Diffraction tomography and the Sinc Basis Moment Method

To obtain high-resolution quantitative images of acoustic parameters (such as sound speed and absorption coefficient) in tissue from measurements of the interaction of incident waves of ultrasound with the tissue, diffraction effects must be included. The vast majority of all diffraction tomography algorithms in existence depend on first-order scattering assumptions for their validity which do not hold in practical tissue inverse scattering problems. The present work investigates several aspects of a higher-order algorithm, the Sinc Basis Moment Method. First, conventional tomography algorithms, including those having straight-path and first-order scattering assumptions, are compared. Close attention is given to a comparison of the mathematical meaning of the first Born and Rytov approximations. Then, the equations of the higher-order sinc basis method are explained, and signal processing details for its implementation are given. The scatterer chosen for most of this work is the circular cylinder, for which exact scattered field data resulting from an incident cylindrical wave may be calculated independently of the reconstruction equations. Object parameters such as sound speed and absorption contrast and size, as well as algorithm parameters such as sampling density, grid size, and relaxation constants, are varied to determine behavior and limitations of the algorithm. The algorithm itself was modified to use knowledge about the problem structure to maximize computational efficiency. In addition, an interesting use of the FFT which significantly reduces the order of computation is described. The first iteration of the Sinc Basis Method is shown to be equivalent to a typical first-order, Born-approximation-based solution. Finally, use of a minisupercomputer has helped make evident a fundamental limitation of the algorithm, the size of the phase shift of a wave passing through the object. An abrupt threshold of reconstruction quality exists near {dollar}pmpi{dollar}. The reason appears to be the existence of multiple solutions arising from the periodicity in phase representations. The iterative (perturbation) technique settles upon the closest solution to the starting point, resulting in erroneous reconstructions when the closest solution is no longer the desired solution (which is true for phase shift magnitudes greater than {dollar}pi{dollar}). For the ambiguity to be resolved, either the initial starting points for both the object function and the field must be substantially improved, or somehow the information contained in unwrapped phase measurements must be preserved in the computations.