The Hotelling—Like $T^2$ Control Chart Modified for Detecting Changes in Images having the Matrix Normal Distribution

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.