A novel sampling method for multiple multiscale targets from scattering amplitudes at a fixed frequency

Abstract

A sampling method by using scattering amplitude is proposed for shape and location reconstruction in inverse acoustic scattering problems. Only matrix multiplication is involved in the computation, thus the novel sampling method is very easy and simple to implement. With the help of the factorization of the far field operator, we establish an inf-criterion for characterization of underlying scatterers. This result is then used to give a lower bound of the proposed indicator functional for sampling points inside the scatterers. While for the sampling points outside the scatterers, we show that the indicator functional decays like the bessel functions as the sampling point goes away from the boundary of the scatterers. We also show that the proposed indicator functional continuously dependents on the scattering amplitude, this further implies that the novel sampling method is extremely stable with respect to errors in the data. Different to the classical sampling method such as the linear sampling method or the factorization method, from the numerical point of view, the novel indicator takes its maximum near the boundary of the underlying target and decays like the bessel functions as the sampling points go away from the boundary. The numerical simulations also show that the proposed sampling method can deal with multiple multiscale case, even the different components are close to each other.

Analysis of new direct sampling indicators for far-field measurements

“…One drawback is that Theorems 2.1 and 3.2 only give that the new indicators decay as the sampling point moves away from the scatterer which may result in low contrast reconstructions. In [22] it is seen that this can be overcome by raising the new indicators to the power p > 1 to sharpen the resolution. To recover the scatterer one can take a level curve of the indicator W p .…”

Target Reconstruction with a Reference Point Scatterer using Phaseless Far Field Patterns

“…Inserting this into (3.1), integrating by parts and using the well-known Funk-Hecke formula [4,29], we deduce that where µ α = 2πi −α , n = 2, 4πi −α , n = 3 and f α (t) = J α (t), n = 2, j α (t), n = 3…”

“…We thus expect that G (and therefore A) decays like Bessel functions as the sampling points away from the boundary of the scatterer. Then one may look for the scatterers by using the following indicators [27,29,37] with phased far field patterns, where A and G are given in (3.1). In [29], it has been showed that the indicator I 2 has a positive lower bound for sampling points inside the scatterer, and decays like Bessel functions as the sampling points away from the boundary.…”

“…Then one may look for the scatterers by using the following indicators [27,29,37] with phased far field patterns, where A and G are given in (3.1). In [29], it has been showed that the indicator I 2 has a positive lower bound for sampling points inside the scatterer, and decays like Bessel functions as the sampling points away from the boundary. If the size of the scatterer D is small enough Using the Phase Retrieval Scheme proposed in the previous section, we obtain the approximate phased far field pattern u ∞ D .…”

On a novel inverse scattering scheme using resonant modes with enhanced imaging resolution

“…Here, it is noted that Ω is unknown, but with the far-field data given in (1.3), one can apply, e.g. the direct sampling method in [21], to easily find a point inside Ω. Then associated with the far-field data given in (1.3) and for each k ∈ I, one can solve the discretized linear integral equation (2.2) by the Picard's theorem.…”