REVIEW 3 major objections 5 minor 38 references
Medical Image Segmentation based on Deep Active Contour and Mean Curvature Loss Function
T0 review · 3 major / 5 minor · reviewed 2026-07-15 · grok-4.5
Pith's one-line read A loss that adds mean-curvature geometry to active-contour training yields smoother, more connected organ segmentations on liver CT and spleen MRI.
desk verdict Incremental geometric loss that posts small Dice gains on two tiny 2-D sets; the SOTA claim is real relative to the four baselines shown, but the “mean-curvature” term is a fixed 3 imes3 filter whose geometric fidelity is never checked. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The DACMC loss: a region term taken from the Mumford-Shah/Chan-Vese model plus a mean-curvature penalty approximated by a fixed 3x3 convolution kernel, evaluated on the network's soft prediction mask.
What would settle it
Replace the kernel approximation with an exact discrete mean-curvature operator (or a higher-order finite-difference scheme) on the same networks and datasets; if Dice and surface metrics then drop or become unstable, the kernel-surrogate premise fails.
Extended reading notes
Core claim
The authors claim that embedding a convolution-kernel approximation of mean curvature into a Chan-Vese-style region loss produces a single training objective (DACMC) that measurably improves organ segmentation accuracy and boundary quality over several established geometric and pixel-wise losses.
Load-bearing premise
The fixed convolution kernel is assumed to be a faithful enough stand-in for true mean curvature that the geometric penalty it produces reliably enforces connectivity and smoothness without systematic bias.
Editorial extensions
If this is right
- Any encoder-decoder network can adopt the same loss without architectural change and obtain smoother, more connected organ masks.
- The single scalar weight on the curvature term becomes a practical dial for trading boundary regularity against region fidelity.
- Because the kernel is GPU-friendly, the geometric prior can be used at full resolution during training rather than only at post-processing.
- The same construction can be applied to other organs or modalities whose topology is known a priori to be simply connected.
Reading between the lines
- If the kernel approximation is the dominant source of error, replacing it with a learned or higher-order curvature estimator should yield further gains without altering the rest of the loss.
- The method may be especially useful for few-shot or noisy-label regimes where geometric priors can compensate for scarce or imperfect annotations.
- Extending the curvature term to multi-class or 3-D volumes would test whether the same cheap approximation continues to enforce topology at higher dimension.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes DACMC, a supervised loss for medical image segmentation that combines a Chan-Vese/Mumford-Shah region term with a geometric regularizer intended to encode mean curvature. The continuous mean-curvature definition is stated (Definition 1, Eq. 9), but the implemented term (Eqs. 16-19) replaces it by a fixed 3x3 convolution kernel of Gong applied to both prediction and ground-truth masks, followed by a normalized absolute Hadamard product. The loss is trained with U-Net (and claimed with CE-Net) on a small liver CT set (100 train / 15 val / 16 test) and a spleen MRI subset of CHAOS (120/20/25). Table 1 reports the highest DSC among CE, DC, AC and ACE baselines (0.9426 liver, 0.8443 spleen) and improved surface metrics on the spleen set; qualitative figures illustrate smoother, more connected masks.
Significance. If the geometric term were a faithful, differentiable surrogate for mean curvature and the gains held under stronger baselines and larger cohorts, the work would supply a simple, drop-in regularizer that injects classical geometric priors into modern segmentation networks. The idea of marrying active-contour energies with deep losses is already present in the literature (AC, ACE, MS-driven losses), so the incremental contribution rests entirely on the curvature approximation and the empirical numbers. The paper does not ship code, analytic error bounds, or machine-checked derivations; its value is therefore purely empirical and currently limited by the narrow experimental design.
major comments (3)
- Section IV, Eqs. (16)-(19) versus Definition 1 / Eq. (9): the implemented regularizer is not mean curvature. Definition 1 correctly writes H(u)=div(nabla u / sqrt(1+|nabla u|^2)), yet the loss replaces this by | (Kernel * u) * (Kernel * v) / (Kernel * v + eps) | with a fixed 3x3 stencil. No derivation, consistency proof, or numerical comparison to analytic or finite-difference mean curvature is supplied. Consequently the claim that performance gains arise from 'mean-curvature geometric constraints' (abstract, highlights, ablation narrative) is unsupported; any local smoother could produce the same effect. A quantitative validation of the approximation (or an explicit re-statement that the term is merely a local filter) is load-bearing for the central mechanistic claim.
- Table 1 and Section V.A: experimental evidence is too thin to support a 'new state-of-the-art' claim. Test sets contain only 16 liver and 25 spleen images; only four classical losses are compared; CE-Net results are promised in the text but never tabulated; lambda is hand-tuned without sensitivity curves or cross-validation ranges; no statistical significance tests or multi-run standard deviations beyond the reported variances are given. These limitations make the numerical superiority fragile and non-generalizable.
- Section VI.C (Ablation Study): the narrative attributes successive DSC gains to 'deep active contour regularization, mean curvature loss, and their synergistic combination,' yet the only tabulated numbers are the five complete loss functions. No controlled experiment isolates the kernel term from the region term, nor compares the Gong kernel against a true discrete mean-curvature operator or against a simple total-variation / length term. Without that isolation the causal attribution remains conjectural.
minor comments (5)
- Equation numbering is inconsistent: the Mumford-Shah model is labeled (3) then later referred to as (2); the hybrid loss is (5) while the ACE loss is (7); the DACMC loss appears both as (10) and (16).
- Notation for the predicted mask switches among u_bar_theta, u_tilde_theta, u_i,j without definition of the relationship between continuous and discrete forms.
- Figures 3 and 4 lack quantitative per-image metrics or zoomed boundary insets that would let a reader verify the claimed connectivity/smoothness improvements.
- Several references carry future or mismatched years (e.g., 2025-2026 arXiv entries, 'CVPR 2026 Forthcoming'); these should be cleaned or replaced by stable citations.
- The abstract and highlights claim 'several segmentation datasets' while only two organs are evaluated; the wording should be tightened.
Circularity Check
No circularity: DACMC is an empirical loss design (region term + fixed kernel) trained and scored on held-out images; nothing is forced by construction or self-citation.
full rationale
The paper defines a composite loss (Eqs. 16–19) that adds a fixed 3 imes3 convolution kernel (taken from independent prior work of Gong) to a Chan-Vese/Mumford-Shah region term, then trains standard U-Net/CE-Net architectures and reports Dice/HD95/etc. on held-out liver CT and spleen MRI slices (Table 1). No parameter is fitted to a subset of the evaluation data and then re-used as a “prediction”; the kernel coefficients are constant, c1/c2 are either recomputed from the current mask or fixed to the binary constants 1/0, and the SOTA claim is simply the numerical ranking of those experimental numbers against CE/DC/AC/ACE. Definition 1 and the continuous mean-curvature formula (Eq. 9) are never algebraically recovered from the implemented Hadamard-product term, but that is a correctness/approximation issue, not a circular derivation. There are no load-bearing self-citations of uniqueness theorems, no ansatz smuggled from the authors’ own prior papers, and no renaming of a known empirical pattern. The entire chain is therefore self-contained and non-circular.
Assumptions & free parameters
free parameters (3)
- λ (curvature weight) =
not numerically reported
- learning rate =
0.0001
- c1, c2 (region means) =
1 and 0
assumptions (3)
- standard math Mean curvature of a graph surface is given by the classical divergence formula H(u) = div(∇u / √(1+|∇u|²))
- ad hoc to paper A fixed 3×3 convolution kernel supplies a sufficiently accurate and differentiable approximation to mean curvature for back-propagation
- domain assumption Medical organ masks benefit from an explicit mean-curvature geometric prior that promotes connectivity and smoothness
invented entities (1)
-
DACMC loss function
Cite this review
Pith. "Pith review of Medical Image Segmentation based on Deep Active Contour and Mean Curvature Loss Function." pith.science (2026). https://pith.science/paper/PMGNTDEM
@misc{pith2026260712586,
author = {Pith},
title = {Pith review of: Medical Image Segmentation based on Deep Active Contour and Mean Curvature Loss Function},
year = {2026},
howpublished = {\url{https://pith.science/paper/PMGNTDEM}},
note = {Machine review of arXiv:2607.12586}
}
read the original abstract
Medical image segmentation is a crucial task in the field of clinical analysis and applications. Though deep learning techniques recently play a crucial role in several scenarios, the training at the individual pixel level leads to a lack of geometric prior information. Scholars proposed to integrate the Chan-Vese model into the loss function for training which can take into account the region and length of the region inside and outside the segmentation process and then improve the performance in medical image segmentation. However, these methods still lack an effective characterization of the segmented region. To overcome this problem, we introduce the mean curvature as a geometric natural constraint and propose a Deep Active Contour and Mean Curvature (DACMC) loss function where the convolution kernel is used to approximate the mean curvature to save computational cost. We have validated the performance of our method on the liver and spleen dataset. Our proposed method demonstrates new state-of-the-art performance on several segmentation datasets.
Figures
Reference graph
Works this paper leans on
-
[1]
Deep learning for computational imaging: from data-driven to physics-enhanced approaches
Wang F, Czarske JW, Situ G. Deep learning for computational imaging: from data-driven to physics-enhanced approaches. Adv Photonics. 2025;7(5):054002. doi:10.1117/1.AP.7.5.054002
-
[2]
Ronneberger, P
O. Ronneberger, P. Fischer, and T. Brox. U-Net: Convolutional networks for biomedical image segmentation. In: Medical Image Computing and Computer-Assisted Intervention (MICCAI), 234–241, 2015
2015
-
[3]
Badrinarayanan, A
V. Badrinarayanan, A. Kendall, and R. Cipolla. Segnet: A deep convolutional encoder- decoder architecture for image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(12):2481-2495, 2017
2017
-
[4]
Alzubaidi L, Zhang J, Humaidi AJ, et al. A comprehensive review of convolutional neural networks: architectures, training methods, and recent advances. Neural Comput Appl. 2026;38(4):827. doi:10.1007/s00521-025-11827-w
-
[5]
A survey of the recent architectures of deep convolutional neural networks
Khan A, Sohail A, Zahoora U, Qureshi AS. A survey of the recent architectures of deep convolutional neural networks. Artif Intell Rev. 2025;58(5):190. doi:10.1007/s10462-020-09825- 6
-
[6]
Regional Hausdorff distance losses for medical image segmentation
Guzzi L, Zuluaga MA, Taiello R, Lareyre F, Di Lorenzo G, Goffart S, Chierici A, Raffort J, Delingette H. Regional Hausdorff distance losses for medical image segmentation. In: Cui Z, Rekik I, Suk HI, Ouyang X, Sun K, Wang S, editors. Machine Learning in Medical Imaging – MLMI 2025. Lect Notes Comput Sci. 2026;16241:63-73. doi:10.1007/978-3-032-09513-8_7
-
[7]
MedMamba-UNet: pure Mamba-based U-shaped architecture for efficient medical image segmentation
Yue Y, Li J, Shen X, et al. MedMamba-UNet: pure Mamba-based U-shaped architecture for efficient medical image segmentation. Med Image Anal. 2026;95:103612. doi:10.1016/j.media.2025.103612
-
[8]
Zhang J, Guo W. A new regularization for deep learning-based segmentation of images with fine structures and low contrast. Sensors. 2023;23(4):1887. doi:10.3390/s23041887 13
Show all 38 references
-
[9]
Spatially continuous dual optimization on compactness function for image segmentation
Tan L, Ma S, Xu J, Pan J, Yuan J, Kervadec H, Pelillo M. Spatially continuous dual optimization on compactness function for image segmentation. Pattern Recognit. 2026;172:112613. doi:10.1016/j.patcog.2025.112613
2026 doi
-
[10]
ITSRS: an inverse Taylor series adaptive loss based on synergized regional-structural information for medical image segmentation
Han R, Cheng Z, Zhao J, Ma B, Wang S. ITSRS: an inverse Taylor series adaptive loss based on synergized regional-structural information for medical image segmentation. J Imaging Inform Med. 2026
2026
-
[11]
Recent advances in medical imaging segmentation: a survey
Bougourzi F, Hadid A. Recent advances in medical imaging segmentation: a survey. arXiv:2505.09274. 2025. Preprint
2025 arXiv
-
[12]
Proximal splitting algorithms for convex optimization: a tour of recent advances, with new twists
Condat L, Kitahara D, Contreras A, Hirabayashi A. Proximal splitting algorithms for convex optimization: a tour of recent advances, with new twists. SIAM Rev. 2023;65(2):375-435. doi:10.1137/20M1379344
2023 doi
-
[13]
A comprehensive survey of loss functions and metrics in deep learning
Terven J, Cordova-Esparza DM, Romero-Gonzá lez JA, Ramí rez-Pedraza A, Chá vez-Urbiola EA, et al. A comprehensive survey of loss functions and metrics in deep learning. Artif Intell Rev. 2025;58:195. doi:10.1007/s10462-025-11198-7
2025 doi
-
[14]
Topology-preserving image segmentation with spatial-aware feature learning
Wen X, Zhang J, et al. Topology-preserving image segmentation with spatial-aware feature learning. arXiv:2412.02076. 2025. Preprint
2025
-
[15]
Deep convolutional neural networks meet variational shape compactness priors for image segmentation
Zhang K, Li L, Liu H, Yuan J, Tai XC. Deep convolutional neural networks meet variational shape compactness priors for image segmentation. Neurocomputing. 2025;623:129395. doi:10.1016/j.neucom.2025.129395
2025 doi
-
[16]
Semi-supervised medical image segmentation via anatomy-preserving consistency training
Li S, Yin Y, et al. Semi-supervised medical image segmentation via anatomy-preserving consistency training. Image Vis Comput. 2026;161:105833. doi:10.1016/j.imavis.2025.105833
2026 doi
-
[17]
LMS-Net: a learned Mumford-Shah network for few-shot medical image segmentation
Zhang S, Jia F, Li X, Zhang H, Shi J, Ma L, Ying S. LMS-Net: a learned Mumford-Shah network for few-shot medical image segmentation. arXiv:2502.05473. 2025. Preprint
2025 arXiv
-
[18]
Diff-SegNet: diffusion-guided encoder-decoder network for uncertain region refinement in medical image segmentation
Peng Y, Zhang Q, Liu X, et al. Diff-SegNet: diffusion-guided encoder-decoder network for uncertain region refinement in medical image segmentation. In: Proceedings of CVPR 2026
2026
-
[19]
EfficientMedNeXt: multi-receptive dilated convolutions for medical image segmentation
Roy S, Koehler G, Ulrich C, et al. EfficientMedNeXt: multi-receptive dilated convolutions for medical image segmentation. In: Proceedings of MICCAI 2025. Lect Notes Comput Sci
2025
-
[20]
A hybrid framework integrating active contour and deep learning for optic disc and optic cup segmentation
Hajrasooliha M, Naghsh-Nilchi AR. A hybrid framework integrating active contour and deep learning for optic disc and optic cup segmentation. Signal Image Video Process. 2026;20:350. doi:10.1007/s11760-026-05378-3
2026 doi
-
[21]
A novel deep neural architecture for efficient and scalable multi-domain image classification
Wong WK, Jiang X, Chan KC, et al. A novel deep neural architecture for efficient and scalable multi-domain image classification. Sci Rep. 2025;15:10517. doi:10.1038/s41598-025- 10517-w
2025 doi
-
[22]
GLAC-UNet: global-local active contour loss with an efficient U-shaped architecture for multiclass medical image segmentation
da Silva SPP, Tran TT, Nham DHN, Lo MT, Pham VT. GLAC-UNet: global-local active contour loss with an efficient U-shaped architecture for multiclass medical image segmentation. J Imaging Inform Med. 2025;38:3198-3220. doi:10.1007/s10278-025-01387-9
2025 doi
-
[23]
Context-driven active contour (CDAC): a novel medical image segmentation method based on active contour and contextual understanding
da Silva SPP, Ivo RF, Barroso CB, Fernandes JCN, Portela TF, Medeiros AG, de Sousa PHF, Song H, Rebouç as Filho PP. Context-driven active contour (CDAC): a novel medical image segmentation method based on active contour and contextual understanding. Sensors. 2025;25(9):2864. d...
2025 doi
-
[24]
Topology-guaranteed image segmentation: enforcing connectivity, genus, and width constraints
Li Z, Tai XC, et al. Topology-guaranteed image segmentation: enforcing connectivity, genus, and width constraints. SIAM J Imaging Sci. 2026;19(1):238-268. doi:10.1137/25M1765870
2026 doi
-
[25]
CurvDrop: data-efficient learning for medical image segmentation via curvature-based sample selection
Liu Y, Zhang K, Tai XC, et al. CurvDrop: data-efficient learning for medical image segmentation via curvature-based sample selection. Comput Biol Med. 2025;189:108742. doi:10.1016/j.compbiomed.2025.108742
2025 doi
-
[26]
Optimal approximations by piecewise smooth functions and associated variational problems
Mumford D, Shah J. Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math. 1989;42(5):577-685. doi:10.1002/cpa.3160420503
1989 doi
-
[27]
The Potts model with different piecewise constant representations and fast algorithms: a survey
Tai XC, Li L, Bae E. The Potts model with different piecewise constant representations and fast algorithms: a survey. In: Chen K, Schö nlieb CB, Tai XC, Younes L, editors. Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer; 2023:1887-
2023
-
[28]
doi:10.1007/978-3-030-98661-2_90
-
[29]
Oktay, J
O. Oktay, J. Schlemper, L. Folgoc, et al. Attention u-net: Learning where to look for the pancreas. The first of Conference on Medical Imaging with Deep Learning, 1-10, 2018
2018
-
[30]
Learning Euler's elastica model for medical image segmentation
Chen X, Luo X, Weng Y, et al. Learning Euler's elastica model for medical image segmentation. arXiv:2011.00526. 2021. Preprint
2011 arXiv
-
[31]
Robust variational model based tailored UNet: leveraging edge detector and mean curvature for improved image segmentation
Qi K, Huang Z, Yang W. Robust variational model based tailored UNet: leveraging edge detector and mean curvature for improved image segmentation. arXiv:2512.07590. 2025. Preprint
2025
-
[32]
Discriminative curvature regularization loss for boundary segmentation in microscopy cell images
Asha SB, Gopakumar G. Discriminative curvature regularization loss for boundary segmentation in microscopy cell images. SN Comput Sci. 2026;7:64. doi:10.1007/s42979-025- 04633-7
2026 doi
-
[33]
EDU-Net: retinal pathological fluid segmentation in OCT images with multiscale feature fusion and boundary optimization
Zhang Y, Wang H, Chen L, et al. EDU-Net: retinal pathological fluid segmentation in OCT images with multiscale feature fusion and boundary optimization. arXiv:2604.20918. 2026. Preprint
2026 arXiv
-
[34]
A median filter scheme for mean curvature flow
Laux T, Swanson D. A median filter scheme for mean curvature flow. SIAM J Numer Anal. 2025;63(4):1823-1847. doi:10.1137/24M1705811
2025 doi
-
[35]
Active contour models driven by hyperbolic mean curvature flow for image segmentation
Hu S, He C, Zhang J, Kong D, Huang S. Active contour models driven by hyperbolic mean curvature flow for image segmentation. arXiv:2506.06712. 2025. Preprint
2025
-
[36]
Variational and PDE-based static and video image segmentation
Tai XC, Chan TF, Osher S, et al. Variational and PDE-based static and video image segmentation. In: Scherzer O, editor. Handbook of Mathematical Models in Computer Vision and Image Processing. Springer; 2025:145-178. doi:10.1007/978-3-031-89576-0_5
2025 doi
-
[37]
CHAOS challenge: combined (CT- MR) healthy abdominal organ segmentation
Kavur AE, Gezer NS, Barış M, Aslan S, Conze PH, et al. CHAOS challenge: combined (CT- MR) healthy abdominal organ segmentation. Med Image Anal. 2021;69:101950. doi:10.1016/j.media.2020.101950
2021 doi
-
[38]
Z. Gu, J. Cheng, H. Fu, K. Zhou, H. Hao, Y. Zhao, T. Zhang, S. Gao, J. Liao, et al. CE-Net: context encoder network for 2D medical image segmentation. IEEE Transactions on Medical Imaging, 38(10):2281–2292, 2019. 15 Declarations Funding This research received no external fundi...
2019
Reviewed July 15, 2026 · model on record in the stance chip above.
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