High resolution ultrasonic imaging may be achieved via the synthetic aperture focusing technique, provided that the synthetic antenna is correctly spatially sampled. We show that spatial undersampling is possible, using a digital spotlighting technique that permits to recover the correct reconstructed images from aliased data. The proposed method is applied to experimental ultrasonic data.

Typically, the linear image restoration problem is an ill-conditioned, underdetermined inverse problem. Here, stabilization is achieved via the introduction of a first-order smoothness constraint which allows the preservation of edges and leads to the minimization of a nonconvex functional. In order to carry through this optimization task, we use stochastic relaxation with annealing. We prefer the Metropolis dynamics to the popular, but computationally much more expensive, Gibbs sampler. Still, Metropolis-type annealing algorithms are also widely reported to exhibit a low convergence rate. Their finite-time behavior is outlined and we investigate some inexpensive acceleration techniques that do not alter their theoretical convergence properties; namely, restriction of the state space to a locally bounded image space and increasing concave transform of the cost functional. Successful experiments about space-variant restoration of simulated synthetic aperture imaging data illustrate the performance of the resulting class of algorithms and show significant benefits in terms of convergence speed.

The synthetic aperture image formation process can be formulated as a space-variant 2D convolution. The recovery of the original reflection density is an ill-posed inverse problem which is both underdetermined and ill-conditioned. Its stabilization is achieved via concave stabilizers that are well adapted to the preservation of discontinuities. This leads to the minimization of a non-convex functional, a task which is successfully carried out using a Metropolis-type annealing algorithm. For improved performance, we investigate some inexpensive acceleration techniques which do not alter the theoretical convergence results; their efficiency is demonstrated through restorations from simulated data.