9 December 1997 Numerical scheme for solving the acoustical inverse scattering problem
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Abstract
In current public discussions, much attention is paid to the danger of breast cancer. X-ray methods for cancer detection like mammography are sometimes considered inappropriate. Recently, novel optical and acoustical methods are being examined. One of these is image reconstruction from time- harmonic ultrasound data, which requires determining the coefficient f in the Helmholtz equation (Delta) u + k2(l + f)u equals 0 from boundary measurement of various solutions u of the equation for different boundary conditions. In this equation, f determines the physical properties of the scatterer, while k is the wave number and u is the time-harmonic sound wave. Born- and Rytov- approximations have been used for some time to accomplish this task, but are not precise enough. All functions may be either a 2D or a 3D function, depending on the application. To overcome the complexity problems, an iterative algorithm that works on finite data and incorporates finite boundary conditions instead of radiation condition will be presented. Two main ingredients are used to accomplish this. First, an adjoint-field type iteration scheme is employed. Second, an initial value solver is used to solve the direct problem of computing a sound wave, given boundary conditions and a scatterer. The algorithm allows us to solve the problem with realistic parameters for simulated data on regular workstations in a few minutes.
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Frank Wuebbeling, Frank Natterer, "Numerical scheme for solving the acoustical inverse scattering problem", Proc. SPIE 3171, Computational, Experimental, and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Applications, (9 December 1997); doi: 10.1117/12.284719; https://doi.org/10.1117/12.284719
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