Decomposition of the time reversal operator (DORT) for extended objects

Abstract

The decomposition of the time reversal operator, known by the French acronym DORT, is widely used to detect, locate, and focus on scatterers in various domains such as underwater acoustics, medical ultrasound, or nondestructive evaluation. In the case of point scatterers, the theory is well understood: the number of nonzero eigenvalues is equal to the number of scatterers, and the eigenvectors correspond to the scatterers’ Green function. In the case of extended objects, however, the formalism is not as simple. We show that, in the Fresnel approximation, analytical solutions can be found, and that the solutions are functions called prolate spheroidal wavefunctions. These functions have been studied in optics and information theory as a basis of band‐limited and time‐limited signals. The theoretical solutions are compared to simulation results and previous experimental results Most importantly, we justify the intuition that for an extended object, the number of nonzero eigenvalues is proportional to the number of resolution cells in the object. The case of 3‐D objects imaged by a 2‐D array will also be described.