The prolate spheroidal wave functions as invariants of the time reversal operator for an extended scatterer in the Fraunhofer approximation

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, and 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's function. In the case of extended objects, however, the formalism is not as simple. It is shown here that, in the Fraunhofer approximation, analytical solutions can be found and that the solutions are functions called prolate spheroidal wave-functions. These functions have been studied in information theory as a basis of band-limited and time-limited signals. They also arise in optics. The theoretical solutions are compared to simulation results. Most importantly, the intuition that for an extended objects, the number of nonzero eigenvalues is proportional to the number of resolution cell in the object is justified. The case of three-dimensional objects imaged by a two-dimensional array is also dealt with. Comparison with previous solutions is made, and an application to super-resolution of scatterers is presented.

Spatio-temporal invariants of the time reversal operator

“…3, three lobes for the third͔. This is very similar to what is observed for extended objects 17,18 with the singlefrequency DORT method. The formalism is actually very similar.…”

“…The number of nonzero singular values for one point-scatterer is roughly equal to the time bandwidth product. There is therefore no longer an identification between the number of nonzero singular values and the number of point like scatterers, which was an interesting property of the single-frequency DORT method ͑even though this prop- erty is no longer true if the objects are not point-like 17,18 or if they have both compressibility and density contrast 23 ͒. There can be numerous nonzero singular values and it may not be easy to find which one correspond to the first invariant for a given scatterer.…”

Smart optical coherence tomography for ultra-deep imaging through highly scattering media

“…In the present case, the USAF target is an extended object. It is thus associated with a large number M of eigenstates, M scaling as the number of resolution cells contained in the object [41]. To estimate the rank M of the object, one can compute the standard deviation of the image, | n σ n U n • V n |, as a function of the number n of eigenstates considered for the imaging process.…”

Smart optical coherence tomography for ultra-deep imaging through highly scattering media

“…In the present case, the USAF target is an extended object. It is thus associated with a large number M of eigenstates, with M scaling as the number of resolution cells contained in the object ( 41 ). To estimate the rank M of the object, one can compute the SD of the image, ${\sum }_{\mathrm{italici}=1}^{\mathit{n}}{\mathrm{\sigma }}_{\mathit{i}}false|{\mathbf{U}}_{\mathbf{i}}\circ {\mathbf{V}}_{\mathbf{i}}false|$, as a function of the number n of eigenstates considered for the imaging process.…”