A frame-based approach for wideband correlation-inversion of lossless scatterers

Presented here is an ultra-wideband-correlation-based scheme for imaging and inversion of an unknown weak and lossless scatterer embedded in a known background medium. The scheme uses an excitation and reception of ultra wideband/short-pulsed fields by an array of transducers located outside the imaging domain. The scatterer image is formed by cross correlating (in the short-pulsed domain or via spectral integration in the ultra wideband domain) the numerically/analytically back-propagated, measured, and scattered data set with the forward-propagation excitations. It is shown that in the ultra wideband domain, the forward-backward propagation functions form a frame set in a finite Hilbert space. Within the weak scattering assumption (Born approximation) the scatterer's image and object function (velocity profile) are related via the corresponding frame operator. Therefore, an exact inversion scheme of the frame operator is readily available to yield the object function via an iterative scheme or using the dual frame set. Numerical examples that demonstrate the performance of the imaging and inversion schemes for scatterers with various velocity profiles are presented. It is shown that the scatterer image is generally of poor resolution. However, on inversion, a high-quality velocity profile is obtained that captures the scatterer fine details.