  In image detection, one problem is to test whether the set, though mostly
consisting of uniformly scattered points, also contains a small fraction of
points sampled from some (a priori unknown) curve, for example, a curve with
$C^{\alpha}$-norm bounded by $\beta$. One approach is to analyze the data by
counting membership in multiscale multianisotropic strips, which involves an
algorithm that delves into the length of the path connecting many consecutive
"significant" nodes. In this paper, we develop the mathematical formalism of
this algorithm and analyze the statistical property of the length of the
longest significant run. The rate of convergence is derived. Using percolation
theory and random graph theory, we present a novel probabilistic model named
pseudo-tree model. Based on the asymptotic results for pseudo-tree model, we
further study the length of the longest significant run in an "inflating"
Bernoulli net. We find that the probability parameter $p$ of significant node
plays an important role: there is a threshold $p_c$, such that in the cases of
$p<p_c$ and $p>p_c$, very different asymptotic behaviors of the length of the
significant are observed. We apply our results to the detection of an
underlying curvilinear feature and argue that we achieve the lowest possible
detectable strength in theory.
