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 (), and shear modulus
(). 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.

9/18/2001