## 6 Conclusions  
In this paper, we presented an experimental comparison of five closed path determination methods for the use of GCSM to detect the TC of an OV beam. These five methods are the previously proposed MAICM and FC-PRCM, and three newly proposed methods: the WTM, WT-MAICM, and SC-PRCM. These methods come from two strategies: WTM, MAICM, and WT-MAICM belong to the strategy of ridge extraction, and FC-PRCM and SC-PRCM are derived from the strategy of PRC assignation. The codes for the algorithms are available from the authors by email. The methods were tested with an optical setup that used a LCOS-SLM as an OV beam, and an aberration generator and an SH-WFS located at the far-field plane to measure the OV beams. Two types of evaluation experiments were performed. One was under the condition that the OV beam had hardly any aberration, and the other was under the condition that the OV beam was distorted by simulated atmospheric turbulence. The experimental results indicate that the methods with ridge extraction outperform those based on PRC assignation in terms of both the well-determined detection rate and the processing speed.  
Figure 10: Closed path determination processes and results using the WTM (top row), MAICM (middle row), and WT-MAICM (bottom row).  
Under the condition of an aberration-free OV beam, the WTM, MAICM, and WT-MAICM performed excellently and achieved WTCMR values of 100%. On the other hand, the WTCMR values of the FC-PRCM and SC-PRCM were not always 100% and the performance was reduced with the increase of the TC value. What is more, the SC-PRCM was found to be the worst method in terms of both the WTCMR and the processing speed. There are a number of reasons that can explain the poor performance of the PRC assignation strategy, but the major factors are that the generated circle is prone to deviate from the OV center and go through regions with low intensity as well as invalid phase slope data, which are caused by nonuniformity in the intensity sum map along the azimuthal direction. The performance of the SC-PRCM was worse than that of the FC-PRCM, indicating that simply shifting the center of the PRC is not a suitable solution, and the maximum absolute TC value hunting criterion in the PRC assignation strategy lacks rationality.  
In the case of measuring distorted OV beams, the WTM, MAICM, and WT-MAICM show certain differences in term of the WTCMR, especially when the turbulent strength is high. Among these three methods, the WT-MAICM shows the strongest robustness against distortion, and the WTM was found to be the weakest. The limits of \(r_{0}/D\) to achieve WTCMR \(>\) 90% were around 0.75-1, 0.5-0.75, and 0.25-0.5 for the WTM, MAICM, and WT-MAICM, respectively, in terms of normalized correlation length. Overall, these results reveal that adaptively determining the closed path is necessary for GCSM in detecting the TC from a distorted OV beam.  
Conceptualization, D.W. and H.H.; funding acquisition, H.L.; methodology, D.W. and H.H.; project administration, H.H.; resources, H.T.; supervision, H.L. and H.T.; validation, D.W. and H.H.; writing--original draft, D.W.; writing--review and editing, D.W. and H.H.  
Shenzhen Innovation Funding (No: JCYJ20170818164343304, JCYJ[20170816172431715]; National Natural Science Foundation of China (No: U1809204, 61525106, 61427807, 61701436); National Key Technology Research and Development Program of China (No: 2017 YFE0104000, 2016 YFC1300302).  
We gratefully acknowledge [PERSON] and [PERSON] for their support and encouragement throughout this study. [PERSON] thanks the exchange program between Hamamatsu Photonics K.K. and Zhejiang University. The support from the above funding organizations is also gratefully acknowledged.  
The authors declare no conflict of interest.