Application of the NUFFT for reconstruction problems in diffraction tomography

Ultrasound tomography with diffracting sources is an important type of acoustic imaging. Since the used wavelengths are comparable to the object feature dimensions, the straight ray tomography theory is no longer applicable. Particularly, the Fourier slice theorem should be replaced by the Fourier diffraction theorem. Reconstruction methods used beforehand addressed the problem as straightforward approximation of the inverse nonuniform Fourier transform (NUFT) and involved frequency interpolation, which is liable to introduce significant inaccuracies. More accurate and computationally efficient methods were proposed for forward and inverse 1D NUFT. Recently, fast and accurate approximation of the forward nonuniform fast Fourier transform was introduced by Fessler and Sutton (IEEE Trans. SIGPRO, 2001). Inverse NUFFT can be achieved iteratively in this framework. We adopt this approach for iterative reconstruction in diffraction tomography, combining it with total variation regularization in order to suppress noise while preserving the sharpness of edges. Simulation studies with the Shepp-Logan phantom show that the proposed algorithm significantly outperforms the frequency interpolation methods.