We investigate the use of direct-Fourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobian-weighted 2-D periodic sinc-kernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms utilizing optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the least-squares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentric-squares sampling scheme, DF interpolation can be performed accurately and efficiently...

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 ...

Abstract not found

A number of imaging technologies reconstruct an image function from its Radon projection using the convolution backprojection method. The convolution is an O(N² log N ) algorithm, where the image consists of N x N pixels, while the backprojection is an O(N³) algorithm, thus constituting the major computational burden of the convolution backprojection method. An O(N² log N ) multilevel backprojection method is presented here. When implemented with a Fourier-domain postprocessing technique, also presented here, the resulting image quality is similar to or superior than the image quality of the classical backprojection technique.

2008 i A novel systematic framework of medical image analysis, weighted Fourier series (WFS) analysis is introduced. WFS is a combination of Fourier series and heat kernel smoothing. WFS effectively reduces the Gibbs phenomenon, improves the signal to noise ratio, and increases normality of the estimated errors in the WFS-based generalized linear models. In estimating the parameters of WFS, the least squares estimation of WFS has been widely used but it is computationally inefficient. To address the computational inefficiency in the least squares estimation, much faster but less accurate iterative residual fitting (IRF) method has been proposed. The proposed adaptive iterative regression (AIR) technique inherits the computational efficiency of IRF and improves accuracy of IRF. AIR partitions the function space into a set of subspaces, and performs an extra orthogonalization procedure to reduce the bias of IRF estimation. A complimentary tool, the fast weighted