Imaging and Computing Seminar
Paul Barbone, Mathematics, Boston University
Title:
Direct imaging via time-reversal, Krylov methods, and
sparse signals.
Abstract:
Multiple SIgnal Classification (MUSIC) has been used to form images
and identify sound sources since 1986 [Schmidt, R.O, "Multiple
Emitter Location and Signal Parameter Estimation," IEEE Trans.
Antennas Propagation, Vol. AP-34 (March 1986), pp.276-280.] In
active imaging of point targets, the MUSIC method can be used to
estimate the range of the time-reversal operator. Kirch's
factorization method extends these ideas to extended targets.
Typical implementations of these methods utilize measurements of the
entire time-reversal operator. They then require computations of
its eigenvalues and eigenvectors, or an estimate of its
pseudoinverse.
By contrast, we show how iterative Krylov space methods can be used
to compute these images with relatively few measurements. With the
Lanczos technique, no eigenvalues, eigenvectors, nor psuedoinverses
need be computed. Rather, an orthonormal basis for the range of the
time-reversal operator can be constructed directly from the received
data. Most of the necessary computing is performed by the array
itself, performing as a kind of ``analog computer." Furthermore, we
show that useful images can be formed from one ``iteration" (i.e.
measurement)
to the next, while the data are being collected, and automatically stops
when the full space is spanned.
Used in this way, the Lanczos method provides a perfect
reconstruction of the target with a minimal number of measurements. As such,
this approach provides a new mode of compressed sensing
without an $\ell^1$ constraint.
Contributors: G.R. Feijoo, A.A. Oberai, and P.E. Barbone