{"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":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00180-024-01460-x","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","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：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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