^{1,a)}and Norbert J. Pelc

^{2}

### Abstract

**Purpose:**

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. SPIE671, 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.

**Methods:**

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.

**Results:**

In the equal-angle case, the domain of the zero region is for which , where and are the frequency variables associated with the detector and view angular positions, respectively, is the radial support of the object, and 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.

**Conclusions:**

Interesting properties of the fan-beam sinogram exist and may be exploited in some applications.

The authors would like to thank Priti Balchandani, Rafe Mazzeo, Lei Zhu, and Akshay Nanduri for useful discussions. The authors thank Jiang Hsieh from GE Healthcare for providing the head CT data. Funding was provided by GE Healthcare and the Lucas Foundation.

I. INTRODUCTION

II. METHOD AND MATERIALS

II.A. Equal-angle case

II.B. Equal-spaced case

II.C. Simulation

II.D. Fan-beam double-wedge filter

II.E. Region of interest iterative reconstruction

III. RESULTS

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Computed tomography
- 37.0
- Fourier transforms
- 14.0
- Image reconstruction
- 10.0
- Medical image reconstruction
- 4.0
- Integral transforms
- 3.0

## Figures

Fan-beam CT geometry. Parallel beam, equal-angle, and equal-spaced fan-beam variables are shown.

Fan-beam CT geometry. Parallel beam, equal-angle, and equal-spaced fan-beam variables are shown.

2D Fourier space of equal-angle fan-beam sinogram for a 1 mm circular object at . (a) Magnitude image for (vertical axis) vs (horizontal axis). (b) Magnitude profiles at for theoretical, Bessel approximation, and simulated data. is in . (c) Magnitude profiles at .

2D Fourier space of equal-angle fan-beam sinogram for a 1 mm circular object at . (a) Magnitude image for (vertical axis) vs (horizontal axis). (b) Magnitude profiles at for theoretical, Bessel approximation, and simulated data. is in . (c) Magnitude profiles at .

2D Fourier space of equal-spaced fan-beam sinogram for a 1 mm circular object at . (a) Magnitude image for (vertical axis) vs (horizontal axis). (b) Magnitude profiles at for theoretical, Bessel approximation, and simulated data. is in . (c) Magnitude profiles at .

2D Fourier space of equal-spaced fan-beam sinogram for a 1 mm circular object at . (a) Magnitude image for (vertical axis) vs (horizontal axis). (b) Magnitude profiles at for theoretical, Bessel approximation, and simulated data. is in . (c) Magnitude profiles at .

Magnitude profiles at for the equal-angle fan-beam 2D Fourier transformed sinogram. is in . (a) (b)

Magnitude profiles at for the equal-angle fan-beam 2D Fourier transformed sinogram. is in . (a) (b)

Axial equal-angle fan-beam head CT data. (a) Sinogram data. (b) Image reconstructed using filtered backprojection. Grayscale color bar is in HU. (c) Magnitude image of 2D Fourier transform of sinogram.

Axial equal-angle fan-beam head CT data. (a) Sinogram data. (b) Image reconstructed using filtered backprojection. Grayscale color bar is in HU. (c) Magnitude image of 2D Fourier transform of sinogram.

Double-wedge filtered head CT data. (a) Magnitude image of the 2D Fourier transformed sinogram after the double-wedge filter has been applied. (b) Reconstructed image after double-wedge filter. (c) Difference from the original reconstruction, the circle marks a 50 mm radius from isocenter. Grayscale color bar is in HU.

Double-wedge filtered head CT data. (a) Magnitude image of the 2D Fourier transformed sinogram after the double-wedge filter has been applied. (b) Reconstructed image after double-wedge filter. (c) Difference from the original reconstruction, the circle marks a 50 mm radius from isocenter. Grayscale color bar is in HU.

SART algorithm used to estimate (a) the entire object using all of the original data (b) 60 mm radius region using the original data (c) 60 mm radius region using a 50 mm radius double-wedge filter on the sinogram data. All images truncated to the 50 mm radius ROI. Grayscale color bar is in HU.

SART algorithm used to estimate (a) the entire object using all of the original data (b) 60 mm radius region using the original data (c) 60 mm radius region using a 50 mm radius double-wedge filter on the sinogram data. All images truncated to the 50 mm radius ROI. Grayscale color bar is in HU.

## Tables

Simulation parameters for both the equal-angle and equal-spaced geometries.

Simulation parameters for both the equal-angle and equal-spaced geometries.

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