在近代工業製程中，為了增進製程的效率、產品品質、製程安全等原因，基於數據的統計方法逐漸成為了工業製程中的一項重要技術。由於現代製程多是龐大且複雜的系統，製程中經常會有很多突發狀況導致數據受到很大程度的干擾，例如數據中存在的離群點(outliers)，使得傳統中廣泛受到使用的多變數線性分析方法-主成分分析（Principal Component Analysis，PCA）並不能得到很好的監控效果。因此本研究中嘗試使用一種名為「穩定主成分追蹤 (Stable Principal Component Pursuit，SPCP)」的方法來做為新的數據統計方法。除了嘗試取代PCA應用於實際製程的監控中期望能獲得更穩健的結果之外，本研究中更進一步將穩定主成分追蹤方法應用於非破壞性檢測之領域，針對現今廣泛應用於各大產業中的碳纖維補強之高分子複合材料(Carbon Fiber Reinforced Polymer，CFRP) 進行缺陷檢測。由於該材料成本昂貴與製程複雜，產品須結合非破壞性檢測 (Non-Destructive Testing，NDT)，以確保品質穩定。脈衝式熱成像法(Pulsed Thermography，PT)以快速獲取材料檢測結果廣受青睞，但往往因快速加熱所形成之不均背景與圖像雜訊，導致缺陷訊號被掩蓋，檢測效果不彰，因此本研究中嘗試使用穩定主成分追蹤方法來將圖像中各部分的訊號分離並且檢測出缺陷區域。

In process industries, data-based statistical methods have become important techniques for ensuring product quality and operation safety, where principal component analysis (PCA) may be the most commonly used method among them. However, PCA assumes that the training data matrix only contains an underlying low-rank structure with some dense noise. When gross sparse errors, i.e. outliers, exist, the results of PCA are seriously affected. In this thesis, a robust matrix recovery method called stable principal component pursuit (SPCP) is utilized to solve this problem. By replacing PCA by SPCP in process modeling and monitoring, robust process monitoring is achieved. In addition, SPCP is also applied to non-destructive testing (NDT) of carbon fiber reinforced polymer (CFRP) for eliminating noise and non-uniform backgrounds contained in thermographic images. In doing so, the performance of NDT is enhanced and the defects can be better detected.

1. Jolliffe, I., Principal component analysis. 2002: John Wiley & Sons, Ltd. Hoboken, New Jersey.
2. Yao, Y., T. Chen, and F. Gao, Multivariate statistical monitoring of two-dimensional dynamic batch processes utilizing non-Gaussian information. Journal of Process Control, 2010. 20(10): p. 1188-1197.
3. Nomikos, P. and J.F. MacGregor, Multivariate SPC charts for monitoring batch processes. Technometrics, 1995. 37(1): p. 41-59.
4. Golub, G. and W. Kahan, Calculating the singular values and pseudo-inverse of a matrix. Journal of the Society for Industrial & Applied Mathematics, Series B: Numerical Analysis, 1965. 2(2): p. 205-224.
5. Zhou, Z., X., Li, J., Wright, E., Candes, and Y., Ma, Stable principal component pursuit. in: IEEE ISIT Proceedings, June 2010 (pp. 1518-1522).
6. Stanimirova, I., M. Daszykowski, and B. Walczak, Dealing with missing values and outliers in principal component analysis. Talanta, 2007. 72(1): p. 172-178.
7. Fischler, M.A. and R.C. Bolles, Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM, 1981. 24(6): p. 381-395.
8. Gnanadesikan, R. and J.R. Kettenring, Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics, 1972: p. 81-124.
9. De La Torre, F. and M.J. Black, A framework for robust subspace learning. International Journal of Computer Vision, 2003. 54(1-3): p. 117-142.
10. Huber, P.J., Robust statistics. 2011: Springer-Verlag Berlin Heidelberg..
11. Ke, Q. and T. Kanade. Robust L 1 norm factorization in the presence of outliers and missing data by alternative convex programming. in Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on Computer Vision and Pattern Recognition. Vol.1 p.739-746, San Diego, CA.
12. Candès, E.J., X.D., Li, Y. Ma and John Wright, Robust principal component analysis? Journal of the ACM (JACM), 2011. 58(3).
13. Ge, Z., Z. Song, and F. Gao, Review of recent research on data-based process monitoring. Industrial & Engineering Chemistry Research, 2013. 52(10): p. 3543-3562.
14. Zhu, J., Z. Ge, and Z. Song, Robust modeling of mixture probabilistic principal component analysis and process monitoring application. AIChE Journal, 2014. 60(6): p. 2143-2157.
15. Wold, S., P., Geladi, K., Esbensen, and J., Öhman, Multi‐way principal components‐and PLS‐analysis. Journal of chemometrics, 1987. 1(1): p. 41-56.
16. Ku, W., R.H. Storer, and C. Georgakis, Disturbance detection and isolation by dynamic principal component analysis. Chemometrics and intelligent laboratory systems, 1995. 30(1): p. 179-196.
17. Lee, J.-M., C.-K., Yoo, S.-W., Choi, Vanrolleghem, Peter A and Lee, I.-B., Nonlinear process monitoring using kernel principal component analysis. Chemical Engineering Science, 2004. 59(1): p. 223-234.
18. Cai, J.-F., E.J. Candes, and Z. Shen, A Singular Value Thresholding Algorithm for Matrix Completion. Siam Journal on Optimization, 2010. 20(4): p. 1956-1982.
19. Cai, J.-F. and S. Osher, Fast singular value thresholding without singular value decomposition. UCLA CAM Report, 2010. 5.
20. Ganesh, A., Z.-C., Lin, J., Wright, L.-Q., Wu, M.-M. Chen, and Y., Ma, Fast algorithms for recovering a corrupted low-rank matrix. in Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2009 3rd IEEE International Workshop on. 2009. IEEE.
21. Rousseeuw, P.J. and C. Croux, Alternatives to the median absolute deviation. Journal of the American Statistical Association, 1993. 88(424): p. 1273-1283.
22. Montgomery, D.C., G.C. Runger, and N.F. Hubele, Engineering statistics. 2009: John Wiley & Sons. Hoboken, New Jersey.
23. Nomikos, P. and J. MacGregor, Multivariate SPC charts for monitoring batch processes. Technometrics, 1995. 37: p. 41-59.
24. Downs, J.J. and E.F. Vogel, A plant-wide industrial process control problem. Computers & Chemical Engineering, 1993. 17(3): p. 245-255.
25. Yin, S., S. X., Ding, A., Haghani, H.-Y., Hao, and P., Zhang, A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process. Journal of Process Control, 2012. 22(9): p. 1567-1581.
26. Raišutis, R., E., Jasiūnienė, R., Šliteris, and A., Vladišauskas, The review of non-destructive testing techniques suitable for inspection of the wind turbine blades. Ultragarsas (ultrasound), 2008. 63(1): p. 26-30.
27. Lau, S., D. Almond, and J. Milne, A quantitative analysis of pulsed video thermography. NDT & E International, 1991. 24(4): p. 195-202.
28. Maillet, D., A. S., Houlbert, S., Didierjean, A. S., Lamine, and A., Degiovanni, Non-destructive thermal evaluation of delaminations in a laminate: Part I—Identification by measurement of thermal contrast. Composites science and technology, 1993. 47(2): p. 137-153.
29. Shepard, S.M. Advances in pulsed thermography. in Aerospace/Defense Sensing, Simulation, and Controls. 2001. International Society for Optics and Photonics. p.511-515.
30. Busse, G., D. Wu, and W. Karpen, Thermal wave imaging with phase sensitive modulated thermography. Journal of Applied Physics, 1992. 71(8): p. 3962-3965.
31. Liu, J., Y., Wang, H., Liu, and Y., He, Research on CFRP materials nondestructive testing by IR lock-in thermography. in International Symposium on Photoelectronic Detection and Imaging 2009. (pp. 73833U-73833U). International Society for Optics and Photonics.
32. Pickering, S. and D. Almond, Matched excitation energy comparison of the pulse and lock-in thermography NDE techniques. NDT & E International, 2008. 41(7): p. 501-509.
33. Maldague, X. and S. Marinetti, Pulse phase infrared thermography. Journal of Applied Physics, 1996. 79(5): p. 2694-2698.
34. Vavilov, V., X., Maldague, B., Dufort, and F., Robitaille, Thermal nondestructive testing of carbon epoxy composites: detailed analysis and data processing. NDT & E International, 1993. 26(2): p. 85-95.
35. Pilla, M., M., Klein, X., Maldague, and A., Salerno, New absolute contrast for pulsed thermography. In Proceedings of QIRT (Vol. 2, pp. 53-58). 2002.
36. Shepard, S.M., System for generating thermographic images using thermographic signal reconstruction. 2004, U.S. Patent No. 6,751,342. 15 Jun. 2004.
37. Shepard, S. M., J. R., Lhota, B. A., Rubadeux, T., Ahmed, and D., Wang, Enhancement and reconstruction of thermographic NDT data. in AeroSense 2002. International Society for Optics and Photonics. p. 531-535.
38. Bracewell, R., The Fourier Transform and IIS Applications. New York, 1965.
39. Theodorakeas, P., N. P., Avdelidis, C., Ibarra-Castanedo, M., Koui, and X., Maldague, Pulsed thermographic assessment of CFRP structures: experimental results and image analysis tools. in SPIE Smart Structures and Materials+ Nondestructive Evaluation and Health Monitoring. 2014. International Society for Optics and Photonics.
40. Rajic, Nikolas. Principal component thermography. No. DSTO-TR-1298. Defence Science and Technology Organisation Victoria (Australia) Aeronautical and Maritime Research Lab, 2002.
41. Wei, B.-J., Y.-S., Chang, Y., Yao, and J., Fang, Online estimation and monitoring of local permeability in resin transfer molding. Polymer Composites, 2014. DOI:10.1117/12.421032.