An indicator sampling method for solving the inverse acoustic scattering problem from penetrable obstacles

This paper is concerned with the three-dimensional inverse scattering problem of imaging the support of a penetrable object from far-field data of time-harmonic acoustic scattered waves. An indicator function, which has much larger values inside the obstacle than outside the scatterer, is derived. Based on the property of the indicator function, an indicator sampling method is presented and analysed for solving the inverse transmission problem, which is achieved by using the set of points for which the indicator function arrives at its local extrema in a sampling domain containing the unknown penetrable object as a reliable visualization of the penetrable obstacle. The method has the interesting feature of being independent of any prior knowledge about the geometrical and material properties of the penetrable obstacle. Another attractive feature of the approach is that it needs only the knowledge of far-field data of the acoustic scattered waves at a finite number of incidence and observation directions distributed over some limited apertures. Numerical examples for several three-dimensional penetrable obstacles having a variety of shapes are given, showing the performance of the algorithm.