WildEllipseFit: Robust Ellipse Fitting Methods

• WildEllipseFit is a family of algorithms that achieve robust ellipse fitting with minimal parameters under challenging real-world conditions.
• It integrates diverse approaches—including RANSAC, Bayesian, least-squares, and intensity-weighted methods—to handle outlier contamination and missing data.
• The pipelines are applied across fields such as 3D reconstruction, diffraction analysis, and food portion estimation, demonstrating high accuracy and practical impact.

WildEllipseFit is a family of algorithmic approaches and pipelines designed for robust, accurate, and often minimal-parameter ellipse fitting under challenging real-world (“wild”) conditions. The term encompasses a spectrum of methods, each adapted to particular scenarios such as strong outlier contamination, missing data, non-uniform pixel saliency, semantic ambiguities, or the need for three-point minimality. Representative WildEllipseFit pipelines include algebraic, statistical, Bayesian, RANSAC, and deep-learning-guided methods, with applications in fields as diverse as 3D reconstruction, diffraction-ring parameterization, food portion estimation, and generic geometric vision.

1. Minimal-Parameter and RANSAC-Based WildEllipseFit

The RANSAC-based three-point algorithm by Xu is a foundational WildEllipseFit variant for spherical-object projections (Xu, 2015). Here, ellipses are fit directly in algebraic conic form:

$$f(x,y) = A x^2 + 2 B x y + C y^2 + 2 D x + 2 E y + F = 0$$

Given three points, the null space of the resulting measurement matrix defines a three-dimensional solution space. By imposing two cubic constraints that encode the geometry of a projected sphere,

$S_1(p) = d(b d - a e) - e(b e - c d) = 0$

$$S_2(p) = b(a e - b d)f^2 - e(d e - b f)(c d - b e) = 0$$

the problem is reduced to a system of two cubics in two parameters, with up to nine real solutions. RANSAC is employed over the input point set, iteratively:

• Sampling three non-collinear candidates.
• Solving for all real elliptical hypotheses.
• Evaluating consensus via inlier counts (distance threshold).
• Refining the consensus estimate using all inliers under the imposed constraints.

Key features include an exceptionally small minimal sample size (three), efficient hypothesis generation, and robustness to high outlier proportions (often ≳50%) (Xu, 2015).

2. Bayesian and Probabilistic WildEllipseFit for Outlier-Robustness

Bayesian WildEllipseFit, as developed by Zhao et al. (Mingyang et al., 2024), frames ellipse (or higher-dimensional ellipsoid) parameter estimation as posterior maximization. The model treats the parameters $\theta$ as random variables with a uniform prior over the valid ellipsoid domain, ensuring the solution corresponds to an actual ellipsoid:

$$P(\theta \mid X) \propto P(X \mid \theta) \, P(\theta)$$

The likelihood is a mixture:

$$p(x_i \mid \theta) = (1-w) \frac{1}{M} \sum_{j=1}^M \mathcal{N}(x_i; y_j(\theta), \sigma^2 I) + w \frac{1}{V}$$

where the uniform component $w/V$ addresses outlier contamination, and $y_j$ are Monte Carlo samples on the surface of $E(\theta)$. Parameter estimation proceeds via an EM-type algorithm, with closed-form responsibility updates and $\epsilon$-acceleration for fast convergence. The framework naturally extends to $n$-dimensional ellipsoids and provides interpretable statistical diagnostics such as estimated noise variance and outlier rates (Mingyang et al., 2024).

3. Least-Squares and Unconstrained WildEllipseFit

For cases with clean or moderately noisy data, unconstrained, trigonometric least-squares ellipse fitting achieves reliable, high-accuracy estimation (Quan et al., 2023). The general approach parameterizes the ellipse with center, axes, and rotation, associating each observed point $(x_i, y_i)$ to an ellipse phase angle $\phi_i$ derived analytically per point:

$$E(a, b, x_0, y_0, \phi_0) = \sum_{i=1}^N \left\| X_i - \left( (x_0, y_0)^T + R(\phi_0)\, \mathrm{diag}(a,b)\, P(\phi_i) \right) \right\|^2$$

with $P(\phi_i) = [\cos\phi_i, \sin\phi_i]^T/2$, and $\phi_i$ determined as a function of data and current parameters. The objective is minimized via Gauss–Newton, exploiting Jacobians with respect to all parameters. The method gracefully handles incomplete contours and missing data, with no need for explicit angle continuity constraints (Quan et al., 2023).

4. WildEllipseFit under Heavy Outliers and Occlusion

Approaches such as the maximum correntropy criterion with variable center (MCC-VC) address settings with severe outlier contamination (Wang et al., 2022). Here, the fit minimizes a nonconvex cost based on Laplacian-kernel correntropy:

$$J(c, \sigma) = - \sum_{i=1}^N \exp\left(-\frac{|e_i(\theta) - c|}{\sigma}\right)$$

where $e_i(\theta)$ is the algebraic distance of point $i$ from the candidate ellipse. The bandwidth $\sigma$ and center $c$ are alternately optimized (closed-form via Taylor expansion and median-based minimization, respectively), while the ellipse parameters $\theta$ are updated via SOCP with an explicit constraint $B^2 - 4AC < 0$. The algorithm is empirically shown to outperform RANSAC and other recent ellipse-fitting variants in both synthetic and real datasets with up to 55% outliers, and it extends naturally to coupled ellipse sets (e.g., for iris images) using association-variable relaxation (Wang et al., 2022).

5. Median/Mode and Voting-Based WildEllipseFit

Greminger's median/mode parameterization (Greminger, 2015) fits hundreds to thousands of five-point ellipses to randomly selected subsets of data. For each angle $\theta$, the contour radial coordinate is determined as the median or mode of intersection points from the ensemble of ellipses. The final contour is constructed as a function $r^*(\theta)$, with the center set by the componentwise median of all fitted centers. This approach is robust to both outliers and large-scale occlusions as long as a significant portion of the five-point samples are inliers. Median parameterization ensures continuity; mode parameterization can handle multimodality in the ray statistics. This framework enables accurate quantification of irregular or non-elliptical boundaries, as demonstrated on microexploding fuel droplets (Greminger, 2015).

6. WildEllipseFit with Saliency-Weighted and Incomplete Data

For applications such as fitting Debye–Scherrer rings in textured, spotty, or occluded 2D diffractograms, intensity-weighted algebraic fits are employed (Karamched et al., 2021). The method minimizes the sum of squared algebraic distances weighted by pixel intensities,

$$J(p) = \sum_{i=1}^N w_i [F(p, x_i, y_i)]^2, \quad \text{s.t.} \quad 4a c - b^2 = 1$$

solving a generalized eigenvalue problem. The approach is robust to missing angular sectors (“incomplete rings”) and variable angular intensity (strong texture), outperforming “cake-integration” for strain determination. Plug-and-play MATLAB and Python reference implementations are openly available (Karamched et al., 2021).

7. Semantic and Multi-Stage WildEllipseFit in Natural Scenes

Recent instantiations in food image analysis combine zero-shot semantic segmentation with robust geometric fitting (Pathiranage et al., 2024). WildEllipseFit here first detects candidate “plate” and “bowl” regions using open-set object detectors such as GroundingDINO. Within each bounding region, high-curvature, low-noise rim segments are selected by filtering based on local curvature proxies, edge continuity, and geometric relation to the food region. After grouping and convex hull extraction, standard least-squares ellipse fitting under discriminant constraints provides parameter estimates. Semantic validation using area overlap and center-food distance metrics rejects false positives. On the Yummly-ellipse dataset, the method achieves lower Chamfer distances than state-of-the-art arc-adjacency-based methods, demonstrating tight integration of high-level semantic cues with classical edge geometry (Pathiranage et al., 2024).

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References:

• (Xu, 2015) RANSAC based three points algorithm for ellipse fitting of spherical object's projection
• (Mingyang et al., 2024) A Bayesian Approach Toward Robust Multidimensional Ellipsoid-Specific Fitting
• (Quan et al., 2023) Optimised Least Squares Approach for Accurate Polygon and Ellipse Fitting
• (Wang et al., 2022) Robust Ellipse Fitting Based on Maximum Correntropy Criterion With Variable Center
• (Greminger, 2015) Median and Mode Ellipse Parameterization for Robust Contour Fitting
• (Karamched et al., 2021) Weighted ellipse fitting routine for spotty or incomplete Debye-Scherrer rings on a 2D detector
• (Pathiranage et al., 2024) In The Wild Ellipse Parameter Estimation for Circular Dining Plates and Bowls