a suitable scaling' produces a first-order macroscopic model whose solutions remain faithful to the original particle dynamics for segmentation purposes is load-bearing for the central contribution, yet no scaling parameter, limit procedure, or error estimate is supplied; without these the reduced-complexity claim cannot be assessed.

    Authors: We agree that the abstract does not specify the scaling parameter or procedure. In the manuscript body (Section 3), the scaling is introduced via a small parameter ε that balances the interaction strengths in position and feature space; the first-order macroscopic model is obtained formally by letting ε → 0, yielding a transport equation for the feature density. This is a standard formal limit argument from kinetic theory rather than a rigorous convergence result with error bounds. We will revise the abstract to mention the scaling parameter explicitly and add a short clarifying paragraph in the introduction noting the formal character of the limit and its motivation from the underlying particle system. These changes will make the reduced-complexity claim easier to assess without altering the overall contribution. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract and described chain present a standard particle-to-kinetic-to-macroscopic derivation via 'suitable scaling' followed by an application of the reduced model to segmentation via particle optimization. No quoted equations show a fitted parameter renamed as prediction, no self-citation load-bearing the central claim, and no self-definitional loop where the output is used to define the input. The scaling and fidelity assumptions are stated as modeling choices without reduction to the segmentation results themselves. This is the common honest case of an independent derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no concrete free parameters, axioms, or invented entities; all such items remain unknown.

pith-pipeline@v0.9.1-grok · 5681 in / 1014 out tokens · 35686 ms · 2026-06-29T13:09:55.289766+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

45 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    G. Albi, L. Pareschi. Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul. , 11:1–29, 2013

    2013

  2. [2]

    G. Albi, M. Herty, L. Pareschi. Kinetic description of optimal control problems and applica- tions to opinion consensus, Commun. Math. Sci. , 13: 1407–1429, 2015

    2015

  3. [3]

    G. Albi, L. Pareschi, M. Zanella. Boltzmann-type control of opinion consensus through lead- ers, Phil. Trans. R. Soc. A , 372, 20140138, 2014

    2014

  4. [4]

    G. Albi, L. Pareschi, M. Zanella. Opinion dynamics over complex networks, kinetic modelling and numerical methods, Kinet. Relat. Models , 10:1–32, 2017

    2017

  5. [5]

    Arridge, R

    S. Arridge, R. Barbano, B. Jin and R. Tanno. Uncertainty quantification in medical image synthesis, Biomedical Image Synthesis and Simulation: Methods and Applications , 601-641 (2022)

    2022

  6. [6]

    Agosti, E

    A. Agosti, E. Shaqiri, M. Paoletti, F. Solazzo, N. Bergsland, G. Colelli, G. Savini, S. Muzic, F. Santini, X. Deligianni, L. Diamanti, M. Monforte, G. Tasca, E. Ricci, S. Bastianello and A. Pichiecchio. Deep learning for automatic segmentation of thigh and leg muscles, Magn. Reson. Mater. Phys. Biol. Med. , 35, 467-483, 2021

    2021

  7. [7]

    G. A. Bird. Direct simulation and the Boltzmann equation, Phys. Fluids, 13: 2676-2681, 1970

    1970

  8. [8]

    Babovsky, R

    H. Babovsky, R. Illner. A convergence proof for Nanbu’s simulation method for the full Boltzmann equation, SIAM J. Numer. Anal. , 26:45–65, 1989

    1989

  9. [9]

    A. K. Boyat, B. K. Joshi. A review paper: Noise models in digital image processing, Preprint arXiv:1505.03489, 2015. 24

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  10. [10]

    Bertozzi, S

    A. Bertozzi, S. Esedoglu, A. Gillette. Analysis of a Two-Scale Cahn-Hilliard Model for Binary Image Inpainting. Multiscale Model. Simul. , 6(3): 913936, 2007

    2007

  11. [11]

    Bungert, T

    L. Bungert, T. Roith, P. Wacker. Polarized consensus-based dynamics for optimization and sampling. Math. Program., 211: 125-155, 2025

    2025

  12. [12]

    Bailo, A

    R. Bailo, A. Barbaro, S. N. Gomes, K. Riedl, T. Roith, C. Totzeck, U. Vaes. CBX: Python and Julia packages for consensus-based interacting particle methods, J. Open Source Softw. , 9: 6611, 2024

    2024

  13. [13]

    Cucker, S

    F. Cucker, S. Smale. Emergent behavior in flocks, IEEE Trans. Autom. Control , 52:852-862, 2007

    2007

  14. [14]

    R. F. Cabini, A. Pichiecchio, A. Lascialfari, S. Figini, M. Zanella. A kinetic approach to consensus-based segmentation of biomedical images, Kinet. Relat. Models, 18: 286–311, 2025

    2025

  15. [15]

    R. F. Cabini, H. Tettamanti, M. Zanella. Understanding the impact of evaluation metrics in kinetic models for consensus-based segmentation, Entropy, 27: 149, 2025

    2025

  16. [16]

    J. A. Carrillo, F. Hoffmann, A. M. Stuart, U. Vaes. Consensus-based sampling, Stud. Appl. Math., 148:1-34, 2022

    2022

  17. [17]

    J. A. Carrillo, M. Fornasier, J. Rosado, G. Toscani. Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal. , 42: 218–236, 2010

    2010

  18. [18]

    J. A. Carrillo, S. Jin, L. Li, Y. Zhu. A consensus-based global optimization method for high dimensional machine learning problems, ESAIM: Control Optim. Calc. Var. , 27, S5, 2021

    2021

  19. [19]

    G. B. Coleman, H. C. Andrews. Image segmentation by clustering, Proceedings of the IEEE , 67: 773–785, 1979

    1979

  20. [20]

    Cordier, H

    N. Cordier, H. Delingette, N. Ayache. A patch-based approach for the segmentation of patholo- gies: application to glioma labelling. IEEE Trans. Med. Imaging , 35: 1066–1076, 2015

    2015

  21. [21]

    Cordier, L

    S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys. , 120: 253–277, 2005

    2005

  22. [22]

    Coupé, J

    P. Coupé, J. V. Manjón, V. Fonov, J. Pruessner, M. Robles, D. L. Collins. Patch-based segmentation using expert priors: application to hippocampus and ventricle segmentation, NeuroImage, 54: 940-954, 2011

    2011

  23. [23]

    Dimarco, L

    G. Dimarco, L. Pareschi. Numerical methods for kinetic equations, Acta Numer., 23: 369–520, 2014

    2014

  24. [24]

    S.-Y. Ha, E. Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models , 1:415–435, 2008

    2008

  25. [25]

    R. M. Haralick, L. G. Shapiro. Image segmentation techniques, Computer Vision, Graphics, and Image Processing , 29: 100-132, 1985

    1985

  26. [26]

    Hegselmann, U

    R. Hegselmann, U. Krause. Opinion dynamics and bounded confidence models, analysis, and simulation, J. Artif. Soc. Soc. Simul. , 5, 2002

    2002

  27. [27]

    Herty, L

    M. Herty, L. Pareschi, G. Visconti. Mean field models for large data-clustering problems, Netw. Heterog. Media , 15: 463–487, 2020

    2020

  28. [28]

    Isensee, P

    F. Isensee, P. F. Jaeger, S. A. Kohl, J. Petersen, K. H. Maier-Hein. Nnu-net: a self configuring method for deep learning-based biomedical image segmentation, Nature Methods, 18: 203-211, 2021. 25

    2021

  29. [29]

    A. Jain, M. Murty, P. Flynn. Data clustering: a review, ACM Computing Surveys , 31: 264– 323, 1999

    1999

  30. [30]

    S. Kayal. Unsupervised image segmentation using the Deffuant-Weisbuch model from social dynamics, Signal, Image and Video Processing , 11: 1405–1410, 2017

    2017

  31. [31]

    Kakchingtabam, S

    P. Kakchingtabam, S. B. K, P. Jadav, H. Harshavardhan. A comprehensive survey of transfer learning techniques and applications across domains, Proceedings of the International Con- ference on Soft Computing for Security Applications , 1186-1195, 2025

    2025

  32. [32]

    Oliviero, S

    A. Oliviero, S. Cacace, G. Visconti. Agent-based optimal control for image processing. Appl. Math. Model , 158: 116967, 2026

    2026

  33. [33]

    Kumwilaisak, P

    W. Kumwilaisak, P. Lasang, T. Piriyatharawet, N. Thatphithakkul. Image denoising with deep convolutional neural and multi-directional long short-term memory networks under Pois- son noise environments, IEEE Access, 8: 86998-87010, 2020

    2020

  34. [34]

    Pareschi, G

    L. Pareschi, G. Russo. Time relaxed Monte Carlo methods for the Boltzmann equation, SIAM J. Sci. Comput. , 23: 1253-1273, 2001

    2001

  35. [35]

    Pareschi, G

    L. Pareschi, G. Russo. An introduction to Monte Carlo method for the Boltzmann equation, ESAIM: Proceedings, 10: 35-75, 2001

    2001

  36. [36]

    Pareschi, G

    L. Pareschi, G. Toscani. Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013

    2013

  37. [37]

    Pareschi, M

    L. Pareschi, M. Zanella. Structure preserving schemes for nonlinear Fokker-Planck equations and applications, J. Sci. Comput. , 74: 1575–1600, 2018

    2018

  38. [38]

    Pareschi, G

    L. Pareschi, G. Toscani, A. Tosin, M. Zanella. Hydrodynamic models of preference formation in multi-agent societies, J. Nonlin. Sci. , 29: 2761–2796, 2019

    2019

  39. [39]

    Ronneberger, P

    O. Ronneberger, P. Fischer, T. Brox. U-net: convolutional networks for biomedical image segmentation, Med. Image Comput. Comput. Assist. Interv. , 558–590, 2015

    2015

  40. [40]

    G. Toscani. Kinetic models of opinion formation, Commun. Math. Sci. , 4: 481–496, 2006

    2006

  41. [41]

    Tosin, M

    A. Tosin, M. Zanella. Boltzmann-type description with cutoff of follow-the-leader traffic mod- els. In: Albi, G., Merino-Aceituno, S., Nota, A., Zanella, M. (eds) Trails in Kinetic Theory . SEMA SIMAI Springer Series, vol 25. Springer, Cham, 2021

    2021

  42. [42]

    E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics , Springer Berlin, Heidelberg, 2009

    2009

  43. [43]

    E. F. Toro, S. A. Tokareva. Rusanov-type schemes for hyperbolic equations: wave-speed estimates, monotonicity and stability, Comput. Fluids , 309:107000, 2026

    2026

  44. [44]

    Zanella, Kinetic models for epidemic dynamics in the presence of opinion polarization, Bull

    M. Zanella, Kinetic models for epidemic dynamics in the presence of opinion polarization, Bull. Math. Biol. , 85, 2023

    2023

  45. [45]

    Z. Zhou, M. M. Rahman Siddiquee, N. Tajbakhsh and J. Liang. Unet++: a nested U-net architecture for medical image segmentation, Deep Learning in Medical Image Analysis and Multimodal Learning for Clinical Decision Support , 3-11, 2018. 26

    2018