Statistics & Analytical Sciences
Given a convex-shape inhomogeneous region embedded in a noisy image, we consider the conditions under which such an embedded region is detectable. The existence of low order-of-complexity detection algorithms is also studied. The main results are (1) an analytical threshold (of a statistic) that specifies what is detectable, and (2) the existence of a multiscale detection algorithm whose order of complexity is roughly the optimal O(n(2) log(2) (n)).
Our analysis has two main components. We first show that in a discrete image, the number of convex sets increases faster than any finite degree polynomial of the image size n. Hence the idea of generalized likelihood ratio test cannot be directly adopted to derive the asymptotic detectability bound. Secondly, we show that the maximally embedded hv-parallelogram is at least 2/9 of the convex region (in area). We then apply the results of hv-parallelograms in Arias-Castro, Donoho, and Huo (2005) on detecting convex sets. Numerical examples are provided.
Our results have potential applications in several fields, which are described with corresponding references.
Huo, X., & Ni, X. (2009). Detectability of convex-shaped objects in digital images, its fundamental limit and multiscale analysis. Statistica Sinica, 19(4), 1439-1462.