AN ELLIPTICAL ORBIT BACKPROJECTION FILTERING ALGORITHM FOR SPECT

Transcription

1 1102 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 40, NO. 4, AUGUST 1993 AN ELLIPTICAL ORBIT BACKPROJECTION FILTERING ALGORITHM FOR SPECT Grant. T. Gullberg and Gengsheng L. Zeng, Department of Radiology, University of Utah, Salt Lake City, Utah Abstract An orbit geometry where the central projection ray of the detector is not constrained to pass through the center-of-rotation for all projection angles can be useful to reduce uniformity artifacts and to avoid truncation when imaging non centrally located organs. We present a backprojection filtering algorithm for parallel geometry with an elliptical orbit acquisition and verify the algorithm by computer simulations. We show that the point response function is shift invariant and has an analytical expression which is not the common l/r- but is a function of the orbit geometry. 1. INTRODUCTION In single photon emission computed tomography (SPECT), the resolution of the collimated detector deteriorates with increased distance from the face of the collimator. Thus, it is desirable to place the detector as close as possible to the patient to reduce the blurring caused by the distance-dependent system response function and to minimize loss of resolution. To accomplish this, it has been common practice by camera manufacturers to design noncircular orbits where the detector follows the body contour but with the central projection ray of the detector always passing through the center-of-rotation. This simplifies the reconstruction algorithm and maintains full body cross-sectional viewing for all projection angles. However, using orbits where the central projection ray is allowed to translate instead of remaining fixed to the center-of-rotation, can reduce uniformity artifacts. These orbits require the development of new algorithms that bring forth some interesting new insights into the mathematics of computed tomography. The effects of detector uniformity on image quality has been well understood in rotating gamma camera tomography [l]. For orbits where the central projection ray of the detector is constrained to pass through the center-of-rotation for all projection angles, uniformity variations in the rotating gamma camera will result in reconstructed ring artifacts in tomographic images. In most modern day gamma cameras these artifacts are reduced by nonlinearity and energy uniformity corrections performed in the camera electronics. In addition, flood images are acquired and stored in computer memory to numerically correct for any residual uniformity variations due to the collimator. We previously showed [ 11 that uniformity artifacts could also be reduced in reconstructed tomograms if the detector is allowed to follow an orbit where the central ray (in reference to the detector only) is not constrained to pass through the center-of-rotation. In our work, we demonstrated the reduction of uniformity artifacts for an elliptical orbit and other central-ray translating orbits. These orbits reduce the artifact amplitude by distributing the ring artifact over path lengths which are longer than those for orbits were the central ray is fixed to the center-of-rotation. Even though orbits are general body contouring curves in SPECT, we chose to investigate an elliptical orbit shown in Figure 1 to study algorithm requirements for central-ray translating orbits. A parallel-hole collimated detector rotates around the patient following an elliptical contour with the center of the detector always tangent to the ellipse. One can see from Figure 1 that the central projection ray of the camera is not fixed to the center-of-rotation. The development in this paper assumes that the ray from the center-of-rotation to the center of the detector rotates at equal increments of the angle p. As we will show, this assumption becomes important in deriving the backprojection filtering algorithm and projections acquired with this orbit cannot be reconstructed by the same filter function as for a circular orbit geometry. In this paper, we develop a backprojection filtering algorithm for parallel geometry to reconstruct data acquired from an elliptical orbit with equal angular projections in p (Figure 1). We first derive the point response function for the projection operation followed by a backprojection operation. We show that the point response function is shift invariant but is not equal to llr as it is for circular orbit geometry. In computer simulations, we first backproject the projection data, take the two-dimensional (2D) Fast Fourier Transform (FFT) of the backprojected image, multiply the transformed image by the 2D filter function (the reciprocal of the 2D FFT of the analytically derived point response function), and finally take the inverse 2D FFT to obtain the reconstructed image. It is shown that if the common ramp filter is applied to the backprojected image, artifacts appear. These artifacts are removed using the newly derived filter /93$ EEE -1 -

2 2.THEORY g(p) = rsin(+-6 P ) -rosin(+o-8p), (6) Consider a parallel-beam geometry and an elliptical orbit with A and B being the semi-axes in the x and y directions, respectively. The projections p(s, p) are P(S9 PI = Ilf(X,Y) 6[S-d(8 ) +xsin8 -yc0~8~]dxdy(1) where P P and p* is a single root in the interval (0,~) such that Using Eq. (2), we can rewrite g(p) as 1103 g(p*) = 0. (7) A2 sin p g(p) = (x-xxg) da4sin2p + B4cos2P B2 cos p + (Y -Yo),!A4sin2P + B4cos2P For a point source at 6 (x - xo) 6 (y - yo), the projection of the point source is p (S, PI = 6 (S - d (ep) - rosin ($o - ep). The backprojection is defined as x (3) b(r,$) = j pwp) +rsin(+-bp),p)dp. (4) 0 For elliptical geometry, the point response bo (r, $) of the projection operation followed by the backprojection operation for a point source at (ro, $o) is obtained by substituting Eq. (3) into Eq. (4): n bo (r, $1 = 16 [rsin (+ - ep) - rosin (q0-0 I I dp The derivative of g(p) with respect to beta is given by 4 2 (x-xo)a2b4cos~-(y-yo)a B sinp g'(p) = (9) (A4sin2p + B4cos2P) 3'4 Setting the right hand side of Eq. (8) to zero, we can show that at g(p*) = 0 We use the relationship in Eq. (10) to obtain expressions for sin@*) and cos(p*). Substituting these expressions into Eq. (9), we obtain the following expression for g' <P*> : where Using Eq. (1 1) to evaluate Eq. (5), we have center-of-rotation Figure 1. The scanning geometry. The detector rotates in an elliptical orbit with its face is always tangent to the ellipse. The detector is mounted with a parallel hole collimator.

3 1104 Equation (1 2) implies that the projection-backprojection operator has the following shift invariant point response function: A ~ B ~ h (x, y) = dm A4.r2 + B4y2 3. ALGORITHM Since the point response function of the projectorbackprojector is shift invariant, the image can be reconstructed by deconvolving h(x,y) from the backprojected image. Here, we propose the following hackprojection filtering algorithm: Backproject the projection measurements p (s, p) acquired from the elliptical orbit scanning, obtaining h(x, y) [Eq. (4)l. Take the two-dimensional FFT of h(x,y), obtaining B(u, 1)). Take the two-dimensional FFT of the point response function h(x,y) given by Eq. (13), obtaining H(u, 13). Compute the ratio of B(u,v) and H(u,\,), yielding F(u, v) = B(u, v) / H(u, 1)). (v) Take the inverse two-dimensional FFT of F(u, ti) to obtain the reconstructionf(x, y). From Eq.(13), we see that the factor on the right depends upon the equation of the elliptical orbit. When A = B, the point response function h(x, y) reduces to the well-known tomographic point response function (,~~+y~)- ~, which can be deconvolved by applying a twodimensional ramp filter to the backprojected image. We see from inspection of Eq.( 13) that the tomographic point response function (~~+y*)- ~ is modulated by the orbit geometry. 4. COMPUTER SIMULATIONS The backprojection filtering algorithm was verified by computer simulations for an elliptical orbit with parallel-beam geometry. The filter function h in Eq. (1 3) was sampled at discrete points and stored in a 256x256 array. Then we took the FFT of the 256x256 array of h. We divided the FFT of the backprojection image in (iv) by this result. Figure 2 shows the results of the computer simulation study. The elliptical orbit had axes A and B such that A = 2B = 120 (pixels). The number of projection views over 360 degrees were 256. The projections were backprojected into a 256x256 array before performing the 2D Fourier transform. The reconstruction was displayed as a 128x128 image. Figure 2. Computer simulations with parallel-beam geometry (128x128): (a) the ideal Shepp-Logan image generated by the computer, (b) the backprojection filtering reconstruction with a 2D ramp filter, and (c) the backprojection filtering reconstruction with the newly derived filter in Eq.( 13). -1

4 1105 Figure 2(a) is the ideal Shepp-Logan phantom with major axis of 11 8 and minor axis of 88 pixels. Figure 2(b) is the backprojection filtering reconstruction using a twodimensional ramp filter. Artifacts can be observed in the reconstruction. Figure 2(c) shows the backprojection filtering reconstruction with the newly developed twodimensional filter, which was calculated in Step (iii) in the algorithm outlined above. The results indicate that backprojection filtering works for elliptical orbits if the proper filter function is applied. 5. DISCUSSION In this paper, a backprojection filtering algorithm was presented for a parallel collimator geometry where the collimator traverses an elliptical orbit with the face of the collimator tangent to the ellipse. The point response function of the projector-backprojector was derived for parallel geometry and was shown to be shift invariant. The point response function was used to derive a filter for the backprojection filtering algorithm and was shown to give good results in computer simulation studies. Central-ray translating orbits would be useful in SPECT to reduce uniformity artifacts. However, a disadvantage is that to image the entire body these orbits would require a larger crystal area than that is presently being used on clinical SPECT systems and thus is not the most efficient use of the detector area. These orbits would be more useful for imaging non centrally located organs like the heart with converging collimators. Allowing the central ray to fix on the organ instead of the center-ofrotation would reduce the potential of projection truncation which is a common problem in imaging with converging collimators. To design central-ray translating orbits would require detector gantries which could either translate or rotate the central projection ray during detector rotation. These orbits cannot easily be implemented with present clinical SPECT systems, and to implement would require manufacturers to make major modifications to present detector gantries. This illustrates the importance of designing for a particular imaging application through integration of both hardware and algorithm concepts to obtain optimum image quality. Backprojection filtering algorithms are not necessarily the optimum reconstruction approach in computed tomography. Backprojection filtering is not commonly used because of the requirement to backproject into an array of twice the dimensions to prevent wraparound effects in the application of Fourier filtering. Also, the backprojection is truncated leaving some small fraction of the total integral outside the calculated backprojected image, which results in a small underestimation of the total counts. However, we point out that it is not clear that a filtered backprojection algorithm can be derived for the elliptical orbit tomography presented in this paper. There are other examples in computed tomography where a backprojection filtering algorithm exists but a filtered backprojection algorithm has not been derived [2,3]. Backprojection filtering algorithms have been proposed for several geometries including parallel [4,5], fan-beam [6], three-dimensional x-ray [7], and cone-beam geometries [8]. However, one example of where backprojection filtering will not work is in the reconstruction of exponential Radon projection data [9]. The projector-backprojector gives a space invariant kernel [lo]; however, the kernel diverges and cannot be deconvolved from the backprojected image. In SPECT, we see the application of central-ray translating orbits being used not with elliptical but applied to general body contouring orbits. We see from our analysis of elliptical scanning that the common l/r point response for circular orbits is modified by the geometry of the orbit. For general body contouring orbits, it may be difficult analytically to derive the point response function. Also, it is not clear at this time in our research whether in general the point response will be shift invariant. It is hoped that in most cases the point response would be shift invariant and that it could be calculated at least numerically by digitally projecting and backprojecting a point source. The FFT of the calculated point response would be obtained and the reciprocal of this result would be used as the filter function in Step (iii). Also, we point out that in applying any algorithm in SPECT it is important to consider attenuation. It is anticipated that the proposed algorithm would be applied either without attenuation correction, as is presently done in clinical cardiac SPECT, or correcting for attenuation using preprocessing [ 111 or postprocessing [ 121 attenuation correction techniques. Orbits where the central projection ray is not constrained to pass through the center-of-rotation are useful to reduce uniformity artifacts. This particular tomographic application introduces a new tomographic inverse problem. We showed that a backprojection filtering algorithm can be derived for parallel geometry with elliptical orbit scanning, but at this time we can only speculate that backprojection filtering would apply for parallel geometry with arbitrary body contouring orbits that are used in SPECT. The SPECT application presents

5 1106 an interesting inverse problem for which much of the mathematical formulation is still unsolved, especially for converging collimator geometries. ACKNOWLEDGMENTS The research work presented in this manuscript was partially supported by NIH Grant R01 HL 39792, the Whitaker Foundation, and Ohio Imaging of Picker Intemational. We also thank Biodynamics Research Unit, Mayo Foundation for use of the Analyze software package. Also, we thank Rolf Clack for his helpful suggestions. [lo] G. T. Gullberg and T. E Budinger, The use of filtering methods to compensate for constant attenuation in single-photon emission computed tomography, IEEE Trans. on Biomed. Eng., vol. BME- 28, pp , [l ] T. E Budinger, G. T. Gullberg, and R. H. Huesman, Emission Computed Tomography, in Topics in Applied Physics, Editor: G. T. Herman, Springer- Verlag, New York, pp , [ 1 ] L. T. Chang, A method for attenuation correction in radionuclide computed tomography, IEEE Trans. Nucl. Sci., vol. 25, pp , REFERENCES [l] G. T. Gullberg, An analytical approach to quantify uniformity artifacts for circular and noncircular detector motion in single photon emission computed tomography imaging, Med. Phys., vol. 14, pp ,1987. [2] H. Hu, G. T. Gullberg, R. A. Kruger, Convolutional reconstruction algorithm for fan beam with concave and convex circular detectors, IEEE Trans. on Med. Imag., vol. 7, pp , [3] R. V. Denton, B. Friedlander, and A. J. Rockmore, Direct three-dimensional image reconstruction from divergent rays, IEEE Trans. Nucl. Sci., vol. NS-26, pp , R. H. T. Bates and T. M. Peters, Towards improvements in tomography, New Zealand J. Sei., vol. 14, pp , T. F. Budinger and G. T. Gullberg, Reconstruction by two-dimensional filtering of simple superposition transverse-section image, in Technical Digest: Image Processing for 2-0 and 3-0 Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences, August 4-7, Opt. Soc. Amer., Washington, D.C., [6] G. T. Gullberg, The reconstruction of fan-beam data by filtering the back-projection, Computer Graphics and Image Processing, vol. 10, pp , [7] S. S. Orlov, Theory of three-dimensional reconstruction. 11. The recovery operator, Sov. Phys. Crystallogr., vol. 20, no. 4, pp , [SI E C. Peyrin, The generalized back projection theorem for cone beam reconstruction, IEEE Trans. onnucl. Sci., vol. NS-32, pp , [9] 0. J. Tretiak and C. Metz, The exponential Radon transform, SIAM J. Appl. Math., vol. 39, pp , 1980.