P0-7 Computational Time Reversal Ultrasonic Array Imaging of Multipoint Targets

In computational time reversal (CTR) ultrasonic imaging of point scatterers, the singular value decomposition of the response matrix is of critical importance. Determination of response matrix rank using singular value decomposition (SVD) is the first step in obtaining the null subspace projection operator which is used to quantify the contribution of a test illumination vector to the measured response matrix. The null subspace projection operator is formed from the summed outer products of the singular vectors associated with singular values of zero magnitude. For imaging in noisy environments or inhomogeneous media, singular values do not generally attain zero magnitude and we face the problem of correctly determining the identities of the singular vectors that span the signal subspace. Failure to correctly determine response matrix rank will result in a projection operator that either incompletely spans the actual noise subspace, or erroneously spans part of the signal subspace. This paper will provide physical justification for the observed loss of rank in the response matrix at low frequencies and develop a robust, efficient algorithm to determine the correct subspace dimensionality for generation of the null subspace projection operator in DORT-based CTR imaging algorithms. However, while successive singular values of the decomposition are guaranteed to be monotonically non-increasing, numerical simulations typically show that singular values do not correlate well to the actual scatterer reflectivities. The computed singular values may decrease in value very rapidly, so that even for a system with M equally reflective targets the Mth singular value may take on small values near zero. In general, the rank of the response matrix at a given frequency is only maximally given by the number of point targets being probed, and a robust and computationally efficient thresholding method is required to maximize imaging efficiency of CTR algorithms. We demonstrate using n- - umerical simulations the efficiency of the thresholding algorithm. Imaging error introduced by suboptimal rank determination of the response matrix is quantified. We also quantify the effect of signal noise on image quality through its effect on rank determination accuracy.