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