Decomposition of the time reversal operator: Detection and selective focusing on two scatterers

Abstract

The decomposition of the time reversal operator ͑DORT͒ method is a selective detection and focusing technique using an array of transmit-receive transducers. It relies on the theory of iterative time reversal mirrors which was presented by Prada et al. ͓C. Prada, J. L. Thomas, and M. Fink, J. Acoust. Soc. Am. 97, 62-71 ͑1995͔͒. The time reversal operator was defined as K*()K(), where is the frequency, * means complex conjugate, and K͑͒ is the transfer matrix of the array of L transducers insonifying a time invariant scattering medium. It was shown that this time reversal operator can be diagonalized and that for ideally resolved scatterers of different reflectivities, each of its eigenvectors of nonzero eigenvalue provides the phase law to be applied to the transducers in order to focus on one of the scatterers. The DORT method consists in determining these eigenvectors and using them for the selective focusing. This paper presents a complete analysis of this method in the case of two scatterers. The mathematical expressions of the eigenvectors are given and several experimental results are described. In particular, the effectiveness of the method to focus selectively through an inhomogeneous medium is established.

Time-Reversal Microwave Imaging Based on Random Configuration of Transmitters or Receivers

“…Eigenspace analysis of the TRO provides information about the scattering scenario under study. Specifically, in the case of multiple scatterers, selective focusing on the desired scatterers is possible via the decomposition of the TRO method (DORT under its French acronym) [5,6]. Similarly, TR-MUSIC (Multiple signal Classification) method developed in [7] utilizes TRO to obtain TR-based imaging for the localization of target(s) [8].…”

D.O.R.T. method as applied to electromagnetic subsurface sensing

“…method [Prada and Fink, 1994;Prada et al, 1996], developed in Acoustics to synthesize an incident time-harmonic wave focusing on a selected target. The D.O.R.T.…”

Nonlinear Time Reversal Tomography of Structural Defects

“…DORT is an approximative method which substitutes the iterative time-reversal process. With use of this method, it is possible to extract echoes corresponding to well-separated defects present in the media (their Green's functions), which can be used to focuse on these defects [6]. Moreover, for each defect, information about its frequency dependant apparent reflectivity is obtained.…”

“…Thus, the apparent reflectivity of the scatterers where their corresponding Green's functions g( f ) are given by the TRO eigenvectors [6]. For each frequency component f 0 and scatterer…”

“…These methods, already used in several applications (NDT and medicine), represent effective methods to focus acoustic energy into the complex media (noisy, scattering, complex shape etc.). Invariant properties of TRA methods were mathematically formulated, and DORT (Décomposition de l'Opérateur Retournement Temporel) analysis was developed in order to localize scatterers and acoustic sources in a linear medium using the concept of the Time Reversal Operator and Green's functions [6].…”