{"title":"Fitting concentric elliptical shapes under general model","authors":"","doi":"10.1007/s00180-024-01460-x","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Fitting concentric ellipses is a crucial yet challenging task in image processing, pattern recognition, and astronomy. To address this complexity, researchers have introduced simplified models by imposing geometric assumptions. These assumptions enable the linearization of the model through reparameterization, allowing for the extension of various fitting methods. However, these restrictive assumptions often fail to hold in real-world scenarios, limiting their practical applicability. In this work, we propose two novel estimators that relax these assumptions: the Least Squares method (LS) and the Gradient Algebraic Fit (GRAF). Since these methods are iterative, we provide numerical implementations and strategies for obtaining reliable initial guesses. Moreover, we employ perturbation theory to conduct a first-order analysis, deriving the leading terms of their Mean Squared Errors and their theoretical lower bounds. Our theoretical findings reveal that the GRAF is statistically efficient, while the LS method is not. We further validate our theoretical results and the performance of the proposed estimators through a series of numerical experiments on both real and synthetic data.</p>","PeriodicalId":55223,"journal":{"name":"Computational Statistics","volume":"40 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00180-024-01460-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
Fitting concentric ellipses is a crucial yet challenging task in image processing, pattern recognition, and astronomy. To address this complexity, researchers have introduced simplified models by imposing geometric assumptions. These assumptions enable the linearization of the model through reparameterization, allowing for the extension of various fitting methods. However, these restrictive assumptions often fail to hold in real-world scenarios, limiting their practical applicability. In this work, we propose two novel estimators that relax these assumptions: the Least Squares method (LS) and the Gradient Algebraic Fit (GRAF). Since these methods are iterative, we provide numerical implementations and strategies for obtaining reliable initial guesses. Moreover, we employ perturbation theory to conduct a first-order analysis, deriving the leading terms of their Mean Squared Errors and their theoretical lower bounds. Our theoretical findings reveal that the GRAF is statistically efficient, while the LS method is not. We further validate our theoretical results and the performance of the proposed estimators through a series of numerical experiments on both real and synthetic data.
摘要 拟合同心椭圆是图像处理、模式识别和天文学中一项重要而又具有挑战性的任务。为了解决这一复杂问题，研究人员通过施加几何假设引入了简化模型。这些假设通过重新参数化使模型线性化，从而扩展了各种拟合方法。然而，这些限制性假设在现实世界中往往不成立，限制了它们的实际应用性。在这项工作中，我们提出了两种放宽这些假设的新型估计方法：最小二乘法（LS）和梯度代数拟合法（GRAF）。由于这些方法都是迭代法，我们提供了数值实现方法和策略，以获得可靠的初始猜测。此外，我们还利用扰动理论进行了一阶分析，得出了它们的均方误差前导项及其理论下限。我们的理论研究结果表明，GRAF 在统计上是高效的，而 LS 方法则不然。我们通过对真实数据和合成数据进行一系列数值实验，进一步验证了我们的理论结果和所提估计方法的性能。
期刊介绍：
Computational Statistics (CompStat) is an international journal which promotes the publication of applications and methodological research in the field of Computational Statistics. The focus of papers in CompStat is on the contribution to and influence of computing on statistics and vice versa. The journal provides a forum for computer scientists, mathematicians, and statisticians in a variety of fields of statistics such as biometrics, econometrics, data analysis, graphics, simulation, algorithms, knowledge based systems, and Bayesian computing. CompStat publishes hardware, software plus package reports.
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