Probabilistic Detection and Estimation of Conic Sections From Noisy Data

Inferring unknown conic sections on the basis of noisy data is a challenging problem with applications in computer vision. A major limitation of the currently available methods for conic sections is that estimation methods rely on the underlying shape of the conics (being known to be ellipse, parabola, or hyperbola). A general purpose Bayesian hierarchical model is proposed for conic sections and corresponding estimation method based on noisy data is shown to work even when the specific nature of the conic section is unknown. The model, thus, provides probabilistic detection of the underlying conic section and inference about the associated parameters of the conic section. Through extensive simulation studies where the true conics may not be known, the methodology is demonstrated to have practical and methodological advantages relative to many existing techniques. In addition, the proposed method provides probabilistic measures of uncertainty of the estimated parameters. Furthermore, we observe high fidelity to the true conics even in challenging situations, such as data arising from partial conics in arbitrarily rotated and nonstandard form, and where a visual inspection is unable to correctly identify the type of conic section underlying the data. Supplementary materials for this article are available online.

基于带噪声数据推断未知圆锥曲线（conic sections）是一项极具挑战性的问题，在计算机视觉领域具有重要应用价值。当前主流的圆锥曲线处理方法存在一项核心局限：其参数估计流程依赖于已知的圆锥曲线基础形态（即预先明确其为椭圆、抛物线或双曲线）。本文提出了一款面向通用场景的贝叶斯分层模型（Bayesian hierarchical model），并验证了基于带噪声数据的对应估计方法，即便在圆锥曲线具体类型未知的情况下仍可正常运行。该模型可实现对潜在圆锥曲线的概率性检测，并完成对圆锥曲线关联参数的推断。通过大量预设真实圆锥曲线未知的仿真实验，本文证实所提方法相较于诸多现有技术，兼具实践实用性与方法学优越性。此外，所提方法还可输出估计参数不确定性的概率度量指标。进一步研究表明，即便在极具挑战性的场景中，该方法对真实圆锥曲线的拟合精度仍保持较高水准——例如源自任意旋转非标准形式的部分圆锥曲线数据，以及目视检测无法准确识别数据所对应圆锥曲线类型的场景。本文的补充材料可在线获取。