Precision improved spot centroid-locating algorithm based on ...

Precision improved spot centroid-locating algorithm based on profile regularization

Feng QIAN1, a, Xiao-Pei ZHANG2, b 1 University of Chinese Academy of Sciences, Beijing, 100049, China

2 Henry Samueli School of Engineering and Applied Science, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095, USA

aEmail: [email protected], bEmail: [email protected]

Keywords: centroid detection; profile regularization; center of weight

Abstract. The spot centroid localization is one of the key problems in various fields of scientific research and engineering applications. This paper proposed a novel algorithm based on profile regularization of the detecting spots. The algorithm first roughly eliminates the background and random noise errors by an a priori statistically estimated threshold; then determines the profile edge of the spot region by using morphological filtering technique; last calculates the centroid position with a center of weight algorithm. The field experiments were performed to validate the presented algorithm. The experimental results have demonstrated the superiorities of the proposed centroid detection algorithm on precision, stability and repeatability. This study is of value in improving the precision of center location algorithms.

Introduction

Centroid detection of spot images acts as a key role in measuring wave-front distortion by Shack-Hartmann wave-front sensor for adaptive optics[1][2], optical testing[3][4] and many other area such as pattern recognition for point object[5], freeform surface measurement[6], lithography system[7], etc. The measuring precision of the centroid for each spot directly affects the accuracy of measurement or assessment in such cases. The conventional centroid detection algorithm is based on the gray level of the spot image[8]. However, when there is noise in the spot image and the noise is not negligible compared to the signals, the detection of an accurate center position becomes difficult by solely using the conventional algorithm. Considering that the background noise may have influence to detect the center position, the modified center of weight algorithms have been developed. The modified versions take power of the gray level of the spot image intensity[9] instead of the gray level itself as the weighting factor. The general method to eliminate the background noise is threshold method[10] and windowing algorithm[11]. In the threshold method, a threshold is used to determine the boundary between spot signal and noise. In the windowing algorithm, the size of a detection window is changed to reduce the influence of noises. Image filtering technique is also introduced to remove the noise, such as smoothing filter, high-frequency removal filter, etc.

This study proposed a novel algorithm of locating spot centroids based on profile regularization. The algorithm estimates the profile edge of a pot more precisely based on the intrinsic spatial correlation of the spot, rather than an optimal estimated threshold, which is usually implied in segmentation algorithms. A center of weight algorithm is applied on the regularized spot region for centroid location. In the experiments section, the performance of the presented algorithm and its comparison with the reference method has been discussed. The conclusion is in the last section.

Principles of Centroid Detection

Center of Weight algorithms. The conventional center of weight algorithm and its modified versions can be uniformly

expressed as

4th International Conference on Electrical & Electronics Engineering and Computer Science (ICEEECS 2016)

Copyright © 2016, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Advances in Computer Science Research, volume 50

849

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Field Experiments and Discussions

The field experiments were performed in the laser testing range. The receiving camera was set fixed directing to the target board and captured the sequential video images. The laser emitter shot every 4~5 seconds. The laser spot shot on the target board lasted for more than 40 seconds and then began to disappear. We capture the multiple frames right after the occurrence of the spot on the target board for repeated tests. The frame sampling frequency of the commercial camera is usually between 24 to 30 frames/s, resulting in that the capture time cost of one frame is 0.03 to 0.04 second. Since the acquisition duration was much shorter than the time of a laser spot lasted until it began to disappear. We considered that the differences between every acquired sequential images were mainly due to sample error and noise error rather than the fading effect of the laser spot itself. Because the true positions of the spot centroids cannot be obtained, we defined an assessment criterion for evaluating the precision of the centroid detection.

Assessment Criterion. Take the procedure of locating the centroid position of spot as one detection test. The detected

position of spot on the -th test is denoted as , yc cx

, where 1 K , K is the number

of laser spots on the target board; 1 N , N is the number of total tests. The assessment criterion is defined as

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As shown in Fig.4, the centroid distribution of our method is more compact than that of the reference method. It demonstrates that our method has superior stability for centroid positioning. The detected centroid positions with their confidence intervals ( 0.05 ) and assessment criterion are shown in Table 1.

Table 1 The statistical result of centroid detection for one certain spot. Method Centroid (α= 0.05) Mξ

Proposed Method （7.9296±0.0613，7.9495±0.0711） 0.1727 Reference Method （7.7134±0.0825，8.1669±0.0997） 0.5043

As shown in Table 1, both of the confidence intervals and the assessment criterion value demonstrate the superiority of the proposed method on centroid location precision. Considering the influence of the threshold selection, we recalculated the centroid with the threshold variance from 11 to 20. For each threshold, we also took 20 tests for statistic. The distribution of detected centroids with threshold variance is shown in Fig. 5.

Fig. 5 Distribution of the detected spot centroids with threshold variance from 11 to 20.

In Fig. 5, the proposed method also shows a more compact distribution than the reference method, which demonstrates that our method has superior stability. The detailed result is as shown in Table 2.

Table 2 The result of centroid location with threshold variance for one certain spot.

Threshold Proposed Method Reference Method

Centroid (α= 0.05) Mξ Centroid (α= 0.05) Mξ

11 (7.9296±0.0613,7.9495±0.0771) 0.1727 (7.7134±0.0825,8.1669±0.0997) 0.5043

12 (8.1044±0.0196,7.7405±0.0217) 0.2183 (7.7723±0.0508,7.9424±0.0571) 0.3671

13 (8.1044±0.0196,7.7405±0.0217) 0.2183 (7.7723±0.0508,7.9424±0.0571) 0.3671

14 (8.0691±0.0167,7.7239±0.0201) 0.1368 (7.8365±0.0531,7.8558±0.0493) 0.3475

15 (8.0691±0.0167,7.7239±0.0201) 0.1368 (7.8365±0.0531,7.8558±0.0493) 0.3475

16 (7.9929±0.0158,7.6901±0.0219) 0.2330 (7.9276±0.0312,7.7052±0.0364) 0.3050

17 (7.9929±0.0158,7.6901±0.0219) 0.2330 (7.9276±0.0312,7.7052±0.0364) 0.3050

18 (7.9957±0.0221,7.5983±0.0218) 0.3835 (8.0092±0.0284,7.63113±0.0238) 0.4005

19 (7.9957±0.0221,7.5983±0.0218) 0.3835 (8.0092±0.0284,7.63113±0.0238) 0.4005 20 (8.0009±0.0363,7.5025±0.0189) 0.4482 (7.9960±0.0365,7.5564±0.0464) 0.5064

As shown in Table 2, the confidence intervals and assessment criterion values of the proposed method are smaller than those of the reference method, which also indicates a validation of the superior stability of the former method.

For further validation of repeatability, we calculated the centroid positions for different laser spots. The laser spots shot on the target board and their detected centroid positions by both of the proposed and reference methods are shown in Fig. 6.


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922±0.0334)

495±0.0771)

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References

[1] Z. Jiang, S. Gong, and Y. Dai, “Numerical study of centroid detection accuracy for Shack-Hartmann wavefront sensor,” Optics & Laser Technology, vol. 38, 614-619, 2006.

[2] M. Nicolle, T. Fusco, G. Rousset, and V. Michau, “Improvement of Shack–Hartmann wave-front sensor measurement for extreme adaptive optics,” OPTICS LETTERS, vol. 29, no. 23, pp. 2743-2745, 2004.

[3] J. E. Greivenkamp, D. G. Smith, R. O. Gappinger, and G. A. Williby, “Optical Testing Using Shack-Hartmann Wavefront Sensors,” Proceedings of SPIE, vol. 4416, pp. 260-263, 2001.

[4] J. Schwiegerling, and E. DeHoog, “Problems testing diffractive intraocular lenses with Shack-Hartmann sensors,” Applied Optics, vol. 49, no. 16, pp. D62-D68, 2010/06/01, 2010.

[5] H. Chen, and C. Rao, “Accuracy analysis on centroid estimation algorithm limited by photon noise for point object,” Optics Communications, vol. 282, pp. 5, 2009.

[6] W. Guo, L. Zhao, C. S. Tong, C. I-Ming, and S. C. Joshi, “Adaptive centroid-finding algorithm for freeform surface measurements,” APPLIED OPTICS, vol. 52, no. 10, pp. D75-D83, 2013.

[7] D. W. Kang, M. Kang, and J. W. Hahn, “Measuring two-dimensional profiles of beam spots in a high-density spot array for a maskless lithography system,” APPLIED OPTICS, vol. 53, no. 36, pp. 8507-8513, 2014.

[8] X. Yu, D. Zhao, and L. Chen, “Adaptation of adaptive optics system.,” Proceedings of SPIE, vol. 3126, pp. 432, 1997.

[9] S.-H. Baik, S.-K. Park, C.-J. Kim, Y.-S. Seo, and Y.-J. Kang, “A new centroid algorithm for shack-hartmann wavefront sensor,” Proceedings of SPIE, vol. 4926, pp. 251-260, 2002.

[10] S.-K. Park, and S.-H. Baik, “A study on a fast measuring technique of wavefront using a Shack–Hartmann sensor,” Optics & Laser Technology, vol. 34, no 8, pp. 687-694, 2002.

[11] J. F. Ren, C. H. Rao, and Q. M. Li, “An Adaptive Threshold Selection Method for Hartmann-Shack Wavefront Sensor,” Opto-electronic Engineering, vol. 29, no. 1, 1-5, 2002.

[12] H. Li, H. Song, C. Rao, and X. Rao, “Accuracy analysis of centroid calculated by a modified center detection algorithm for Shack–Hartmann wavefront sensor,” Optics Communications, vol. 281, pp. 750-755, 2008.

[13] R. C. Gonzalez, and R. E. Woods, Digital image processing, p.^pp. 520-525, Beijing: Publishing House of Electronics Industry, 2007.


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