Mathematical foundations of imaging, tomography and wavefield inversion. (Q2902056)

The cover illustration of any scientific book tries to draw attention of potential readers and, at same time, to show one of the most pre-eminent results of all contents in the text. In this case, the illustration shows the result of the time reversal MUSIC algorithm applied to the following problem: A set of transmitting and receiving elements (for instance, antennas) located at the boundary of a circle are used to generate an incident wave propagating into the system and to measure the resulting scattered wave produced by a set of scatterers distributed within the circle. The result is a colored density plot of the pseudo-spectrum. Estimates of the locations of the scattering centers occur at the peaks of the pseudo-spectrum.NEWLINENEWLINEClosely related with this is the decomposition of the time reversal operator (DORT) method where the goal is to generate a set of incident waves having the property that they each focus on a different scatterer in the system without prior knowledge of the scatterer locations.NEWLINENEWLINEThese are the kind of problems the book deals with. Problems arising in the fields of optics, acoustics and elastic wave propagation: forward and inverse problems encountered in propagation, radiation, and scattering of waves. The book is mostly about linearized formulations of inverse scattering. The book goes from first principles to the current state of the art always with mathematical rigor. Additional resources including MATLAB codes and solutions are available online.NEWLINENEWLINEOne of the main characteristics of the text is that the Fourier based inversion schemes have been replaced in the book by the much more powerful singular value decomposition. This new approach provides a uniform framework for treating virtually all of the linearized inverse problems associated with the wave and Helmholtz equations. Other tools are the use of the Born approximation -- which results from dropping all terms in the infinite Born series except for the first two and then seeing if this is adequate to describe the observed scattering -- and the use of the Green function. A warning should be made. Despite the use of the term tomography, the reader will not find applications to computed tomography and medical imaging. For example, the Radon transform is not covered.