A Bayesian Approach Toward Robust Multidimensional Ellipsoid-Specific Fitting

TL;DR

This paper tackles the challenge of robustly fitting multidimensional ellipsoids to noisy, outlier-contaminated point clouds by casting ellipsoid fitting as a Bayesian parameter estimation problem. It imposes an ellipsoid-specific constraint via a uniform prior over the ellipsoidal domain and models outliers with an additional uniform component in the predictive distribution, leading to a MAP/ML formulation solvable by an $\varepsilon$-accelerated EM algorithm. The method generalizes to spaces $\mathbb{R}^n$, is robust to large axis ratios and heavy disturbances, and supports practical tasks such as 2D ellipse fitting, 3D ellipsoid reconstruction, and higher-dimensional shape fitting, with competitive efficiency. Empirical results across synthetic and real data demonstrate superior accuracy and robustness compared to algebraic, geometric, and robust baselines, and the approach shows promise for applications in microscopy, 3D reconstruction, and calibration. The work also provides theoretical justification for the uniform component via Kullback–Leibler considerations and outlines avenues for future extensions to other conics and automatic model selection.

Abstract

This work presents a novel and effective method for fitting multidimensional ellipsoids to scattered data in the contamination of noise and outliers. We approach the problem as a Bayesian parameter estimate process and maximize the posterior probability of a certain ellipsoidal solution given the data. We establish a more robust correlation between these points based on the predictive distribution within the Bayesian framework. We incorporate a uniform prior distribution to constrain the search for primitive parameters within an ellipsoidal domain, ensuring ellipsoid-specific results regardless of inputs. We then establish the connection between measurement point and model data via Bayes' rule to enhance the method's robustness against noise. Due to independent of spatial dimensions, the proposed method not only delivers high-quality fittings to challenging elongated ellipsoids but also generalizes well to multidimensional spaces. To address outlier disturbances, often overlooked by previous approaches, we further introduce a uniform distribution on top of the predictive distribution to significantly enhance the algorithm's robustness against outliers. We introduce an ε-accelerated technique to expedite the convergence of EM considerably. To the best of our knowledge, this is the first comprehensive method capable of performing multidimensional ellipsoid specific fitting within the Bayesian optimization paradigm under diverse disturbances. We evaluate it across lower and higher dimensional spaces in the presence of heavy noise, outliers, and substantial variations in axis ratios. Also, we apply it to a wide range of practical applications such as microscopy cell counting, 3D reconstruction, geometric shape approximation, and magnetometer calibration tasks.

Figures (27)

• Figure 1: Geometric shape approximation using elliptical curves and ellipsoidal surfaces based on the developed method. (a) Ellipse fitting to a 2D brain MRI slice in the presence of heavy outliers (i.e., other brain structures). Note that we perform ellipse fitting on the edge map directly without any preprocessing step such as arc detection or structure removal. (b) Ellipsoid fitting to the anomalous 3D medical femur data DataBone.
• Figure 2: Visualization of using RDOS to initialize $M$ and $w$ simultaneously, where red points indicate the inferred outliers.
• Figure 3: Left: Comparison of the probability $p(x)$ with ($w\neq0.0$) or without ($w=0.0$) the uniform distribution in 1D space. Right: The corresponding negative log-likelihood function of $p(x)$, where bounded influence functions are ensured by our method when $w\neq0.0$.
• Figure 4: Quantitative evaluations on noisy measurements with respect to 2D ellipse fitting, utilizing the log-scale $y$ axis to enhance the readability of the statistical results. We gradually increase the noise level from $10\%$ to $200\%$ and report the MSE of each compared method based on 100 tests. Remarkably, BayFit demonstrates superior fitting precision across all noise levels.
• Figure 5: Qualitative results of 2D ellipse fitting on different noise levels. The zoomed-in sub-figures suggest that BayFit has superior fitting precision than competitors, especially under the existence of heavy noise.

Theorems & Definitions (7)

• Definition 1
• Proposition 1: Axis-Ratio-Dependent Ellipsoid-Specificity kesaniemi2017direct
• Definition 2: Ellipsoid in $\mathbb{R}^n$
• Theorem 1: Convergence Assurance
• Definition 3
• Theorem 2: Convergence Consensus of the $\varepsilon$-Accelerated EM wang2008acceleration
• Theorem 3: Nonequivalence between $p(\mathbf{x}_i|\mathcal{E}(\bm{\theta}))$ and $f(\mathbf{u},\mathbf{\xi})$