A robust methods of fitting plane based on LS-WTLS
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摘要: 为了解决平面数据点位精度差异性及平面模型常数项解算精度较低的问题，提出了一种基于最小二乘-加权总体最小二乘(LS-WTLS)的稳健平面拟合方法。该方法采用加权最小二乘模型与稳健估计IGGⅢ方案相结合的方式对平面模型误差项参量进行解算，然后通过设置阈值剔除粗差数据，利用最小二乘法对平面模型常数项进行解算，以此进一步提高了平面模型各参量的解算精度。结果表明, 新方法相对于最小二乘(LS)法、总体最小二乘(TLS)法、LS-TLS法、IGGⅢ-LS-TLS法，其单位权中误差分别提高了53.6%, 195.0%, 47.5%和5.1%，平面拟合精度分别提高了49.4%, 179.3%, 48.7%和46.99%，表现出了良好的抗粗差干扰能力。该研究验证了新方法在平面拟合领域的优越性和可靠性。Abstract: In order to solve the positional accuracy difference of plane data and the low calculation precision of plane model's constant term, a robust plane fitting method based on least squares-weighted total least square (LS-WTLS) was proposed. This method uses least squares model and robust estimation of IGGⅢ scheme to calculate the error parameters of plane model. Meanwhile, after rejecting the gross error data by setting the threshold, the constant term of plane model was calculated by using least square model. And based on this model, the accuracy of plane parameters was further improved. The new method shows favorable resistant to gross errors in experiments of fit the simulated plane data, meanwhile, the observed plane data fitting experiments show that compared with LS method, TLS method, LS-TLS method, IGGⅢ-LS-TLS method, the new method's mean square error of unit weight increased by 53.6%, 195.0%, 47.5%, and 5.1%, respectively, and its plane fitting accuracy increased by 53.6%, 195.0%, 47.5%, and, 5.1%, respectively. The results effectively verify this new method's superiority and reliability in the field of plane fitting.
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Table 1 Plane parameters and fitting precision of simulated data
proportion of σ Δ|a| Δ|b| Δ|c| ˆσ0 ˆσp 0% LS 0 0 0.0007 0.0152 0.0029 TLS 0 0 0.0009 0.0152 0.0029 LS-TLS 0 0 0.0007 0.0030 0.0029 IGGⅢ-LS-TLS 0 0 0.0007 0.0024 0.0029 LS-WTLS 0 0 0.0003 0.0023 0.0022 5% LS 0.0020 0 0.0047 3.5046 0.6770 TLS 0.0158 0.0064 15.2664 6.4890 1.2569 LS-TLS 0.0020 0 0.0047 3.5046 0.6770 IGGⅢ-LS-TLS 0.0003 0.0002 0.3013 0.2115 0.6843 LS-WTLS 0.0001 0 0.0028 0.0565 0.0556 10% LS 0.0036 0.0008 1.7493 4.1745 0.8067 TLS 0.0193 0.0081 19.1327 7.4930 1.4523 LS-TLS 0.0035 0.0008 1.7493 4.1745 0.8067 IGGⅢ-LS-TLS 0 0.0001 0.7808 0.1841 0.8314 LS-WTLS 0 0.0001 0.0718 0.0823 0.0810 20% LS 0.0113 0.0020 2.6627 11.0755 2.1427 TLS 0.1209 0.0527 124.3950 44.9742 8.8819 LS-TLS 0.0111 0.0019 2.5019 2.1755 2.1427 IGGⅢ-LS-TLS 0.0016 0.0006 4.6832 0.94298 2.2148 LS-WTLS 0.0014 0.0005 0.4423 0.51715 0.5080 
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Table 2 Observed data of plane
X Y Z 1 11.2 36.0 -5.0 2 10.0 40.0 -6.8 3 8.5 35.0 -4.0 4 8.0 48.0 -5.2 5 9.4 53.0 -6.4 6 8.4 23.0 -6.0 7 3.1 19.0 -7.1 8 10.6 34.0 -6.1 9 4.7 24.0 -5.4 10 11.7 65..0 -7.7 11 9.4 44.0 -8.1 12 10.1 31.0 -9.3 13 11.6 29.0 -9.3 14 12.6 58.0 -5.1 15 10.9 37.0 -7.6 16 23.1 46.0 -9.6 17 23.1 50.0 -7.7 18 21.6 44.0 -9.3 19 23.1 56.0 -9.5 20 19.0 36.0 -5.4 21 26.8 58.0 -16.8 22 21.9 51.0 -9.9
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Table 3 Plane parameters and fitting precision of observed data
ˆa ˆb ˆc ˆσ00 ˆσp LS -0.2710 0.0085 -4.2789 2.1536 1.9316 TLS -0.2558 0.2616 -15.9565 4.1357 3.6093 LS-TLS -0.3108 0.0215 -4.2807 2.0680 1.9219 IGGⅢ-LS-TLS -0.2111 0.0279 -5.8982 1.4735 1.8997 LS-WTLS -0.2384 0.0388 -5.8379 1.4018 1.2924
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