ELLIPSE Dataset: Benchmark for Elliptical Algorithms

• ELLIPSE Dataset is a curated collection of data instances featuring elliptical features for benchmarking algorithms in detection, overlap computation, and parameter estimation.
• Key methodologies include canonical transformation, quartic polynomial solving for intersections, and arc-support line segment clustering for precise ellipse fitting.
• The dataset supports applications in satellite imaging calibration, statistical modeling with data ellipses, and deep learning evaluation using metrics like F-measure and recall.

An ELLIPSE Dataset is a collection of data instances—images, mathematical entities, or simulation parameters—in which elliptical features, calculations, or geometric properties play a central role. It serves as a practical testbed or benchmark for algorithms involving ellipses, such as ellipse detection, overlap computation, parameter estimation, geometric modeling, or statistical interpretation. ELLIPSE Datasets are relevant in fields ranging from computer vision, physics, and engineering to statistical modeling and machine learning, where the precise handling of elliptical geometry is required to address real-world problems.

1. Algorithmic Foundations for Ellipse Overlap and Segmentation

Central to many ELLIPSE Datasets is the computation of geometric relationships, most notably the overlap area between two general ellipses. The foundational algorithm, as described in "Calculating ellipse overlap areas" (Hughes et al., 2011), decomposes the problem into three stages: transformation to canonical form, intersection determination via quartic polynomials, and partitioning of the overlap region using elliptical segment and polygonal areas. The fundamental formulas include the sector area of an ellipse between two parametric angles $\theta_1, \theta_2$:

$$\text{Area}_{\text{sector}} = \frac{AB}{2}(\theta_2 - \theta_1)$$

and the segment area adjustment:

$$\text{Area}_{\text{segment}} = \frac{AB}{2}(\theta_2 - \theta_1) \pm \frac{1}{2}|x_1 y_2 - x_2 y_1|$$

where the sign depends on the integration angle. Intersection points between ellipses are robustly found by solving the resultant quartic equation using Ferrari's formula, with a case-wise analysis for all possible intersection configurations. These geometric primitives are implemented efficiently in C, enabling large-scale simulations where overlap queries are ubiquitous, such as in satellite imaging system calibration or pedestrian dynamics modeling.

2. Dataset Structure, Annotations, and Statistical Geometry

ELLIPSE Datasets emerge in both computational vision and statistical contexts. In machine learning and data analysis, an ellipse may represent a confidence region (as in covariance-based statistical estimation (Friendly et al., 2013)) or a summary ellipse fitted to observed data points. The geometric characterization is encoded via quadratic forms, with data ellipses standardized as:

$$\mathcal{E}_c(\bar{\mathbf{y}}, \mathbf{S}) = \left\{ \mathbf{y} : (\mathbf{y} - \bar{\mathbf{y}})^{\mathsf{T}} \mathbf{S}^{-1} (\mathbf{y} - \bar{\mathbf{y}}) \leq c^2 \right\}$$

Labeling protocols in computer vision benchmarks, such as the Yummly-ellipse dataset (Pathiranage et al., 12 May 2024), provide ground-truth ellipse parameter annotations—center, axes lengths, orientation—against which automated detection and fitting methods can be quantitatively assessed, using metrics such as Chamfer distance or mean overlap ratio. Datasets may also include contour sequences, bounding boxes (from semantic segmentation models), or hybrid forms combining semantic and geometric cues.

3. Methodologies for Ellipse Detection, Fitting, and Grouping

Contour-based and arc-support line segment-based algorithms form the backbone of many image-based ELLIPSE Datasets. The method in (Lu et al., 2018) first extracts arc-support line segments (LSs) conveying curvature and geometric directionality, forms arc-support groups by linking segments with shared convexity, and applies hierarchical clustering in a 5D parameter space (center, orientation, semi-axes) to refine ellipse candidates. The algorithm incorporates the superposition principle in matrix fitting, using design matrix $D$ and scatter matrix $S = D^{\mathsf{T}} D$ for fast least-squares fitting. Synthetic and real-world datasets—traffic signs, PCB inspection images—are used to benchmark detection performance with F-measure, recall, and precision statistics.

Wildcard approaches, such as WildEllipseFit (Pathiranage et al., 12 May 2024), merge edge-based contour extraction with zero-shot semantic bounding box predictions (GroundingDINO), using multi-stage contour filtering, grouping (based on Euclidean fitting error thresholds), and final semantic validity checks (e.g., bounding box area fraction and center proximity). The combination of geometric and semantic priors enables accurate ellipse fitting in complex real-world scenarios.

4. Advanced Geometric and Analytic Properties

ELLIPSE Datasets support research into invariant properties and analytic expansions. For example, pedal-like curves derived from ellipses exhibit area invariance for particular locii of pedal points (Reznik et al., 2020), with explicit closed-form area expressions for pedal, contrapedal, rotated, and hybrid pedal curves. These invariants provide signature features for shape recognition and classification.

Recent analytic results establish that geometric relations such as the normal and point-to-ellipse distance can be expressed via Fourier series whose coefficients are themselves power series in ellipse eccentricity and normalized distance (Nilsson, 18 Jun 2025). For instance,

$$\phi - \psi = \sum_{n=1}^\infty \left( \sum_{k=1}^\infty \left( \sum_{l=\max(n,k)}^\infty c_{n,k,l}^\phi e^{2l} \right) \rho^k \right) \cdot \sin(2n\psi)$$

where $\psi$ is the angle of the external point, $e$ is the eccentricity, and $\rho = a/\rho_p$. These expansions replace quartic solvers with rapidly convergent, differentiable forms suitable for large-scale geometric computation or analytic manipulation.

5. Performance Evaluation, Benchmarks, and Applications

ELLIPSE Datasets serve as standard benchmarks for both traditional and deep learning-based algorithms. For instance, in robotics, datasets such as the AirLab Elliptic Target Detection Dataset (Keipour et al., 2021) provide frames annotated with ellipse presence and parameters to compare detection, tracking, and robustness under variable lighting and partial occlusion, with performance measured by F1 score (up to 0.981).

In ellipsometry, the EllipBench dataset (Ma et al., 25 Jul 2024) encompasses millions of simulated thin-film measurement instances, supporting residual connection and self-attention-based deep models for inverse mapping $$(\Psi, \Delta, n_3, k_3, \lambda) \rightarrow (n_2, k_2, d)$$, offering 98% accuracy on optical constants and 82.56% on thickness, outperforming shallow learners.

Applications span calibration (satellite optics), model-based image segmentation (semi-supervised fetal head detection with ellipse-constrained label refinement (Zhou et al., 27 Aug 2025)), geometric object recognition (PCB pads, traffic signs), and statistical modeling (data ellipse visualization for uncertainty assessment and regression inference (Friendly et al., 2013)). In all cases, the ELLIPSE Dataset paradigm supports rigorous evaluation, comparative algorithm development, and the integration of geometric priors for robust data interpretation.

6. Limitations and Prospects for ELLIPSE Datasets

Key limitations arise in degeneracy handling (e.g., near-identical or tangent ellipses (Hughes et al., 2011)), numerical instability at extreme parameter regimes, and annotation noise. The sensitivity to intersection point precision necessitates careful merging and tolerance setting in implementations. Potential future directions include high-performance optimization (vectorized, parallel algorithms), extension to three-dimensional ellipsoids and quasi-elliptical curves, and hybrid semantic-geometric data curation (e.g., foundational model-informed object class segmentation). The pursuit of error-bounded, analytic, and physically interpretable representations remains an active area, particularly for applications requiring uncertainty quantification or symbolic geometric computing.

7. Summary and Impact

The ELLIPSE Dataset concept encapsulates the convergence of geometric computation, data annotation, and algorithmic benchmarking where elliptical features are essential. Foundational C-based algorithms for overlap calculation, advanced statistical visualizations, arc-support detection schemes, and analytic power/Fourier series expansions collectively populate a rich methodological landscape. These datasets underpin modern approaches in computational geometry, statistical inference, robotics, and computer vision, providing the structure and quantitative rigor needed for algorithm validation, physical modeling, and practical deployment across numerous scientific and engineering domains.