Resolution of Born Scattering in Curve Geometries: Aspect-Limited Observations and Excitations

Abstract

In inverse scattering problems, the most accurate possible imaging results require plane waves impinging from all directions and scattered fields observed in all observation directions around the object. Since this full information is infrequently available in actual applications, this paper is concerned with the mathematical analysis and numerical simulations to estimate the achievable resolution in object reconstruction from the knowledge of the scattered far-field when limited data are available at a single frequency. The investigation focuses on evaluating the Number of Degrees of Freedom (NDF) and the Point Spread Function (PSF), which accounts for reconstructing a point-like unknown and depends on the NDF. The discussion concerns objects belonging to curve geometries, in this case, circumference and square scatterers. In addition, since the exact evaluation of the PSF can only be accomplished numerically, an approximated closed-form evaluation is introduced and compared with the exact one. The approximation accuracy of the PSF is verified by numerical results, at least within its main lobe region, which is the most critical as far as the resolution discussion is concerned. The main result of the analysis is the space variance of the PSF for the considered geometries, showing that the resolution is different over the investigation domain. Finally, two numerical applications of the PSF concept are shown, and their relevance in the presence of noisy data is outlined.

Performance of the Linear Model Scattering of 2D Full Object with Limited Data

“…For the sake of illustration, we recall three modalities from [ 30 ] to combine the impinging angles of the plane waves (transmitter) and observation directions (receiver) for the aspect-limited data case. First, it is supposed that the transmitters and receivers are located on the same side in the receiving mode, where the back-scattered fields are only available; it may be useful in GPR imaging and oil and gas exploration.…”

“…A resolution analysis of the linear inverse scattering problem at a single frequency has been proposed in [ 20 , 26 , 30 ] by the estimation of the NDF of the relevant operator and the evaluation of the PSF in the object domain in the 2D scalar geometry. First, particularly in [ 20 , 26 ], for the case of round angle observation and impinging plane waves directions, curve investigation domain geometries (i.e., consisting of the boundaries of a 2D domain, as strips or arc of circumferences) were examined, since this allows one to evaluate in closed-form the NDF according to scattering object length.…”

“…Next, the case of aspect-limited excitations and observations has been considered in [ 30 ] and applied to three different sensing modalities for the same geometries as in [ 20 , 26 ]. Then, the NDF cannot be computed in closed-form; however, a PSF analysis has been performed with the help of a different approximate analytical evaluation.…”

“…In fact, it is important to examine the 2D full geometry to show the differences between the full and the void curve geometries of the previous papers. In this case, the NDF can be estimated in closed form; it means that the approximate PSF in the object domain is the same as in [ 30 ], since it depends only on the angular ranges of the available data in the aspect-limited configuration and not on the investigation domain geometry.…”

“…Again, for the sake of illustration, the receiving, transmission, and angle modes are recalled as in [ 30 ] to simulate the actual sensing configurations and to show the difference between their NDF and resolution. The main results to be mentioned are that they change when the range of plane waves and observation directions vary.…”