二次曲线拟合算法的统计性能分析与改进

Statistical Performance Analysis and Improvement of Conic Fitting Algorithm

传统的二次曲线拟合使用标准特征值分析算法。通过统计分析技术 ,可知该技术在拟合数字二次曲线时 ,存在估计偏差大、均方误差大的缺点。其产生原因是数据噪声的有色性和自相关函数矩阵的条件数过大 ,因此白化数据噪声和正则化变换是提高曲线拟合的有效措施。这从理论上有力地支持了Hartley提出的正则化算法。通过理论分析和计算机仿真实验 ,表明了降维EVD技术固有地同时具备噪声预白化功能和数据正则化功能 ,因此它能给出均方误差相当小的无偏估计。由于它无须进行预白化变换或正则化变换 ,并把最优化过程的维数从 6降为 2 ,所以它还具有计算快速、实现简单方便的优点。

The traditional procedure of conic fitting utilizes the standard Eigen Value Decomposition (EVD) algorithm. By means of the statistical analysis, we know that, when the technique is utilized to fit a digital conic, it has the disadvantages of very big estimation bias and mse. Its reason is that the data noise is not white and the condition number of the ACF matrix of the data observation is extremely big. Thus, the effective measure ment to improve the performance of a conic fitting algorithm is whitening the data noise and regulation transformation. This theoretic analysis has strongly supported the regularized EVD algorithm developed by Hartley. Then, we develop a dimension reduced EVD algorithm. The theoretical analysis and computer simulations have demonstrated that the technique has the advantages of intrinsical functions to whiten the data noise and to regulate the condition number of the ACF matrix of the data observation so that it can give a non biased estimation of conic parameter with very small mse. Furthermore, it has neither whitening transformation nor regulation transformation. At the same time, the dimension number of the optimization procedure is reduced from 6 to 2. Therefore the computation complex is largely simplified.