  Many non-iterative imaging algorithms require a large number of incident
directions. Topological derivative-based imaging techniques can alleviate this
problem, but lacks a theoretical background and a definite means of selecting
the optimal incident directions. In this paper, we rigorously analyze the
mathematical structure of a topological derivative imaging function, confirm
why a small number of incident directions is sufficient, and explore the
optimal configuration of these directions. To this end, we represent the
topological derivative based imaging function as an infinite series of Bessel
functions of integer order of the first kind. Our analysis is supported by the
results of numerical simulations.
