Statistical modeling and monitoring of image data in the presence of temporal and spatial correlations

Document Type : Article


1 Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

2 Assistant Professor, Iranian Research Institute for Information Science and Technology (IranDoc)


In this paper, the regression-based approach was developed to monitor image data under two-scale analysis. In the first scale, wavelet transformation was used to extract the main features of geometric profile created from the images. The next scale was to monitor the small-scale components which could be expressed by correlation in error terms. To monitor correlation in error terms, one parametric and one non-parametric methods were developed. Parameters of the parametric model including spatial correlation coefficient and error term variance were estimated using Ordinary Least Squares (OLS) and Generalized Least Squares (GLS) estimators, respectively. In non-parametric method, no assumption was made about the structure of correlation in error terms. To extract useful information about the nature of correlation in error terms, Functional Principal Component Analysis (FPCA) was used. After extracting features for both scales, some appropriate test statistics were computed. Then, monitoring the process was performed by plotting these test statistics on corresponding control charts. Simulation and industrial case studies were also performed to evaluate the proposed method’s performance in detecting different shifts. The results indicated the proper performance of the proposed method in monitoring industrial processes to detect out-of-control conditions and identify the source of variability.


1. Wang, K. and Tsung, F. "Using profile monitoring techniques for a data-rich environment with huge sample size", Quality and reliability engineering international, 21(7):677–688 (2005).
2. Wells, L. J., Megahed, F. M., Niziolek, C. B., et al. "Statistical process monitoring approach for high density point clouds", Journal of Intelligent Manufacturing, 24(6):1267–1279 (2013).
3. Menafoglio, A., Grasso, M., Secchi, P., et al. "Profile monitoring of probability density functions via simplicial functional pca with application to image data", Technometrics, 60(4):497–510 (2018).
4. Eslami, D., Izadbakhsh, H., Ahmadi, O., et al. "Spatial-nonparametric regression: an approach for monitoring image data", Communications in Statistics-Theory and Methods, pages 1–24 (2021a).
5. Megahed, F. M., Wells, L. J., Camelio, J. A., et al. "A spatiotemporal method for the monitoring of image data", Quality and Reliability Engineering International, 28(8):967–980 (2012).
6. He, Z., Zuo, L., Zhang, M., et al. "An image-based multivariate generalized likelihood ratio control chart for detecting and diagnosing multiple faults in manufactured products", International Journal of Production Research, 54(6):1771–1784 (2016).
7. Koosha, M., Noorossana, R., and Megahed, F. "Statistical process monitoring via image data using wavelets", Quality and Reliability Engineering International, 33(8):2059–2073 (2017).
8. Zuo, L., He, Z., and Zhang, M. "An ewma and region growing based control chart for monitoring image data", Quality Technology & Quantitative Management, 17(4):470–485 (2020).
9. Huang, T., Wang, S., Yang, S., et al. "Statistical process monitoring in a specified period for the image data of fused deposition modeling parts with consistent layers", Journal of Intelligent Manufacturing, pages 1–16 (2020).
10. Amirkhani, F. and Amiri, A. "A novel framework for spatiotemporal monitoring and post-signal diagnosis of processes with image data", Quality and Reliability Engineering International, 36(2):705–735 (2020).
11. Eslami, D., Izadbakhsh, H., Ahmadi, O., et al. "Statistical monitoring of image data using multi-channel functional principal component analysis", Communications in Statistics-Theory and Methods, pages 1–18 (2021b).
12. Noorossana, R. and Nikoo, M. "Phase ii monitoring of geometric profiles", Communications in Statistics-Simulation and Computation, 44(4):1036–1049 (2015).
13. Čížek, P. and Sadıkoğlu, S. "Robust nonparametric regression: A review", Wiley Interdisciplinary Reviews: Computational Statistics, 12(3):e1492 (2020).
14. Nikoo, M. and Noorossana, R. "Phase ii monitoring of nonlinear profile variance using wavelet", Quality and Reliability Engineering International, 29(7):1081–1089 (2013).
15. Cohen, A. and Atoui, M. A. "On wavelet-based statistical process monitoring", Transactions of the Institute of Measurement and Control, 44(3):525-538 (2020).
16. Lim, M. and Bae, S. J. "Spatial monitoring of wafer map defect data based on 2d wavelet spectrum analysis", Applied Sciences, 9(24):5518 (2019).
17. Noorossana, R., Saghaei, A., and Amiri, A. "Statistical analysis of profile monitoring", John Wiley & Sons, volume 865 (2011).
18. Cheng, T. C. and Yang, S. F. "Monitoring profile based on a linear regression model with correlated errors", Quality Technology & Quantitative Management, 15(3):393–412 (2018).
19. Colosimo, B. M. "Modeling and monitoring methods for spatial and image data", Quality Engineering, 30(1):94–111 (2018).
20. Colosimo, B. M., Mammarella, F., and Petro, S. "Quality control of manufactured surfaces", In Frontiers in Statistical Quality Control, pages 55–70. Springer (2010).
21. Paynabar, K. and Jin, J. "Characterization of non-linear profiles variations using mixed-effect models and wavelets", Iie transactions, 43(4):275–290 (2011).
22. Vidakovic, B. "Statistical modeling by wavelets", John Wiley & Sons, volume 503 (2009).
23. Daubechies, I. "Ten lectures on wavelets", CBMS-NSF series, vol 61 (philadelphia: Siam) (1992).
24. Deregowski, K. and Krzysko, M. "Principal components analysis for functional data", In Colloquium Biometricum, volume 41 (2011).
25. Soleimani, P., Noorossana, R., and Amiri, A. "Simple linear profiles monitoring in the presence of within profile autocorrelation", Computers & Industrial Engineering, 57(3):1015–1021 (2009).
26. Littell, R. C., Milliken, G. A., Stroup, W. W., et al. "Sas system for mixed models" (1996).
27. LeSage, J. and Pace, R. "Introduction to spatial econometrics", Chapman & hall/crc (2009).
28. Qiu, P. "Image processing and jump regression analysis", John Wiley & Sons, volume 599 (2005).
29. Marques, O. "Practical image and video processing using MATLAB", John Wiley & Sons (2011).
30. Wei, G., Zhou, Y., Gao, X., et al. "Field-aligned quadrangulation for image vectorization", In Computer Graphics Forum, volume 38, pages 171–180. Wiley Online Library (2019).
31. Reynolds Jr, M. R. and Lou, J. "An evaluation of a glr control chart for monitoring the process mean", Journal of quality technology, 42(3):287–310 (2010).
32. Anselin, L. "Spatial econometrics: methods and models", Springer Science & Business Media, volume 4 (2013).
33. Greene, W. H. "The econometric approach to efficiency analysis", The measurement of productive efficiency and productivity growth, 1(1):92–250 (2008).
34. Zhang, J., Zou, C., and Wang, Z. "A control chart based on likelihood ratio test for monitoring process mean and variability", Quality and Reliability Engineering International, 26(1):63–73 (2010).
35. Ren, H., Chen, N., andWang, Z. "Phase-ii monitoring in multichannel profile observations", Journal of Quality Technology, 51(4):338–352 (2019).