Inverse problems of 3D ultrasonic tomography with complete and incomplete range data

Abstract

The paper focuses on the development of efficient methods for solving inverse problems of 3D ultrasound tomography as coefficient inverse problems for the wave equation. The idea of standard tomographic approaches to solving tomography problems is to analyze the 3D objects by their two-dimensional cross sections. This scheme is perfectly implemented in the case of X-ray tomography. Unlike X-ray tomography, ultrasonic tomography has to deal with diffraction and refraction effects, which limit the possibility of solving 3D problems by analyzing 2D cross sections. We propose efficient methods for solving inverse problems of ultrasound tomography directly in the 3D formulation. The proposed algorithms are based on the direct computation of the gradient of the residual functional. The algorithms are primarily oriented toward the development of ultrasound tomographs for differential diagnosis of breast cancer. Computer simulations demonstrated the high efficiency of the developed algorithms. The algorithms are implemented on GPU-based supercomputers. We analyze various schemes of 3D ultrasonic tomographs including those without rotating elements and with fixed positions of the sources and receivers. The algorithms developed can be used for solving inverse problems of seismology, acoustics, and electromagnetic sounding.