INTRODUCTION

Time-reversal acoustics (Fink, 1997; 1999) has become an important research area in recent years because of the many potential applications associated with it and envisioned for it. In biomedical applications, the prototype is the detection and destruction of kidney stones by ultrasound. By insonifying the kidney with pulses of ultrasound and measuring the return signal, it is possible to localize one or more kidney stones and then to send a return signal back to the scatterer with high enough amplitude to cause the stone to fragment. Then, the remaining small pieces eventually pass harmlessly from the system. Training the sound to find the kidney stone automatically is the main purpose of time-reversal acoustics in this application. Analyses of why it works so well and means of improving its performance have been studied by many authors (Prada et al., 1991; 1995; Prada and Fink, 1994; Mast et al., 1997; Devaney, 1999; Blomgren et al., 2001; Tsogka and Papanicolaou, 2001). Most of these analyses have concentrated on imaging of or focusing on small point-like targets.

Chambers and Gautesen (2001) have recently shown that a single spherical acoustic scatterer has from two to four eigenfunctions associated with it. To understand this result, it might help to put it in a larger context by considering scattering from a single spherical elastic scatterer imbedded in an otherwise homogeneous elastic medium. Then, it is well-known that the principal scattering terms arise from changes in bulk modulus (K), density ($\rho$), and shear modulus ($\mu$). Bulk modulus changes produce monopole scattering; density changes produce dipole scattering; and shear modulus changes produce quadrupole scattering. There is at most one contribution from monopole scattering; at most three from dipole scattering; and, for acoustics, there is no quadrupole scattering, as there are no viable shear waves. The scattering multiplicity found by Chambers and Gautesen is then understood as, at most 1 + 3 = 4, while some of the dipole terms may not be excited because of the scatterer-to-sensor array orientation. The monopole term is the one usually treated in analyses of time-reversal acoustics, but for some situations -- such as air bubbles in liquid -- the dipole terms should also be considered. Nevertheless, we will only consider the monopole contributions in this paper. The generalization of the methods to be presented here in order to incorporate other modes is actually straightforward, as we shall see, and, except for the fact that the trial vectors used for imaging must be tailored to these other modes, the analysis proceeds without any significant alteration.

The fundamental concepts used in time-reversal imaging for acoustics are closely related to concepts in optimal probing and imaging for a variety of physical problems including work by Isaacson (1986), Gisser et al. (1990), Cherkaeva and Tripp (1996a,b), Colton and Kirsch (1996), Kirsch (1998), and Brühl et al. (2001).

The next section presents a review of the relevant issues in acoustic scattering and time-reversal signal processing. Then I discuss the linear subspace methods of imaging including the well-known MUSIC algorithm, as well as some modifications of MUSIC, and related algorithms. The following section introduces the maximum-entropy imaging approach which is often the preferred method when data are sparse. Then I show some examples and the final section discusses conclusions.