## 3 Methods for Closed Path Determination
### Perfectly Round Circle Assignation  
As mentioned above, besides the ridge-extracting strategy, adaptively generating an appropriate perfectly round circle according to the intensity sum profile of an OV beam is also a closed path determination strategy that is worthy of discussion. In general, the center position and radius are two integral parameters to generate a perfectly round circle. However, given an intensity sum profile, it is difficult to directly determine the most appropriate perfectly round circle. Consequently, in our previous research, we proposed to generate a group of concentric perfect circles, and chose the one for which the measured TC value had the maximum absolute value as the determined perfect round circle. The center position of the concentric perfect circles was determined as the nearest lenslet position from the centroid of the intensity sum map, while the lower and upper bounds of the radius variation range was determined by the radii of the inscribed and circumscribed circles of the intensity sum profile [24].  
The performance of this generated perfectly round circle method (PRCM) as the closed path determination method to measure the TC value of a distorted OV beam is generally worse than that of the MAIC method [21; 24]. The reasons are listed as follows:  
* The method determining the center position of the generated perfectly round circles by the centroid of intensity sum map is not optimal, because the nonuniformity along the azimuthal direction in the intensity sum distribution significantly affects the centroid calculation.
* The center position as well as the radius are all forced to be integers.
* The generated circles restricted to perfectly round shapes will unavoidably go through the low-intensity region, which means that invalid phase slope data will be obtained.  
Notwithstanding, the generated circle should be fixed to a perfectly round shape to agree with the annular intensity sum profile and the position. Moreover, the radius should be forced to be an integer. This is because if we adaptively vary the shape or precisely calculate the position and radius and determine the phase slopes of the non-integer position by interpolation, the incurred complexity will go against our original goal of maintaining simplicity in the calculations. However, regarding the center position determination, the performance may be improved by shifting the center position within a proper region, referring to the radius variation. Although this dramatically increases the closed path determination time, we can accept a tradeoff between TC determination speed and accuracy if such a tradeoff manifests an improvement in the determination performance.  
Consequently, in this paper, we propose a new, modified method for the generation of perfectly round circles based on the previously proposed method, where the modification is merely to shift the center position of the concentric perfectly round circles within an advisable region, as is further discussed in Section 5. For the sake of differentiation, we named this method the shifting-center perfectly round circle method (SC-PRCM) and the previously proposed perfectly round circle method the fixed-center perfectly round circle method (FC-PRCM).