Wave Modeling for Inverse Problems with Acoustic, Electromagnetic, and Elastic Waves

Finite element study of ultrasonic imaging

“…The dimensionality of the scatterer will be the same as that of the measured data. Therefore, under the assumption of a weak scatterer, a transient experiment which records the scattered data in a certain pe riod of time would possess a unique solution to the inverse scattering problem [44].…”

“…The object function can be recovered from the projection either via the algorithm of filtered backprojection [56] or utilizing the Fourier slice theorem [44,56,57], which allows the recovering of projection data in the spatial frequency domain of the object function.…”

“…For a far-field measurement surface, the Born approximation immediately yields the algorithm of the far-field Fourier inversion either in angular or frequency diversity, the time domain counterpart of the latter algorithm exhibiting the struc ture of a backprojection scheme [45]. If the far-field approximation does not hold, a generalized holography [44,58] is introduced. The back-propagation principle of " generalized holography then yields the Porter-Bojarski integral equation [59,60] with an image output of minimal energy sources revealing the properties of the scatterer on an Ewald sphere [44,58].…”

“…The back-propagation principle of " generalized holography then yields the Porter-Bojarski integral equation [59,60] with an image output of minimal energy sources revealing the properties of the scatterer on an Ewald sphere [44,58]. The integration of minimal energy output with regard to a diversity variable (angle or frequency) results in either an angular diversity or a fre quency diversity generalized filtered back-propagation algorithm under the weak scat terer approximation [44,46]. For a planar measurement, the Porter-Bojarski integral equation reduces to the Fourier diffraction slice theorem [61,62] …”

A Finite Element Test Bed for Diffraction Tomography

“…For diffraction sources such as ultrasound, the inverse scheme should be based on the wave equation rather than on the assumption of a straight line path from the source to the receiver as in X-ray tomography [5][6]. Based on the Born or Rytov approximation [7], diffraction tomography reconstructs the object function from the planar measurement of the displacement field by back-propagating the displacement to the full-space and summing over the frequency or over the different angles [8][9][10].…”