Detecting the emergence of a signal in a noisy image

We study sequential change-point detection when obser- vations form a sequence of independent Gaussian random fields, and the change-point is the time at which a sig- nal of known functional form involving a finite number of unknown parameters appears. Building on Siegmund and Yakir (2008), which identifies in a simpler problem a detec- tion procedure of Shiryayev-Roberts type that is asymptot- ically minimax up to terms that vanish as the false detec- tion rate converges to zero, we compare easily computed ap- proximations to the Shiryayev-Roberts detection procedure with similar approximations to CUSUM type procedures. Although the CUSUM type procedures are suboptimal, our studies indicate that they compare favorably to the asymp- totically optimal procedures. gradually. Generally speaking, the problem is to detect the signal as soon as possible after it appears, under the con- straint that (false) detection occurs very rarely if no signal appears. To illustrate the principles involved we will assume that the observations consist of a sequence of uncorrelated Gaussian random fields and that the signal is a parameter- ized function of a known form superimposed on the noisy observations at an unknown change-point. The discussion will be formulated in the context of image analysis. Yet the results and the principles that we introduce are meaningful in other settings as well. Following the developments in the companion to this paper (12), we measure detection delay by the expected Kullback-Leibler information accumulated between the change-point and its detection. We begin in the next sub- section with a precise description of the model. In Subsec- tion 1.2 the criterion for asymptotic minimax optimality, which is stated and proved in the companion paper, is re- formulated to fit the current context. The asymptotic minimax policy uses a randomized form of the Shiryayev-Roberts monitoring scheme. The alterna- tive Cumulative Sum (CUSUM) monitoring scheme is a bet- ter known approach. In Section 2 expressions describing the asymptotic performance of optimal CUSUM and optimal Shiryayev-Roberts rules are obtained. Suboptimal formula- tions of these procedures are also assessed. It turns out that the natural candidate for optimal Shiryayev-Roberts and CUSUM rules may require substan- tial computation. In Section 3 we propose alternative rules which are asymptotically equivalent but require less com- putational effort. A simulation study is conducted in order to investigate the finite-sample properties of these simplified rules. The paper concludes with a discussion of related open problems. The analysis of the different detection methods draws on a substantial literature for its justification. The calculations are only sketched here and emphasize new features of the present formulation. See (12) and the references cited there, and (3).