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Given an inhomogeneous chain embedded in a noisy image, we consider theconditions under which such an embedded chain is detectable. Many applications,such as detecting moving objects, detecting ship wakes, can be abstracted asthe detection on the existence of chains. In this work, we provide thedetection algorithm with low order of computation complexity to detect thechain and the optimal theoretical detectability regarding SNR (signal to noiseratio) under the normal distribution model. Specifically, we derive ananalytical threshold that specifies what is detectable. We design a longestsignificant chain detection algorithm, with computation complexity in the orderof $O(n\log n)$. We also prove that our proposed algorithm is asymptoticallypowerful, which means, as the dimension $n \rightarrow \infty$, the probabilityof false detection vanishes. We further provide some simulated examples and areal data example, which validate our theory.

Fast and Asymptotically Powerful Detection for Filamentary Objects in Digital Images