If we consider the structure of the response matrix ,whenever the number of targets is much less than the array size so that N << M, the matrix information is very redundant; the matrix is then rank N, where N is the number of targets to be imaged. Yet, the matrix for M sensors in our array is , with M(M+1)/2 distinct complex elements. The total number of distinct data present is therefore M(M+1), which is to be used to determine the x,y,z-coordinates in 3D of each target and possibly also its scattering strength q. For complex q, this means we need to find at most 5N numbers from M(M+1). In the presence of noise or strong inhomogeneities in the background medium, the redundancy may be needed to resolve several targets. In homogeneous backgrounds, the overdetermined nature of this problem is something we need to consider.
Examples of situations in which limited data are available include: (1) Only one transducer is available. Assuming that the transducer is moveable, then it would be possible to collect data in a ``synthetic aperture'' mode as is commonly done in radar applications (SAR). (2) Only two transducers are available, but some triangulation is then possible. (3) Only one ping is allowed, but many transducers are available (multistatic case). Here, we can collect only one row of the response matrix. (4) Only the primary eigenvector has been found, as in iterative time-reversal processing in the physical domain, but no attempt has been made to find eigenvectors associated with secondary targets.
How much of the information in the scattering data is really needed to solve the inverse problem in these situations? With limited data, how much data is essential to collect to locate and possibly identify the targets of most interest?
Having posed the general problem, we will not try to answer it completely here. Instead, we will show some examples of what can be done to image with restricted data sets.