Inverse source problem by convex optimization with constraints over the object space and signal field

In the acoustical endscopy, due to the physical limitations, the transducer array is composed of a small number of elements and each interspacing is larger than the acoustical wavelength that is called a sparse array system. In such cases, avoiding the ill-posed problems, projection onto convex sets (POCS) methods are used with incorporating constraints about both the signal field and the object space. POCS, however, is based on the alternating projections paradigm, which has a slow-convergence property in general. Furthermore if inconsistency exists in the set of constraints, this POCS algorithm cannot guarantee the convergence to the optimal estimate. The proposed algorithm is based on convex optimization over the direct product of the object space and the observed signal field. By acoustical experiments, it is proved that the proposed algorithm has the following improvements: (1) Targets can be identified when unknown components exist in the transfer function. (2) Transient behavior of the convergence becomes more stable than that of POCS algorithm. (3) Instability caused by the inconsistency in the constraints can be reduced. (4) Artifacts caused by the spurious lobes can be reduced under the condition that the interspacing of transducer elements is larger than the wavelength.