The main results of this article consist of two parts. In the first part, for a fixed value b∈N, we consider the origin O as the observation point, with the observational line given by f(x)=ax^b,a∈Q,b∈N. The observed target is a lattice point array V(m)={├ (i,j)┤| i,j∈N,1≤i≤m,1≤j≤m}. We study the quantity and probability of visible points. We find that the number of visible points, denoted as h_b (m), which is related to the Euler function and the Möbius function, and the probability of visible points is also related to the Riemann zeta function. In the second part, for a fixed value b∈N, we arrange observation points on the x-axis and y-axis. We use f(x)=ax^b,a∈Q as the observational lines in the coordinate system with observation points as the new origin. We study the methods and quantity of complete observation point placements for the target point set V(m×n)={(i,j)│i,j,∈N,1≤i≤m}. We obtain important results as follows: assume m≥6 and T ⊂{1,…,m+1} is a F(m)-cover , r be the smallest prime number greater than m. For the target point set V(m×n), we construct the observation point set S_2 = {(0,0),(0,r)}∪{(t,0) | t∈T}. Then, V(m×n) is S_2-visible. Furthermore, we investigate the placement methods and quantity of observation points for the target point set V(n×m)={(i,j)│i,j,∈N,1≤j≤m} and find that the number of observation points can be significantly reduced.