This paper also proposed another reconstruction method based on a direct approximation of the Fourier inversion formula using a twodimensional (2-D) trapezoidal rule. In addition, the possibility of reconstruction from a concentric-squares raster was discussed. Numerous simple interpolators have been tried in DF reconstruction with the results compared with CBP [33]. In [34] and [35], the concept of angular bandlimiting was used to interpolate the polar data onto a Cartesian grid. In [36], a DF reconstruction using bilinear interpolation for diffraction tomography provided image quality that was comparable to that produced by the CBP algorithm. Very good reconstruction quality was obtained in [37] and [38] using a spline interpolator, or a hybrid type of spline interpolator. The notion of "gridding" was introduced in [39] as a method of obtaining optimal inversion of Fourier data. An optimal gridding function was proposed, and successful results were obtained when applied to the tomographic reconstruction problem. In [40], several different gridding functions were tried for DF reconstruction, and the performances were compared. In [41, 42], the linogram reconstruction method was proposed as a form of DF reconstruction. The data collection grid in the linogram method is the same as in the concentric-squares sampling scheme. The inversion of the Fourier data in [41, 42] was accomplished by first applying the chirp-z transform in one direction and then computing FFTs in the other direction. In CT, many of these attempts at DF reconstruction have given a poorer result than the CBP algorithm, due to the error incurred in the process of the polar-to-Cartesian interpolation. The attraction of DF reconstruction, however, is that it is thought to require less computation than ...

In this paper we analyse a class of DASP (Digital Alias-free Signal Processing) methods for spectrum estimation of sampled signals. These methods consist in sampling the processed signals at randomly selected time instants. We construct estimators of Fourier transforms of the analysed signals. The estimators are unbiased inside arbitrarily wide frequency ranges, regardless of how sparsely the signal samples are collected. In order to facilitate quality assessment of the estimators, we calculate their standard deviations. The optimal sampling scheme that minimises the variance of the resulting estimator is derived. The further analysis presented in this paper shows how sampling instant jitter deteriorates the quality of spectrum estimation. A couple of numerical examples illustrate the main thesis of the paper.