An intuitive method for circle fitting is proposed. Assuming an approximate circle(CA,n) for the fitting of some scattered points, it can be imagined that every point would apply a force to CA,n, which all together fo...An intuitive method for circle fitting is proposed. Assuming an approximate circle(CA,n) for the fitting of some scattered points, it can be imagined that every point would apply a force to CA,n, which all together form an overall effect that "draws" CA,n towards best fitting to the group of points. The basic element of the force is called circular attracting factor(CAF) which is defined as a real scalar in a radial direction of CA,n. An iterative algorithm based on this idea is proposed, and the convergence and accuracy are analyzed. The algorithm converges uniformly which is proved by the analysis of Lyapunov function, and the accuracy of the algorithm is in accord with that of geometric least squares of circle fitting. The algorithm is adopted to circle detection in grayscale images, in which the transferring to binary images is not required, and thus the algorithm is less sensitive to lightening and background noise. The main point for the adaption is the calculation of CAF which is extended in radial directions of CA,n for the whole image. All pixels would apply forces to CA,n, and the overall effect of forces would be equivalent to a force from the centroid of pixels to CA,n. The forces from would-be edge pixels would overweigh that from noisy pixels, so the following approximate circle would be of better fitting. To reduce the amount of calculation, pixels are only used in an annular area including the boundary of CA,n just in between for the calculation of CAF. Examples are given, showing the process of circle fitting of scattered points around a circle from an initial assuming circle, comparing the fitting results for scattered points from some related literature, applying the method proposed for circular edge detection in grayscale images with noise, and/or with only partial arc of a circle, and for circle detection in BGA inspection.展开更多
Forging spur gears are widely used in the driving system of mining machinery and equipment due to their higher strength and dimensional accuracy.For the purpose of precisely calculating the volume of cylindrical spur ...Forging spur gears are widely used in the driving system of mining machinery and equipment due to their higher strength and dimensional accuracy.For the purpose of precisely calculating the volume of cylindrical spur gear billet in cold precision forging,a new theoretical method named average circle method was put forward.With this method,a series of gear billet volumes were calculated.Comparing with the accurate three-dimensional modeling method,the accuracy of average circle method by theoretical calculation was estimated and the maximum relative error of average circle method was less than 1.5%,which was in good agreement with the experimental results.Relative errors of the calculated and the experimental for obtaining the gear billet volumes with reference circle method are larger than those of the average circle method.It shows that average circle method possesses a higher calculation accuracy than reference circle method(traditional method),which should be worth popularizing widely in calculation of spur gear billet volume.展开更多
We propose a two-stage method for detecting circular objects in this paper. In the first stage, curves are divided as linear segments or nonlinear segments. A least square estimator is used to find the estimated cente...We propose a two-stage method for detecting circular objects in this paper. In the first stage, curves are divided as linear segments or nonlinear segments. A least square estimator is used to find the estimated centers and radii of the nonlinear segments in the second stage. The found centers and radii are then evaluated to see if there exist circles in the nonlinear segments. Both of the broken and occluded circular objects are evaluated for the proposed method. From the experimental results, it is seen that the proposed method is efficient.展开更多
The point spread function（PSF） is investigated in order to study the centroids algorithm in a reverse Hartmann test（RHT） system. Instead of the diffractive Airy disk in previous researches, the intensity of PSF be...The point spread function（PSF） is investigated in order to study the centroids algorithm in a reverse Hartmann test（RHT） system. Instead of the diffractive Airy disk in previous researches, the intensity of PSF behaves as a circle of confusion（CoC） and is evaluated in terms of the Lommel function in this paper. The fitting of a single spot with the Gaussian profile to identify its centroid forms the basis of the proposed centroid algorithm. In the implementation process, gray compensation is performed to obtain an intensity distribution in the form of a two-dimensional（2D） Gauss function while the center of the peak is derived as a centroid value. The segmental fringe is also fitted row by row with the one-dimensional（1D） Gauss function and reconstituted by averaged parameter values. The condition used for the proposed method is determined by the strength of linear dependence evaluated by Pearson＇s correlation coefficient between profiles of Airy disk and CoC. The accuracies of CoC fitting and centroid computation are theoretically and experimentally demonstrated by simulation and RHTs. The simulation results show that when the correlation coefficient value is more than 0.9999, the proposed centroid algorithm reduces the root-mean-square error（RMSE） by nearly one order of magnitude, thus achieving an accuracy of - 0.01 pixel or better performance in experiment. In addition, the 2D and 1D Gaussian fittings for the segmental fringe achieve almost the same centroid results, which further confirm the feasibility and advantage of the theory and method.展开更多
基金Project(2013CB035504) supported by the National Basic Research Program of ChinaProject(2012zzts078) supported by the Fundamental Research Funds for the Central Universities of Central South University,ChinaProject(2009ZX02038) supported by the National Science and Technology Major Project of the Ministry of Science and Technology of China
文摘An intuitive method for circle fitting is proposed. Assuming an approximate circle(CA,n) for the fitting of some scattered points, it can be imagined that every point would apply a force to CA,n, which all together form an overall effect that "draws" CA,n towards best fitting to the group of points. The basic element of the force is called circular attracting factor(CAF) which is defined as a real scalar in a radial direction of CA,n. An iterative algorithm based on this idea is proposed, and the convergence and accuracy are analyzed. The algorithm converges uniformly which is proved by the analysis of Lyapunov function, and the accuracy of the algorithm is in accord with that of geometric least squares of circle fitting. The algorithm is adopted to circle detection in grayscale images, in which the transferring to binary images is not required, and thus the algorithm is less sensitive to lightening and background noise. The main point for the adaption is the calculation of CAF which is extended in radial directions of CA,n for the whole image. All pixels would apply forces to CA,n, and the overall effect of forces would be equivalent to a force from the centroid of pixels to CA,n. The forces from would-be edge pixels would overweigh that from noisy pixels, so the following approximate circle would be of better fitting. To reduce the amount of calculation, pixels are only used in an annular area including the boundary of CA,n just in between for the calculation of CAF. Examples are given, showing the process of circle fitting of scattered points around a circle from an initial assuming circle, comparing the fitting results for scattered points from some related literature, applying the method proposed for circular edge detection in grayscale images with noise, and/or with only partial arc of a circle, and for circle detection in BGA inspection.
文摘Forging spur gears are widely used in the driving system of mining machinery and equipment due to their higher strength and dimensional accuracy.For the purpose of precisely calculating the volume of cylindrical spur gear billet in cold precision forging,a new theoretical method named average circle method was put forward.With this method,a series of gear billet volumes were calculated.Comparing with the accurate three-dimensional modeling method,the accuracy of average circle method by theoretical calculation was estimated and the maximum relative error of average circle method was less than 1.5%,which was in good agreement with the experimental results.Relative errors of the calculated and the experimental for obtaining the gear billet volumes with reference circle method are larger than those of the average circle method.It shows that average circle method possesses a higher calculation accuracy than reference circle method(traditional method),which should be worth popularizing widely in calculation of spur gear billet volume.
基金supported by the I-Shou University under Grant No.ISU 102-05-01
文摘We propose a two-stage method for detecting circular objects in this paper. In the first stage, curves are divided as linear segments or nonlinear segments. A least square estimator is used to find the estimated centers and radii of the nonlinear segments in the second stage. The found centers and radii are then evaluated to see if there exist circles in the nonlinear segments. Both of the broken and occluded circular objects are evaluated for the proposed method. From the experimental results, it is seen that the proposed method is efficient.
基金Project supported by the National Natural Science Foundation of China(Grant No.61475018)
文摘The point spread function（PSF） is investigated in order to study the centroids algorithm in a reverse Hartmann test（RHT） system. Instead of the diffractive Airy disk in previous researches, the intensity of PSF behaves as a circle of confusion（CoC） and is evaluated in terms of the Lommel function in this paper. The fitting of a single spot with the Gaussian profile to identify its centroid forms the basis of the proposed centroid algorithm. In the implementation process, gray compensation is performed to obtain an intensity distribution in the form of a two-dimensional（2D） Gauss function while the center of the peak is derived as a centroid value. The segmental fringe is also fitted row by row with the one-dimensional（1D） Gauss function and reconstituted by averaged parameter values. The condition used for the proposed method is determined by the strength of linear dependence evaluated by Pearson＇s correlation coefficient between profiles of Airy disk and CoC. The accuracies of CoC fitting and centroid computation are theoretically and experimentally demonstrated by simulation and RHTs. The simulation results show that when the correlation coefficient value is more than 0.9999, the proposed centroid algorithm reduces the root-mean-square error（RMSE） by nearly one order of magnitude, thus achieving an accuracy of - 0.01 pixel or better performance in experiment. In addition, the 2D and 1D Gaussian fittings for the segmental fringe achieve almost the same centroid results, which further confirm the feasibility and advantage of the theory and method.
