Summary: For a simple polygon \(\mathcal{P}\) of size \(n\), we define weak visibility counting problem (WVCP) as finding the number of visible segments of \(\mathcal{P}\) from a query line segment \(p q\). We present different algorithms to compute WVCP in sub-linear time. In our first algorithm, we spend \(O(n^7)\) time to preprocess the polygon and build a data structure of size \(O(n^6)\), so that we can optimally answer WVCP in \(O(\log n)\) time. Then, we reduce the preprocessing costs to \(O(n^{4 + \epsilon})\) time and space at the expense of more query time of \(O(\log^5 n)\). We also obtain a trade-off between preprocessing and query time costs. Finally, we propose an approximation method to reduce the preprocessing costs to \(O(n^2)\) time and space and \(O(n^{1 / 2 + \epsilon})\) query time.