Imaging and Computing Seminar

Paul Barbone, Mathematics, Boston University

Direct imaging via time-reversal, Krylov methods, and sparse signals.

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