Abstract
Our aim is to propose a control chart for detecting changes in an image sequence. The starting point is the well-known Hotelling \(T^2\) chart for changes in the mean of multivariate Gaussian distributions. However, this chart requires to know (or to be able to estimate from historical data) at least the in-control covariance matrix. Unfortunately, even if very small images, e.g., \(100\times 100\) pixels are vectorized, the covariance matrix is of the size \(10^4\times 10^4\) and its estimation would require \(O(10^8)\) sample images. As a remedy, we propose considering a narrower class of multivariate Gaussian distributions, namely, the so-called matrix normal distributions (MND). The MND class of distributions allows only for interrow and for intercolumn correlations, assuming other correlations to be negligible. This correlation model seems to be adequate for many image sequences, including industrial processes. In this paper we display how the Hotelling \(T^2\) chart looks like, when specialized to the MND. We also invoke known facts about estimating the interrow and the intercolumn covariance matrices. Then, we discuss how to select the threshold of such a chart, putting an emphasis on the case when a(-n) alternative(-s) to in-control behavior is (are) known. This approach has many common features with classifying images in the empirical Bayesian sense, since alternatives are known (see [5, 7]). Extensions to a localized approach are studied, where the images are decomposed in blocks for which the MND distribution is assumed, and the maximum of the Hotelling statistics is then used. It is discussed how to select an appropriate threshold in this setting. We also provide an example of the laser cladding process (3-D printing using metallic powders), monitored by a camera.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arnold, S.F.: The Theory of Linear Models and Multivariate Analysis. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1981)
Assent, I.: Clustering High Dimensional Data, Wiley Interdisciplinary Reviews. Data Min. Knowl. Discov. 2(4), 340–350 (2012)
Birgé, L.: An alternative point of view on Lepski’s method. In: State of the Art in Probability and Statistics Leiden, 1999, Inst. Math. Statist., Beachwood, OH), pp. 113–133 (2001)
Dawid, A.P.: Some matrix-variate distribution theory: notational considerations and a Bayesian application. Biometrika 68(1), 265–274 (1981)
Devroye, L., Gyorfi, L., Lugosi, G.: A Probabilistic Theory of Pattern Recognition. Springer, Berlin (2013)
Duchesne, C., Liu, J.J., MacGregor, J.F.: Multivariate image analysis in the process industries: a review. Chemom. Intell. Lab. Syst. 117, 116–128 (2012)
Fukunaga, K.: Introduction to Statistical Pattern Recognition. Academic, New York (2013)
Glanz, H., Carvalho, L.: An expectation-maximization algorithm for the matrix normal distribution with an application in remote sensing. J. Multivar. Anal. 167, 31–48 (2018)
Jurewicz P., Rafajlowicz W., Reiner J., Rafajlowicz E.: Simulations for Tuning a laser power control system of the cladding process. In: IFIP International Conference on Computer Information Systems and Industrial Management, pp. 218–229. Springer (2016)
Krzyśko M., Skorzybut M., Wolynski W.: Classifiers for doubly multivariate data. Discussiones Mathematicae: Probability & Statistics, pp. 31 (2011)
Krzyéko, M., Skorzybut, M.: Discriminant analysis of multivariate repeated measures data with a Kronecker product structured covariance matrices. Stat. Pap. 50(4), 817–835 (2009)
Manceur, A.M., Dutilleul, P.: Maximum likelihood estimation for the tensor normal distribution: algorithm, minimum sample size, and empirical bias and dispersion. J. Comput. Appl. Math. 239, 37–49 (2013)
Megahed, F.M., Woodall, W.H., Camelio, J.A.: A review and perspective on control charting with image data. J. Q. Technol. 43(2), 83–98 (2011)
Ohlson, M., Ahmad, M.R., Von Rosen, D.: The multi-linear normal distribution: introduction and some basic properties. J. Multivar. Anal. 113, 37–47 (2013)
Pepelyshev, A., Sovetkin, E., Steland, A.: Panel-based stratified cluster sampling and analysis for photovoltaic outdoor measurements. Appl. Stoch. Models Bus. Ind. 33(1), 35–53 (2017)
Prause, A., Steland, A.: Detecting changes in spatial-temporal image data based on quadratic forms. Stochastic Models, Statistics and Their Applications, pp. 139-147. Springer, Cham (2015)
Prause, A., Steland, A.: Sequential detection of three-dimensional signals under dependent noise. Seq. Anal. 36(2), 151–178 (2017)
Prause, A., Steland, A.: Estimation of the asymptotic variance of univariate and multivariate random fields and statistical inference. Electron. J. Stat. 12(1), 890–940 (2018)
Rafajlowicz, E.: Detection of essential changes in spatio-temporal processes with applications to camera based quality control. Stochastic Models, Statistics and Their Applications, pp. 433–440. Springer, Berlin (2015)
Rafajłowicz, E., Rafajłowicz, W.: Iterative learning in optimal control of linear dynamic processes. Int. J. Control 91(7), 1522–1540 (2018)
Rafajłowicz, E.: Data structures for pattern and image recognition with application to quality control Acta Polytechnica Hungarica. Informatics 15(4), 233–262 (2018)
Rafajłowicz, E.: Classifiers for matrix normal images: derivation and testing. International Conference on Artificial Intelligence and Soft Computing, pp. 668-679. Springer, Cham (2018)
Rafajłowicz, W. et al.: Iterative learning of optimal control for nonlinear processes with applications to laser additive manufacturing. IEEE Trans. Control Syst. Technol. 99 (2018)
Skubalska-Rafajłowicz, E.: Sparse random projections of camera images for monitoring of a combustion process in a gas burner. In: IFIP International Conference on Computer Information Systems and Industrial Management, pp. 447–456 Springer (2017)
Skubalska-Rafajłowicz, E.: Random projections and Hotelling’s T2 statistics for change detection in high-dimensional data streams Int. J. Appl. Math. Comput. Sci. 23(2), 447–461 (2013)
Skubalska-Rafajłowicz E.: A change detection in high dimensions using random projection - simulation study. In: 7-th International Workshop on Simulation 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini University of Bologna, Italy, Quaderni di Dipartimento Serie Ricerche no 3, ISSN 1973-9346 (2013)
Sovetkin, E., Steland, A.: Automatic processing and solar cell detection in photovoltaic electroluminescence images. Integrated Computer-Aided Engineering, (Preprint), pp. 1–15 (2018)
Steland, A., von Sachs, R.: Asymptotics for high-dimensional covariance matrices and quadratic forms with applications to the trace functional and shrinkage. Stoch. Process. Appl. 128(8), 2816–2855 (2018)
Steland, A.: Vertically weighted averages in Hilbert spaces and applications to imaging: fixed sample asymptotics and efficient sequential two-stage estimation. Seq. Anal. 34(3), 295–323 (2015)
Steland, A., von Sachs, R.: Large-sample approximations for variance-covariance matrices of high-dimensional time series. Bernouli 23, 2299–2329 (2017)
Tomasi C., Manduchi, R.: Bilateral Filtering for gray and color images. In: Proceedings of the 1998 IEEE International Conference on Computer Vision, Bombay, India (1998)
Werner, K., Jansson, M., Stoica, P.: On estimation of covariance matrices with Kronecker product structure. IEEE Trans. Signal Process. 56(2), 478–491 (2008)
Acknowledgements
The first author expresses his thanks to Professor J. Reiner and to MSc. P. Jurewicz for common research of the laser cladding process control. A small part of the images taken during this research are used in this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Rafajłowicz, E., Steland, A. (2019). The Hotelling—Like \(T^2\) Control Chart Modified for Detecting Changes in Images having the Matrix Normal Distribution. In: Steland, A., Rafajłowicz, E., Okhrin, O. (eds) Stochastic Models, Statistics and Their Applications. SMSA 2019. Springer Proceedings in Mathematics & Statistics, vol 294. Springer, Cham. https://doi.org/10.1007/978-3-030-28665-1_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-28665-1_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-28664-4
Online ISBN: 978-3-030-28665-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)