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图 4中给出了图 2轮廓线平滑降噪的效果。可以看出,平滑降噪后的曲线除中心区域外,其它波峰和波谷处都实现了单值尖峰结构,同时保持了原曲线的轮廓趋势和峰位位置。
图 5中给出了图 4平滑降噪后曲线的取反效果。图形显示,原来的亮环和暗环虽然发生了上下翻转,但是峰位坐标保持不变。为了描述方便,亮环和暗环概念仍表示原始数据的亮环和暗环,而反亮环和反暗环分别表示取反以后的亮环和暗环,即反亮环对应原始数据的暗环,反暗环对应原始数据的亮环。
图 5 平滑降噪后曲线x方向和y方向的取反效果
Figure 5. Reversal effect of x-direction and y-direction after smoothing and noise reduction
为了精确提取圆心和半径,在图 5中先获得A和B两个反亮环的坐标,然后在AA′区域和BB′区域上对取反前后的曲线进行处理。对于暗环来说,依次取得两个区域内所有反亮环最大值的坐标,也就是暗环直径坐标;对于亮环来说,A点和B点就是取反前曲线中心区域往外第一亮环起始的暗点位置。这样,依次取得AA′区域和BB′区域内所有亮环最大值的坐标,也就是亮环直径坐标。对所有直径坐标进行纵横坐标有序组合,通过圆回归拟合即可获得每个考察干涉圆环的半径和圆心坐标。其中,所有暗环给出的圆心坐标记为xd和yd,所有亮环给出的圆心坐标为xb和yb。对应同一张同心圆环图片来说,理论上亮环和暗环给出的坐标是同一个点,因此,最终的圆心坐标为x=0.5(xb+xd), y=0.5(yb+yd)。
图 5中的强度轮廓线显示, 靠近中心区域的内部几个圆环强度和间隔都比较理想,因此,只考虑靠近圆心的内部3个亮环和3个暗环即可。通过霍夫原变换自动获取考察图像参考圆心(x0, y0),然后,以参考圆心为基础分别在其x方向和y方向上[-50, 50]像素范围内每隔5 pixel取一条强度轮廓线,即每个方向上取21条线,按照图 3所示的过程完成每一个圆环的圆心和半径提取,最后,这些亮环圆心坐标和暗环圆心坐标分组分列在表 1中。为了更清晰地研究各个圆环对应圆心的波动情况,分别求出亮环和暗环各自总的圆心坐标平均值,然后以此为基准求出各个圆环所对应圆心坐标的差值,按照图 1c所示的亮暗顺序排列后如图 6所示。由表 1和图 6可知,3个亮环和3个暗环各自的圆心坐标非常接近,相对于总平均值的最大偏差为-3.7 pixel,即相对于平均值的最大相对偏差为-0.15%,重复性很好,这表明本文中的平滑降噪和取反方法具有良好的可行性。
表 1 图 1c内部3个亮环和3个暗环的回归半径和圆心坐标
Table 1. Regression radius and circle center coordinates of three inner bright rings and three dark rings of Fig. 1c
circle 1/pixel circle 2/pixel circle 3/pixel average value/pixel overall mean/pixel bright ring x 2467.0 2470.8 2467.7 2468.5 2468.5 dark ring x 2469.1 2466.7 2469.5 2468.4 bright ring y 2391.8 2384.9 2388.3 2388.3 2386.6 dark ring y 2389.1 2389.3 2388.1 2388.9 研究显示,非定域干涉同心圆环中每一环的半径R与其离开中心点的序数N之间存在如下式所示的关系
$ R_N{}^2=\alpha N $
(1) (1) 式中: α为常数系数。由此可知,相邻两个圆环半径平方比值与对应序数之比成正比。将表 2中的半径值R都按照图 1c所示的明暗位置排列, 首先确定相邻圆环序数之比Rr(后一个圆环比前一个圆环),然后对半径的平方取比值Rs(后一个圆环比前一个圆环),最后取半径平方比值与序数比值的相对误差E,如表 2所示。比较结果显示,圆回归拟合后圆环半径的准确性很好,相对误差在-4.18%~0.36%之间。
表 2 相邻圆环之间半径平方比值的关系
Table 2. Relationship between the radius square ratios of adjacent rings
dark ring 1 bright ring 1 dark ring 2 bright ring 2 dark ring 3 bright ring 3 R/pixel 598.7 828.8 1003.3 1153 1275.8 1400.1 ring number 0.5 1 1.5 2 2.5 3 Rr 2.000 1.500 1.333 1.250 1.200 Rs 1.916 1.465 1.321 1.224 1.204 E/% -4.18 -2.31 -0.70 -2.05 0.36
基于图像取反的同心圆环半径和圆心提取方法
The method to extract concentric ring radius and circle center based on image inversion
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摘要: 为了解决非定域干涉同心圆环图像的圆心和半径的精确提取问题, 提出了一种基于图像取反的暗亮环兼顾的算法, 以点光源非定域干涉同心圆环为对象, 在平滑降噪预处理基础上, 利用霍夫圆变换获得参考圆心, 以参考圆心为基础分别在其x方向和y方向上[-50, 50]像素范围内每隔5 pixel取一条强度轮廓线, 每一条强度轮廓线取反后, 分别实现暗环和亮环峰位坐标的精确识别, 再利用圆回归拟合得到系列圆心和半径, 通过系列圆心和半径的平均运算实现图像最终圆心和半径的精确提取。结果表明, 靠近中心的3个亮环和3个暗环各自的圆心坐标非常接近, 相对于总平均值的最大偏差为-3.7 pixel, 即相对于总平均值的最大相对偏差为-0.15%, 重复性很好; 相邻两个圆环半径平方比值相对于实际比值的误差在-4.18%~0.36%之间变化, 验证了算法的可行性。该研究对实现自动化检测、提高测量精度是有帮助的。Abstract: Aiming at the problem of accurate extraction of circle center and ring radius in non-localization interference concentric ring image, taking point light source non-localization interference concentric rings as the object, an algorithm based on image inversion for simultaneous caring of both dark and bright rings was proposed. The reference center was obtained using the Hough circle transformation after the pre-processing of smooth noise reduction. Then intensity outlines with an interval of 5 pixels in the range of [-50, 50] pixels in its x-direction and y-direction, respectively, were taken. With the inversion of each intensity contour line, accurate identification of dark and bright ring peak coordinates, respectively, was achieved. Finally, the accurate extraction of the circle center and radius of the image was achieved by averaging multiple circle parameters after circle regression fitting of obtained series circle center and radius. The results show that the center coordinates of the three bright rings and the three dark rings are very close to each other and have good reproducibility with the maximum deviation of -3.7 pixels to the mean value, i.e., a maximum relative deviation from the mean is -0.15%. The relative error of the radius-square ratio of two adjacent rings to the actual ratio varies between -4.18% and 0.36%, verifying the feasibility of the algorithm. This research is helpful in realizing automatic detection and improving measurement accuracy.
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表 1 图 1c内部3个亮环和3个暗环的回归半径和圆心坐标
Table 1. Regression radius and circle center coordinates of three inner bright rings and three dark rings of Fig. 1c
circle 1/pixel circle 2/pixel circle 3/pixel average value/pixel overall mean/pixel bright ring x 2467.0 2470.8 2467.7 2468.5 2468.5 dark ring x 2469.1 2466.7 2469.5 2468.4 bright ring y 2391.8 2384.9 2388.3 2388.3 2386.6 dark ring y 2389.1 2389.3 2388.1 2388.9 表 2 相邻圆环之间半径平方比值的关系
Table 2. Relationship between the radius square ratios of adjacent rings
dark ring 1 bright ring 1 dark ring 2 bright ring 2 dark ring 3 bright ring 3 R/pixel 598.7 828.8 1003.3 1153 1275.8 1400.1 ring number 0.5 1 1.5 2 2.5 3 Rr 2.000 1.500 1.333 1.250 1.200 Rs 1.916 1.465 1.321 1.224 1.204 E/% -4.18 -2.31 -0.70 -2.05 0.36 -
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