IMAGING AND INVERSION USING LINEAR SUBSPACE METHODS

One class of imaging methods available for time-reversal imaging of small targets may be called linear subspace methods, of which the best known method is probably MUSIC (Schmidt, 1979; Marple, 1987; Stoica and Nehoral, 1990; Xu and Kaveh, 1996; Stoica and Moses, 1997). The term MUSIC stands for MUltiple SIgnal Classification scheme. The method determines whether or not each vector in a set of vectors is fully or only partially in the range of an operator. If ${\bf T} = {\bf KK}^*$ is the operator of interest (i.e., the time-reversal operator), and the complete set of eigenvectors in the range of the operator (i.e., having nonzero eigenvalue) is given by $\{{\bf V}_n\}$, then we can choose a test vector ${\bf H}_r$ [see (Hr)] and define the square of the direction cosine between ${\bf H}_r$ and any one eigenvector ${\bf V}_n$ to be

^2(V_n,H_r) = V_n^*TH_r^2/H_r^2.   We are assuming in this formula that all the eigenvectors ${\bf V}_n$ are normalized so $\vert{\bf V}_n\vert = 1$,while ${\bf H}_r$ is not necessarily normalized.