The asymptotic rate of the length of the longest significant chain with good continuation in Bernoulli net and its applications in filamentary detection

Abstract

This thesis is devoted to the detectability of an inhomogeneous region possibly embedded in a noisy environment. It presents models and algorithms using the theory of the longest significant run and percolation. We analyze the computational results based on simulation. We consider the length of the significant nodes in a chain with good continuation in a square lattice of independent nodes. Inspired by the percolation theory, we first analyze the problem in a tree based model. We give the critical probability and find the decay rate of the probability of having a significant run with length k starting at the origin. We find that the asymptotic rate of the length of the significant run can be powerfully applied in the area of image detection. Examples are detection of filamentary structures in a background of uniform random points and target tracking problems. We set the threshold for the rejection region in these problems so that the false positives diminish quickly as we have more samples. Inspired by the convex set detection, we also give a fast and near optimal algorithm to detect a possibly inhomogeneous chain with good continuation in an image of pixels with white noise. We analyze the length of the longest significant chain after thresholding each pixel and consider the statistics over all significant chains. Such a strategy significantly reduces the complexity of the algorithm. The false positives are eliminated as the number of pixels increases. This extends the existing detection method related to the detection of inhomogeneous line segment in the literature.