Article

# Fourier rebinning algorithm for inverse geometry CT

Department of Radiology, Stanford University, Stanford, California 94305, USA.

Medical Physics (Impact Factor: 2.64). 12/2008; 35(11):4857-62. DOI: 10.1118/1.2986155 Source: PubMed

**ABSTRACT**

Inverse geometry computed tomography (IGCT) is a new type of volumetric CT geometry that employs a large array of x-ray sources opposite a smaller detector array. Volumetric coverage and high isotropic resolution produce very large data sets and therefore require a computationally efficient three-dimensional reconstruction algorithm. The purpose of this work was to adapt and evaluate a fast algorithm based on Defrise's Fourier rebinning (FORE), originally developed for positron emission tomography. The results were compared with the average of FDK reconstructions from each source row. The FORE algorithm is an order of magnitude faster than the FDK-type method for the case of 11 source rows. In the center of the field-of-view both algorithms exhibited the same resolution and noise performance. FORE exhibited some resolution loss (and less noise) in the periphery of the field-of-view. FORE appears to be a fast and reasonably accurate reconstruction method for IGCT.

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Article: Fourier rebinning algorithm for inverse geometry CT

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**ABSTRACT:**Tibor Tot has given me an unusual opportunity via his request to summarize my work in computed tomography (CT), which indeed since 1977 has been directed toward eliminating the scourge of breast cancer via search and destroy of premetastasis tumors, instead of the predominant, century-old magic bullet approach (Strebhardt and Ullrich 2008). I have had the strange career of a theoretical biologist (Fig. 10.1) on a continent where theoretical biology as a paid discipline died with James F. Danielli, discoverer of the bilayer structure of the cell membrane, and once Director of the Center for Theoretical Biology at the State University of New York at Buffalo and founding editor of the Journal of Theoretical Biology (JTB) (Danielli 1961; Rosen 1985; Stein 1986). Danielli became part of my story, which I will tell in the spirit of the wonderful biography of Louis Pasteur written by his lifetime laboratory assistant (Duclaux 1920). Lacking such a long-term companion to my train of thought, this shall have to be unabashedly autobiographical, with all the risks attendant to that form of literature. I shall try to be honest to you, the reader, and true to myself. If we consider the vast gossamer of activity in science, and CT in particular, my own path is but one thread through that web, but the one I know best. Nevertheless, when I use “I,” please take it as shorthand for “I and my cited collaborators” where appropriate. My world line has crossed that of many others, who have enriched my journey, and made it possible. This includes George Gamow, Mr. World Line himself (Gamow 1970), both of us sitting in on a meteorology course in Boulder, Colorado about 1968, and later his son Igor at Woods Hole and Boulder regarding trying to model the growth of Phycomyces (Ortega et al. 1974). - [Show abstract] [Hide abstract]

**ABSTRACT:**P. R. Edholm, R. M. Lewitt, and B. Lindholm, "Novel properties of the Fourier decomposition of the sinogram," in Proceedings of the International Workshop on Physics and Engineering of Computerized Multidimensional Imaging and Processing [Proc. SPIE 671, 8-18 (1986)] described properties of a parallel beam projection sinogram with respect to its radial and angular frequencies. The purpose is to perform a similar derivation to arrive at corresponding properties of a fan-beam projection sinogram for both the equal-angle and equal-spaced detector sampling scenarios. One of the derived properties is an approximately zero-energy region in the two-dimensional Fourier transform of the full fan-beam sinogram. This region is in the form of a double-wedge, similar to the parallel beam case, but different in that it is asymmetric with respect to the frequency axes. The authors characterize this region for a point object and validate the derived properties in both a simulation and a head CT data set. The authors apply these results in an application using algebraic reconstruction. In the equal-angle case, the domain of the zero region is (q,k) for which / k/(k-q) / > R/L, where q and k are the frequency variables associated with the detector and view angular positions, respectively, R is the radial support of the object, and L is the source-to-isocenter distance. A filter was designed to retain only sinogram frequencies corresponding to a specified radial support. The filtered sinogram was used to reconstruct the same radial support of the head CT data. As an example application of this concept, the double-wedge filter was used to computationally improve region of interest iterative reconstruction. Interesting properties of the fan-beam sinogram exist and may be exploited in some applications. -
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**ABSTRACT:**This article considers the problem of reconstructing cone-beam computed tomography (CBCT) images from a set of undersampled and potentially noisy projection measurements. The authors cast the reconstruction as a compressed sensing problem based on l1 norm minimization constrained by statistically weighted least-squares of CBCT projection data. For accurate modeling, the noise characteristics of the CBCT projection data are used to determine the relative importance of each projection measurement. To solve the compressed sensing problem, the authors employ a method minimizing total-variation norm, satisfying a prespecified level of measurement consistency using a first-order method developed by Nesterov. The method converges fast to the optimal solution without excessive memory requirement, thanks to the method of iterative forward and back-projections. The performance of the proposed algorithm is demonstrated through a series of digital and experimental phantom studies. It is found a that high quality CBCT image can be reconstructed from undersampled and potentially noisy projection data by using the proposed method. Both sparse sampling and decreasing x-ray tube current (i.e., noisy projection data) lead to the reduction of radiation dose in CBCT imaging. It is demonstrated that compressed sensing outperforms the traditional algorithm when dealing with sparse, and potentially noisy, CBCT projection views.