Parametric Probability Distributions for Anomalous Change Detection

The problem of anomalous change detection arises when two (or possibly more) images are taken of the same scene, but at different times. The aim is to discount the “pervasive differences” that occur throughout the imagery, due to the inevitably different conditions under which the images were taken (caused, for instance, by differences in illumination, atmospheric conditions, sensor calibration, or misregistration), and to focus instead on the “anomalous changes” that actually take place in the scene. In general, anomalous change detection algorithms attempt to model these normal or pervasive differences, based on data taken directly from the imagery, and then identify as anomalous those pixels for which the model does not hold. For many algorithms, these models are expressed in terms of probability distributions, and there is a class of such algorithms that assume the distributions are Gaussian. By considering a broader class of distributions, however, a new class of anomalous change detection algorithms can be developed. We consider several parametric families of such distributions, derive the associated change detection algorithms, and compare the performance with standard algorithms that are based on Gaussian distributions. We find that it is often possible to significantly outperform these standard algorithms, even using relatively simple non-Gaussian models. “Just because everything is different doesn’t mean anything has changed.” –Irene Peter 1.0 Introduction Given two or more images of the same scene, taken at different times and under different conditions, what anomalous change detection (ACD) seeks is the “interesting” changes that occurred in the scene. Unfortunately, a mathematics of “interesting” has not been developed1 so our approach is to identify the rare, or anomalous changes. The idea is to distinguish them from the pervasive differences that occur throughout the scene (e.g., see Fig. 1) due to disparities in illumination, calibration, registration, look angle, or even the choice of remote sensing platform. They can also be due to diurnal and seasonal variations [2] in the scene. Part of the motivation for this is the intuition that interesting changes are anomalous. But even when that intuition fails – after all, “anomalous” is not synonymous with “interesting” – then: 1/ since anomalous changes are rare, one will not at least be overwhelmed by uninteresting anomalous changes; and 2/ if pervasive differences are in fact interesting, they will be large enough or plentiful enough that the the analyst can readily find them without the aid of the change detection algorithm. Anomalous changes are assumed to be relatively rare, and occur in only a small part of the image or image archive. Because the nature of the change is not known beforehand, algorithms for anomalous change detection are unsupervised. If the nature of specific changes of interest were already known (and if an adequate and representative sample of those changes were available in the data), then supervised classification might be employed to identify and delineate those changes. 2.0 Probability distributions In this section, we will derive ACD algorithms in terms of probability distributions that characterize both pervasive differences and anomalous changes. An explicit model for anomalous changes seems to defy the meaning of “anomaly” – it is what Rumsfeld would call an unknown unknown [3] – but a number of existing algorithms for anomaly detection and anomalous change detection have effectively employed such a model, even if it was not explicitly stated as part of the model. The use of probability distributions opens up a number of options. The most pragmatic option is to pretend these distributions are Gaussian. This leads to simple closed-form solutions (and in some cases to well established algorithms), and requires only that covariance matrices be estimated. The “purist” option is to make no assumptions about the distribution at all. Following Vapnik’s dictum [4], we would never model the distribution directly, but instead model the boundary that optimally separates the distribution of anomalous changes from the distribution of pervasive differences, and base this model only on the data that are available. As described in Ref. [5] and illustrated in Fig. 2, samples from the pervasive-differences class are given by the data, while samples from the anomalous-changes class are given by resampling either from the data or from a uniform distribution. This approach does have some theoretical advantages, but can also be expensive and is sometimes problematic on the tails. We will take a middle ground, and model the data with a non-Gaussian distribution that can be described by a relatively modest number of parameters. Once we fit these parameters to the data, it is straightforward to plug them into our expressions that involve arbitrary distributions and produce anomalous change detectors. To the extent that these parametric distributions are better descriptors of the observed data, we expect that the resulting algorithms will better detect anomalous changes. In particular, since (detectable) anomalous behavior occurs on the tails of distributions, it will be useful to model data with distributions that better describe the tails. Indeed, some would say that “interesting mathematics” is an oxymoron.