Diffraction Tomography Using INNG Algorithm Combined With TV Regularization

Since the first X-CT machine was invented in the 70s of last century, the theory and technology of CT has made rapid development. However, when the size of measured object and incident source wavelength is equal to or even much smaller, the flow of energy no longer travels straightly in the object. In this condition, it is necessary to consider the impact of refraction or diffraction. Therefore, diffraction tomography (DT) came into being. Now, diffraction tomography has been widely used in ocean exploration, industrial non-destructive testing and geophysical areas.Assumed that the object is weakly scattering, the theoretical basis of DCT, the Fourier Diffraction Theorem(FDT) is derived with the first order Born or Rytov approximation. According to the theory, two-dimensional Fourier transform of the object along a semi-circular arc in the spatial frequency domain can be obtained by collecting the projections of incident wave in the scattered field. However, non-uniform frequency points of the semi-circle trajectory are not suitable for applying the fast inverse Fourier transform for image reconstruction. The purpose of this paper is to study reconstruction algorithms by using non-uniform sampling points in frequency domain.To data, several algorithms have been proposed, including polynomial interpolation, uniform frequency domain (UFR) interpolation, spatial interpolation (usually termed as filtered backpropagation) and non-uniform fast Fourier transform (NUFFT). Generally, polynomial interpolation and UFR interpolation present lowest complexity but largest errors. Spatial interpolation has high computational complexity with slightly better reconstruction results. NUFFT significantly improves reconstruction quality in terms of the use of total variation (TV) regularization but still not perfect.In this paper, an iterative next-neighbor regridding (INNG) algorithm is introduced, which was first developed in MRI reconstruction. The algorithm does not require optimized density compensation functions (DCFs) to avoid profile distortions or singular value decomposition (SVD) regularization to avoid amplification of data imperfections. INNG algorithm is an extension of next-neighboring grid (NNG)algorithm.In INNG algorithm, high quality reconstructed images are obtained from an iterative reconstruction that is performed using matrixes scaled to sizes greater than that of the target image matrix. However, when the scaling factor is large, it is not efficient to implement FFTs, therefore facilitated INNG and block INNG are proposed. The former one utilizes successively increasing scaling factors; the later one partitions the acquired k-space region into several blocks, and then applies the basic (or facilitated) INNG algorithm to each block. On the other hand, gridding itself often introduces errors and artifacts when interpolating the non-uniform data to Cartesian grid. As a result, TV regularization is applied in each iteration in INNG algorithm, considering its merits of eliminating noise and artifacts while preserving image edges.Several simulation experiments were carried out with both ideal and noisy data. Results show that in both conditions, the new hybrid method of INNG algorithm combined with TV regularization has much higher accuracy than the pure INNG algorithm. It is also shown that, within similar duration, the new method got lower reconstruction error than NUFFT algorithm for ideal data. And for noisy data, the advantage over NUFFT algorithm was increasingly apparent with decreasing amount of sampled trajectories. When the amount of sampled trajectories is higher than certain number, the two methods displayed similar quality of reconstruction results.