Transient Wave Imaging

Extensive work has been carried out in the past decade to image the elastic properties of human soft tissues by inducing motion. This broad field, called elasticity imaging or elastography, is based on the initial idea that shear elasticity can be correlated with the pathology of tissues. There are several techniques that can be classified according to the type of mechanical excitation chosen (static compression, monochromatic, or transient vibration) and the way these excitations are generated (externally or internally). Different imaging modalities can be used to estimate the resulting tissue displacements. A very interesting approach to assessing elasticity is to use the acoustic radiation force of an ultrasonic focused beam to remotely generate mechanical vibrations in organs. The acoustic force is generated by the momentum transfer from the acoustic wave to the medium. The radiation force essentially acts as a dipolar source. A spatio-temporal sequence of the propagation of the induced transient wave can be acquired, leading to a quantitative estimation of the viscoelastic parameters of the studied medium in a source-free region. Our aim in this thesis is to provide a solid mathematical foundation for this transient technique and to design accurate methods for anomaly detection using transient measurements. We consider both the acoustic and elastic cases. We develop efficient reconstruction techniques from not only complete measurements but also from limited-view transient data and adapt them in the case of viscous media, where the elastic waves are attenuated and/or dispersed. We begin with transient imaging in a non-dissipative medium. We develop anomaly reconstruction procedures that are based on rigorously established inner and outer time-domain asymptotic expansions of the perturbations in the transient measurements that are due to the presence of the anomaly. It is worth mentioning that in order to approximate the anomaly as a dipole with certain polarizability, one has to truncate the high-frequency component of the far-field measurements. Using the outer asymptotic expansion, we design a time-reversal imaging technique for locating the anomaly. Based on such expansions, we propose an optimization problem for recovering geometric properties as well as the physical parameters of the anomaly. We justify both theoretically and numerically that scale separation can be used to obtain local and precise reconstructions. We show the differences between the acoustic and the elastic cases, namely, the anisotropy of the focal spot and the birth of a near fieldlike effect by time reversing the perturbation due to an elastic anomaly. These interesting findings were experimentally observed before. Our asymptotic formalism clearly explains them. In the case of limited-view transient measurements, we construct Kirchhoff-, back-propagation-, MUSIC-, and arrival time-type algorithms for imaging small anomalies. Our approach is based on averaging of the limited-view data, using weights constructed by the geometrical control method. It is quite robust with respect to perturbations of the non-accessible part of the boundary. Our main finding is that if one can construct accurately the geometric control then one can perform imaging with the same resolution using partial data as using complete data. We also use our asymptotic formalism to explain how to reconstruct a small anomaly in a viscoelastic medium from wavefield measurements. The visco-elastic medium obeys a frequency power-law. For simplicity, we consider the Voigt model, which corresponds to a quadratic frequency loss. By using the stationary phase theorem, we express the ideal elastic field without any viscous effect in terms of the measured field in a viscous medium. We then generalize the imaging techniques developed for a purely quasi-incompressible elasticity model to recover the viscoelastic and geometric properties of an anomaly from wavefield measurements.