Phenomics and Bioinformatics Research Center, School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes, Adelaide, Australia.
PLoS One. 2018 May 15;13(5):e0196902. doi: 10.1371/journal.pone.0196902. eCollection 2018.
Ellipse fitting is a highly researched and mature topic. Surprisingly, however, no existing method has thus far considered the data point eccentricity in its ellipse fitting procedure. Here, we introduce the concept of eccentricity of a data point, in analogy with the idea of ellipse eccentricity. We then show empirically that, irrespective of ellipse fitting method used, the root mean square error (RMSE) of a fit increases with the eccentricity of the data point set. The main contribution of the paper is based on the hypothesis that if the data point set were pre-processed to strategically add additional data points in regions of high eccentricity, then the quality of a fit could be improved. Conditional validity of this hypothesis is demonstrated mathematically using a model scenario. Based on this confirmation we propose an algorithm that pre-processes the data so that data points with high eccentricity are replicated. The improvement of ellipse fitting is then demonstrated empirically in real-world application of 3D reconstruction of a plant root system for phenotypic analysis. The degree of improvement for different underlying ellipse fitting methods as a function of data noise level is also analysed. We show that almost every method tested, irrespective of whether it minimizes algebraic error or geometric error, shows improvement in the fit following data augmentation using the proposed pre-processing algorithm.
椭圆拟合是一个备受研究且成熟的课题。然而，令人惊讶的是，迄今为止，没有任何现有方法在其椭圆拟合过程中考虑数据点的偏心度。在这里，我们引入了数据点偏心度的概念，类比于椭圆偏心度的概念。然后，我们通过经验证明，无论使用哪种椭圆拟合方法，拟合的均方根误差（RMSE）都会随着数据点集的偏心度的增加而增加。本文的主要贡献基于这样一种假设，即如果对点集进行预处理，在高偏心度区域有策略地添加额外的数据点，那么拟合的质量就可以得到提高。通过一个模型场景，从数学上证明了该假设的条件有效性。基于此确认，我们提出了一种算法，对数据进行预处理，以复制具有高偏心度的数据点。然后，通过对植物根系表型分析的 3D 重建的实际应用中的实验，验证了椭圆拟合的改进。还分析了不同基础椭圆拟合方法作为数据噪声水平函数的改进程度。我们表明，几乎所有经过测试的方法，无论其是否最小化代数误差或几何误差，在使用所提出的预处理算法进行数据增强后，拟合效果都有所提高。
