Ellipse-Constrained Pseudo-Label Refinement

• The paper introduces ellipse-constrained pseudo-label refinement that leverages least-squares ellipse fitting and Gaussian heatmap generation to improve segmentation accuracy in medical imaging.
• It employs a dual-scoring approach combining boundary consistency and contour regularity to filter noisy predictions and enforce geometric regularization.
• Quantitative results show Dice score improvements of 1.5–3.5%, significant annotation time reduction, and enhanced boundary adherence in applications such as fetal head and vascular segmentation.

Ellipse-constrained pseudo-label refinement refers to a class of strategies for improving the reliability of pseudo-labels—machine-generated surrogate labels—by exploiting elliptical geometric constraints and probabilistic regularization. This approach is particularly applicable where the underlying targets exhibit near-elliptical shape invariance (e.g., vascular structures or fetal heads in medical imaging), or where geometric regularization improves learning in settings with noisy, unlabeled, or weakly-supervised data. In the recent literature, ellipse-constrained refinement methods integrate techniques such as least-squares ellipse fitting, class-aware geometric regularization, probabilistic heatmaps, and multi-consistency constraints to improve the utility of pseudo-labels for both semi-supervised and unsupervised representation learning.

1. Geometric Foundations of Ellipse Constraints

Ellipse-constrained approaches leverage the shape prior that certain objects or regions—such as vessels in CT scans (Ma et al., 5 Feb 2024) and fetal heads in ultrasound imaging (Zhou et al., 27 Aug 2025)—are well approximated by ellipses in Cartesian image space. Annotation standards mandate the drawing of the minimal covering ellipse, enforcing topological regularity and reducing ambiguous boundary effects. Least-squares conic fitting methods transform discrete annotations into analytical ellipse parameters: center $(x,y)$, axes $(w,h)$, and rotation $\theta$.

Formally, a general ellipse in 2D is given by the conic equation:

$$F_{\alpha}(x) = ax^2 + bxy + cy^2 + dx + ey + f = 0,\quad b^2 - 4ac < 0$$

The result is a robust geometric representation suitable for downstream pseudo-label construction and for constraining the network’s learning process. In ERSR for fetal head segmentation (Zhou et al., 27 Aug 2025), the largest connected component is fit with such an ellipse; all subsequent geometric regularization transforms are computed relative to this parametric prior.

2. Probabilistic Pseudo-label Generation and Refinement

Pseudo-labels generated via ellipse constraints can be binary masks or, more powerfully, probabilistic heatmaps. After ellipse fitting, Gaussian heatmaps are constructed:

$$f(X) = \frac{1}{\sqrt{2\pi|\Sigma|}} \exp\left[ -\frac{1}{2}(X-\mu)^T \Sigma^{-1} (X-\mu) \right]$$

where $\mu$ is the center and $\Sigma$ encodes axis lengths and rotation via covariance.

Discrete Gaussian maps are produced for each image location by coordinate transforms and matrix algebra:

• Center and rotate pixel coordinates to align with the ellipse.
• Compose pixelwise intensity as $G = F_x \otimes F_y$ (element-wise product of Gaussians along principal axes).
• Normalize so that $G \in [0,1]$, with $G>0.5$ inside the ellipse, decaying smoothly at the boundary.

Pseudo-label refinement then consists of aligning DNN outputs $P_X$ to the pseudo-label distribution $P_G$ using KL divergence ($L_{KL}$) and MAE ($L_{Recon}$), enforcing both global distributional similarity and precise local reconstruction (Ma et al., 5 Feb 2024). In ERSR (Zhou et al., 27 Aug 2025), refinement operates by soft-thresholding: central pixels (low elliptical distance) are boosted, while peripheral noise is suppressed polynomially or exponentially.

3. Filtering and Consistency Regularization

Unfiltered pseudo-labels from teacher models often propagate noise. Ellipse-constrained methods utilize pre-filtering strategies (ERSR’s dual-scoring approach) before refinement:

• Boundary consistency ($S_{boundary}$): quantifies smoothness of prediction contours via Sobel gradients.
• Contour regularity ($S_{contour}$): measures local curvature variability using the Laplacian of the EDT.
• Aggregated geometric score:

$$S_{score} = 1 - (\alpha S_{boundary} + (1-\alpha) S_{contour})$$

High-confidence samples (top $K$) are selected for refinement. Subsequently, symmetry-based regularization exploits anatomical symmetry (fetal head, vascular structures) by enforcing multi-image/multi-augmentation consistency losses, including direct loss between predictions and refined pseudo-labels, and cross-consistency losses between symmetric regions (Zhou et al., 27 Aug 2025).

4. Statistical and Energy-based Constrained Refinement

Beyond explicit geometric constraints, statistical regularization is employed for pseudo-label balancing. SEVAL (Li et al., 7 Jul 2024) implements offset-based pseudo-label refinement in imbalanced SSL:

$q = \sigma(z^U - \log \pi)$

where $\pi$ is a learned offset vector (analogous to a logit prior correction) optimized via holdout validation. Class-wise thresholds $\tau_c$ further select only those pseudo-labels achieving a target accuracy.

Energy-based pseudo-label refinement (Kong et al., 2022) constrains learning to low-energy regions of feature space:

$E_w(x) = -\text{LogSumExp}_k(f_w(x)[k])$

and the training objective couples cross-entropy with energy minimization:

$$\min_{w, \hat{y_t}} \sum_t \left[-\sum_k \hat{y}_t^{(k)} \log p_w(k|x_t) + \alpha E_w(x_t) \right]$$

A plausible implication is that ellipse-like constraints could be added in feature or latent space, e.g., using Mahalanobis or elliptical distance-based penalization, to further improve geometric alignment.

5. Implementation Strategies and Quantitative Impact

Ellipse-constrained pseudo-label refinement is usually implemented in training pipelines as a sequence of modules:

1. Efficient annotation (ellipse drawing in ImageJ or equivalent).
2. Automatic least-squares ellipse fitting and parameter extraction.
3. Pseudo-label construction (binary/heatmap).
4. Dual-scoring or confidence-based sample filtering.
5. Refinement loss: distribution (KL/Wasserstein), reconstruction (MAE).
6. Consistency regularization (augmentation and symmetry).
7. (Optional) Incorporation of external unlabeled data via transfer of ellipse fitting to new slices.

Reported metrics show strong improvement over baselines:

• Dice score increases of 1.5–3.5% versus strong-label supervised methods (Ma et al., 5 Feb 2024, Zhou et al., 27 Aug 2025).
• Up to 82% reduction in annotation times.
• Hausdorff distance reduction for boundary adherence (e.g., 68% in vascular segmentation).
• High Dice scores (e.g., 95.36% for fetal head with only 20% labeled data) (Zhou et al., 27 Aug 2025).

6. Comparative Analysis and Application Scope

Ellipse-constrained pseudo-label refinement is effective in domains with known geometric priors—medical imaging (vessels, fetal anatomy), and potentially in other structured tasks (e.g., objects with elliptic boundaries). The approach compares favorably to conventional pixelwise annotation (lower labor and variability), energy-based regularization (greater geometric specificity), and confidence-threshold self-training (greater reliability).

A plausible implication is increased adaptability and lower annotation burden in clinical settings, as rapid ellipse-based labeling facilitates large-dataset scaling. Weakly-supervised frameworks that incorporate pseudo-labels from external, label-agnostic sources offer further generalization power (Ma et al., 5 Feb 2024).

7. Limitations and Prospective Directions

Ellipse constraints presuppose that target structures are approximately elliptical. In scenarios with high shape variance, geometric misfit may suppress valid structure or introduce bias. Hyper-parameter tuning (score thresholds, loss weights) and computational overhead (ellipse fitting, multi-augmentation) can increase training complexity. In highly noisy or non-elliptic cases, further generalization (energy-based or adaptive geometric priors) may be required (Kong et al., 2022).

Future avenues include integration of ellipse-constrained refinement into energy-based models for more abstract latent space regularization, and statistical calibration in imbalanced datasets via offset learning and class-aware thresholding. Robust handling of extreme noise, open-set adaptation, and non-elliptical yet geometrically regular structures remain open challenges.

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The ellipse-constrained pseudo-label refinement paradigm constitutes a principled integration of geometric, probabilistic, and statistical regularization for the generation and utilization of high-quality pseudo-labels, leading to improved reliability and annotation efficiency in diverse semi-supervised and unsupervised learning contexts.