Fan-Beam Reconstruction Algorithm for a Spatially Varying Focal Length Collimator

Transcription

1 IEEE TRANSACTIONS ON MEDICAL IMAGING. VOL. 12, NO. 3, SEPTEMBER Fan-Beam Reconstruction Algorithm for a Spatially Varying Focal Length Collimator Gengsheng L. Zeng, Member, IEEE, Grant T. Gullberg, Member, IEEE, Ronald J. Jaszczak, Senior Member, IEEE, and J. Li, Member, IEEE Abstract-Fan-beam collimators are used in single photon emission computed tomography to improve the sensitivity for imaging of small organs. The disadvantage of fan-beam collimation is the truncation of projection data surrounding the organ of interest or, in those cases of imaging large patients, of the organ itself producing reconstruction artifacts. A spatially varying focal length fan-beam collimator has been proposed to eliminate the truncation problem and to maintain good sensitivity for the organ of interest. The collimator is constructed so that the shortest focal lengths are located at the center of the collimator and the longest focal length is located at the periphery. The variation of the focal length can have various functional forms but in our work it is assumed to increase monotonically toward the edge of the collimator. We have derived a reconstruction algorithm for this type of fan-beam collimation. The algorithm is expressed as an inanite series of convolutions followed by one backprojection. However, simulations show that only a small number of N terms in the series are needed to obtain high quality reconstructions. The weighting and convolution are executed N times, then N convolved projections are summed up and one backprojection is performed to obtain the final reconstructed image. The algorithm was tested for two spatially varying focal length formulations. Through computer simulations it is found that if the focal length function is not smooth, a singular artifact is seen in the reconstruction, whereas for smooth functions, the reconstructions are free of artifacts. I. ~ODUCTION In single photon emission computed tomography (SPECT), a fan-beam collimator is used with a large field-of-view rotating gamma camera to improve sensitivity for imaging small organs like the brain and heart [ 11-[3]. The magnification of the fan-beam geometry can place the radiopharmaceutical distribution in some of the tissue odtside of the field-of-view. In most cases this tissue does not contain the region of interest but does present a low background which when projected is truncated. The application of the reconstruction filter to these truncated projections produces spikes at the truncation edge in the filtered projections, resulting in ring artifacts in the reconstruction [3]. To resolve the truncation problem, it has been proposed to use a spatially varying focal length fan-beam collimator [4], [5]. The central region of interest is imaged with short Manuscript received August 4, 1992; revised February 11, G. L. Zeng and G. T. Gullberg are with the Department of Radiology, University of Utah, Salt Lake City, UT R. J. Jaszczak and J. Li are with the Department of Radiology, Duke University, Durham, NC This work was partially supported by the Whitaker Foundation, NIH Grant R01 HL 39792, NIH RO1 CA 33541, and DOE DE-FG05-91ER IEEE Log Number focal lengths. The focal lengths increase from a minimum at the center to a maximum at the periphery (edge) of the collimator. Thus, the projection rays converge to focal points of focal lengths that increase as the position of the projection ray increases in distance away from the center of the collimator so that tissues at the edge of the body are imaged with nearly parallel rays. The central region of interest is imaged with good sensitivity while at the same time the truncation problem is reduced at the edge of the patient. The spatial variation of the focal length can have various functional forms but as we will show the functional form is important when considering the type of algorithm that is used. As in all tomographic reconstruction problems, it is desirable for computational efficiency to use a convolution backprojection reconstruction algorithm similar to those that have been developed for parallel [6] and various fan-beam geometries [7]-[lo]. Unfortunately, as we will show, a convolution backprojection algorithm cannot be derived for the spatially varying focal length fan-beam geometry. One approach would be to use iterative reconstruction algorithms [ 113 which reconstruct images for this projection geometry; however, this is computationally more time consuming than convolution backprojection algorithms. Another approach would be to use a rebinning method [12] to convert the spatially varying focal length fan-beam projection data into parallel-beam projection data, and then use the parallel-beam convolution backprojection algorithm [6] to reconstruct the image. However, the rebinning method introduces interpolation errors. Still another method would be to backproject the projections and then filter them as has been done for conventional fan-beam geometries [13]. It can be shown that a space invariant point response of the backprojection operation exists for spatially varying focal length geometries [14], although backprojection filtering does require a little more memory than convolution backprojection methods. In this paper, we derive a convolution backprojection algorithm for a spatially varying focal length fan-beam collimator without rebinning. Our approach is similar to the one developed in [15]. In the general case, the algorithm involves an infinite series. However, only a small number of N terms are needed when implemented. The weighting and convolution are executed N times, then N convolved projections are summed up and one backprojection is performed to obtain the final reconstructed image. The algorithm has been tested for two spatially varying formulations /93$ IEEE

2 516 Fig. 1. A spatially varying focal length fan-beam geometry. The focal length D(s) increases as (SI increases. The distance from any point in the object to the collimator is always less than a for each projection angle and D(s) is greater than a. n. THEORY A. Spatially Varying Focal Length Fan-Beam Geometry The geometry for a spatially varying focal length fan-beam collimator is shown in Fig. 1. The minimum focal length (the distance from the focal point to the imaging plane) is denoted as a. From a minimum a at the center of the collimator, the focal length varies as a function D(s) where s is the distance from the projection point to the origin which is the perpendicular projection of all focal points on the detector. The variable s increases toward the edge of the collimator, and has negative and positive values. For tomographic application, the center of rotation is assumed to be a distance R away from the image plane, and lies along the perpendicular line of projection of all focal points. The minimum focal length a is determined so that the distance from any part of the object (patient) to the image plane is always less than a as the detector rotates. If a is too short and lies inside the object, a double image is produced on the detector for those distributions located with distances longer than a, and a single image is produced on the detector for those distributions located with distances shorter than a. To handle this type of projection data, the reconstruction algorithm is more complicated. Therefore, we assume that the minimum focal length a is longer than the distance from any part of the object to the detector. In this paper, we consider two types of spatially varying focal length fan-beam geometries: 1) D(s) = a + Iclsl, and 2) D(s) = a + ICs2, where D is the focal length as a function of the location s along the collimator, measured from the central axis as shown in Fig. 1. The minimal focal length a and parameter k are positive constants. Fig. 2. Geometry relating the fan-beam variables (8, p) to parallel-beam variables (t, e). The focal length for the projection point s is D(s). B. A General Convolution Backpmjection Algorithm A fan-beam geometry is shown in Fig. 2. The fan-beam geometry variables s and /3 are related to the parallel geometry variables t and 6 by D(s) - R t=s " J e = p + tan-' (k) where R is the distance from the center of rotation to the detector. For the parallel geometry, the object can be reconstructed via the filtered backprojection algorithm [61 f(r, 4) = 1" JF, pe(t)h[r cos(+ - e) - (3) where pe(t) is the parallel projection data, f(r, 4) is the reconstructed image in polar coordinates, and the convolver h is the inverse Fourier transform of the truncated ramp filter with the property that 1 h(at) = -h(t). (4) a2 Let Rp(s) be the fan-beam projection data measured by a spatially varying focal length collimator. We use (1) and (2) to change the variables in (3) to obtain the fan-beam reconstruction formula: 4 r 2 ~ roo

3 ZENG e? al.: FAN-BEAM RECONSTRUCTION ALGORITHM where IJI is the Jacobian defined as I ae at I lap apj D3(4 - D2(s)R + s3dys) + srd(s)d (s) - [D2(s) + s2]3/2 (6) and D (s) is the derivative of the variable focal length D(s). We have removed the absolute value operator in (6) because we are only concerned with focal length formulations D(s) for which the numerator is greater than zero. This ensures that the transformation in (1) and (2) is one- to-one. In order to derive a convolution backprojection algorithm, we first simplify the expression for the argument of h in (5): Fig. 3. Fan-beam geometry illustrating the location of the projection point 2 on the detector. Here, 2 is the projection of the point (T, 4) in the image space. where + rs sin (4 - P) - sd(s) + SRI. (8) Using eqs. (4), (6), and (8), eq. (5) can be rewritten as 00 D3(s) - D2(s)R + s3d (s) + srd(s)d (s) dsdp* (9) d- Equation (9) is still not a convolution backprojection algorithm, even though it has been used for reconstruction [16]. Continuing, we factor the argument of h as the product of a function g and (5 - s) in the following fashion: TD(s) cos (4- P) + rssin (4- P) - sd(s) + sr = (5 - s)g(s, T, d - P) (10) where the variable 5 is the projection of the point (r, 4) onto the detector at the projection angle /3 (see Fig. 3), and depends only on r, 4, and P. Here, 5 is the solution to the following equation: -- S rcos(d -PI (11) D( i) - D( 5) - R - T sin (4- P). For most functions of D(s), i is not easy to solve. From (4) and (lo), (9) can be written as W(s) = and 1 g2(s, r, d - P) D3(s) - D2(s)R + s3d (9) + srd(s)d (s) (13) 2 4 q q T S For parallel-beam geometry, D(s) --+ CO,% (14) + l/2, and _gz(,) + 1. For this case, (12) is identical to the classical convofution backprojection algorithm for parallel-beam geonietry. For fan-beam geometry, D(s) = a, % = e,. Then, (12) is identical to the algorithm for fan-beam geometry. In order to obtain a convolution backprojection algorithm, we need to separate l/g2 into terms, each term being a product of a function of s and a function of (r, q5 - P). If we can find an expansion of 1/g2 in the following form: then (12) can be rewritten as a backprojection of an infinite sum of convolutions:

4 + 22M. 578 EEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 12, NO. 3. SEPTEMBER 1993 W(s) = (a + - (a + Ks2) R + s3(2ks) + sr(a + ks2)(2ks) 2d(a + ks2)2 + s2 I, m [d,(~)w(~)rp(~)lh(i - ~ )d~d/3. (16) Implementation of this algorithm requires only a finite number of terms. Also note that in (14), S - s is required in the numerator of the function to be expanded, in order to cancel the simple pole of the expression in the denominator at s = Ei. This ensures that the expansion in (15) exists. C. The Chebyshev Expansion In this section we give the expansion (15) using the Chebyshev polynomials [17], Tn(z), of the first type, where TO(.) = l,tl(x) = x, and Tn+l(z) = 2zTn(x) -Tn-l(x) for n > 0. The representation of a function f(z) by a series of Chebyshev polynomials has the following advantages over the regular power series: 1) The convergence is much more rapid which is the basic theorem proved by Chebyshev [17]. 2) The range of convergence is easier to determine. 3) A more compact representation is possible using telescoping series. 4) A minimax approximation is approached. Here, a telescoping series is an expansion in the form of 00 n=o ~ n ~ ~ T ~ ~ ~ ( z ) for some integer m. (17) Also, by a minimax approximation, we mean to find c; is the solution to For the expansion M n=o N which 1. (18) the coefficients c,(n # 0) can be calculated by using the orthogonality of the Chebyshev polynomials J -1 c, = - f(z)mdx, Tn (x) n = 1,2,3,.... (20) The coefficient CO is equal to one-half of the expression on the right. Unfortunately, there are no closed-form expression of c, for a general function f(x) defined on [-1, 13. In this case, one can use the Gaussian-Chebyshev quadrature formula: 7r - (2 M)! f2m(r) - 1 < -y < 1 (21) to evaluate the integral [18]. Here, cos * T are the zeros of TM(x). When (21) is used to approximate the integral, the first M terms in (21) are used and the last term gives the approximation error. The error depends significantly upon the 2Mth order derivative of the function f(x). The number of terms M is chosen such that the approximation error is small. In our computer simulations, M is chosen as an integer a little greater than N, the number of terms in the Chebyshev expansion. From (20), (21), and the fact that T (cos8) = cosn8, we have n = 1,2,3,.... (22) The coefficient q, is equal to one-half of the expression on the right. Notice that the Chebyshev series converges on [ - 1,1], while our projection data Rp (s) is measured on [ - L, L] for some L such that Rp(s) = 0 as Is1 > L. Therefore, let x = s/l and and the expansion (15) is readily obtained by (22) and d,(s) = T,(s/L). (24) D. Convolution Backprojection Algorithm For D(s) = a+ ks2 When D(s) = a + ICs2, it can be verified from (10) and (11) that with g(s, r, 4 -,L?) = ICs2 + As + B A=kS-kr~0~(4-/3), arcos 4-p) B={ if S # 0, a-r-r ifi=o (25) and (27) [see top of page]. In (26), S is the projection of the point (r,4) onto the detector as shown in Fig. 3, and can be evaluated by solving (1 1): S = /-+ /-+! (28) withq = - $ + e z - a ~ t = -. +e w = 18k 2k 7 3 9k 1 R+rsin(+-p), and z = rcos(4-p). The Chebyshev expansion (14) for 1/(ks2 + As + B)2 is found according to the steps described in Section II-C, giving the following convolution backprojection algorithm:

5 ~ -b ZENG et al.: FAN-BEAM RECONSTRUCTION ALGORITHM 579 g(3,7-,4 - P) = -kslsl+ s[r - a + rsin(4 - p)] + Jslrkcos(+ - p) + arcos(4 - p) k(s( - ksign(s)[e - rcos(4 - p)] + [a - R - rsin(4 - p)] sign(s) # sign(;) S-S (31) sign( s) = sign( i) where n= 1,2,3,..-,N (30) and CO is equal to one-half of the expression on the right when n = 0. E. Convolution Backprojection Algorithm For D(s) = a+ kls1 When D(s) = a + k)s(, (31) applies [see top of page] and (U + kl~))~ - (a - + kl~1)~b + s2k)s) W S ) 2J(a + kls1) Rk(a + k)s))jsj (32) In (31), E is the projection of the point (r,4) onto the detector as shown in Fig. 3, and can be evaluated by solving (11): 3= + Jb2 + 4 k ~ J COS (4- p)) %sign[cos (4- p)] (33) withb= a-r-rsin(c,b-p)-kr(cos(4-p)(.thechebyshev expansion (14) for this situation is found according to the steps described in Section II-C, giving the following convolution backprojection algorithm: where (34) n= 1,2,3,...,N (35) and CO is equal to one-half of the expression on the right when n = EVALUATION USING COMPUTER SIMULATIONS A. Methods A two-dimensional Shepp-Logan head phantom [6] was used in our computer simulations. The computer-generated phantom is shown in Fig. 4. Images were reconstructed in 128 by 128 pixel (25 cm by 25 cm) arrays. The number of projection angles was 128 over 360O. The distance R from the Fig. 4. The computer-generated Shepp-Logan head phantom. center of rotation to the detector was 64 pixels (12.5 cm). The detector size was 160 bins (31.25 cm). For the simulation with D(s) = a+ ICs2, we used IC = 0.03 and the minimum focal length a = 170 pixels (33.2 cm). For the simulation with D(s) = a + kls), we used IC = 0.8 and the minimum focal length a = 250 pixels (48.83 cm). These parameters were chosen so that the projection data were not truncated. The reconstruction algorithms are given in (30) and (34). In the computer simulations, we let L = 80.5, M = 20 for D(s) = a + ks2, and M = 40 for D(s) = a + klsi. Reconstructions were computed for expansions with different number of terms N. To implement the algorithm eq. (29), one needs to 1) pre-weight the projection data Rp(s) by Tn(s/L)W(s) for each n, 2) convolve the pre-weighted projection data with h: 0 k even (36) 3) calculate the Chebyshev coefficients c, eq. (30), 4) weight the convolved data by cn, 5) sum up the data from step 4 for all n, and 6) backproject the summed data to obtain the image. The algorithm, including the projection data generation routine, was coded in Fortran 77 by modifying the Donner Laboratory software package [I91 on a SUN SPARC-I1 computer. In forming the projections, no discretization of the phantom was made, but the projections were formed from the summation of line integrals of ellipses that composed the phantom. The effects of attenuation, scatter, and geometric point response were not included. In order to compare this method with the rebinning method, the rebinning method was also implemented. The projection data from the varying focal length collimator were first rebinned into parallel-beam data, and the standard parallelbeam convolution backprojection algorithm was applied to reconstruct the image.

6 . 580 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 12, NO. 3, SEPTEMBER w ; lo: " ! o (C) (d) (C) (d) Fig. 6. Horizontal profiles across the center of the reconstructed images in Fig. 5. Profile (-) is compared with the ideal profile (...). The orders used in the expansions were (a) N = 0, (b) N = 5, (c) N = 10, and (d) N = 15. B. Results (a) (b) Fig. 7. (a) Reconstructed image using the rebinning method for the spatially varying focal length function D( s) = a + ks2. (b) The reconstruction profile (-) is compared with the ideal profile (...). Fig. 5. Reconstructed images using the spatially varying focal length function D(s) = a + ks2. The orders used in the expansions were (a) N = 0, (b)n = 5, (c) N = 10, and (d) N = 15. Figure 5 shows the reconstructed images for D(s) = a+lcs2 with four expansions of different orders N. In Fig. 5, we have N = 0 (Fig. 5(a)), N = 5 (Fig. 5(b)), N = 10 (Fig. 5(c)), and N = 15 (Fig. 5(d)). Since we did not have a closed-form solution for the Chebyshev coefficients c, (30), we computed them numerically. Therefore the computation time was rather long. It took 3 minutes for the N = 0 case, 18 minutes for N = 5, 31 minutes for N = 10, and 43 minutes for N = 15. The reconstruction appears to have good image quality for N = 10. In order to illustrate the convergence of the Chebyshev expansion, a horizontal profile across the center of each reconstructed image was plotted, and compared with the ideal profile (Fig. 6). In order to compare with the rebinning method, this phantom was also reconstructed via the rebinning method for D(s) = a + ICs2. The reconstructed image is shown in Fig. 7(a). Comparing it with Fig. 5(d), we observed that resolution of the image is degraded by the rebinning step, if we look closely at the three little points at the lower part of the phantom. We also observe some faint ring artifacts in Fig. 7(a). A central profile is shown in Fig. 7(b) for the image reconstructed by the rebinning method. Figure 8 shows the reconstructed images for D(s) = a + klsl : N = 0 (Fig. 8(a)), N = 10 (Fig. 8(b)), N = 20 (Fig. 8(c)), and N = 30 (Fig. 8(d)). For this geometry, the Chebyshev series converged very slowly, and little differences are seen between the image with N = 10 and the image with N = 30, due to the fact that the function D(s) = a+iclsl is not I) 2oo.o m I 1 smooth at s = 0. Even when N is as large as 30, we still can observe a dark spot at the center of the image. In Figs. 9 and 10, 1/g' is plotted for D(s) = a + klsl and D(s) = a + ICs'. In the case of D(s) = a + Ic(s(, the curve is not smooth at s = 0. For this case, it is very difficult to find an expansion which converges at s = 0. On the other hand, for the case D(s) = a + ICs2, the curve is smooth everywhere, and the expansion converges everywhere. IV. DISCUSSION Collimators with a spatially varying focal point have been proposed in SPECT to improve the sensitivity over that of parallel-hole collimators for imaging small organs and to

7 40.0 ZENG et al.: FAN-BEAM RECONSTRUCTION AL.GORITHM I o.oooooojo ' I ' ptxel, Fig. 9. Plot of 1/g2 for D(s) = a + k(sl with a = 250 and k = (C) (4 Fig. 8. Reconstructed images using the spatially varying focal length function D(s) = a + IEJsI. The orders used in the expansions were (a) N = 0, (b) N = 10, (c) N = 20, and (d) N = 30. I k' o~oozo- i \ reduce the truncation problem suffered by higher sensitivity fan-beam collimators, especially when applied to image the heart. For this particular geometry, we were not able to derive a conventional convolution backprojection algorithm. Instead, in this paper we arrived at a reconstruction algorithm in the form of a backprojection of an infinite series of convolutions. It was demonstrated that for the spatially varying focal length function D(s) [Fig. 13 which has a singularity at s = 0, an artifact, seen as a dark hole, appears at the center of rotation. To arrive at an algorithm that would use convolutions before backprojection, we had to expand the function l/g2 [eq. (14)] in an infinite series of orthogonal Chebyshev polynomials. The implementation of the algorithm requires only a finite number of terms to obtain good image quality. In selecting the orthogonal expansion, it is desirable to choose a set of orthogonal functions that would require the smallest number of terms and still obtain an accurate reconstruction. The Chebyshev polynomials were chosen because they form the fastest converging orthogonal expansion for the minimax optimization problem [ 171. The reconstruction times were long because the coefficients of the Chebyshev expansion had to be evaluated using a series of M terms. If l/g2 is evaluated with a series of N Chebyshev polynomials, the first N convolutions are performed on the projection data. Then a series of N terms have to be summed before performing the backprojection operation. In order to reduce the reconstruction time, it is better to have expansions with coefficients that can be evaluated efficiently. Another way to reduce the reconstruction time is to pre-calculate the Fig o.ooooo,o pd\ Plot of l/g2 for D(s) = a + ks2 with a = 170 and k = Chebyshev coefficients c, and store them in a file, so that the on-line reconstruction time can be reduced to at most N times that of the reconstruction time required by the classical convolution backprojection algorithm. For some cases, other orthogonal functions may have an advantage over using a Chebyshev expansion. Orthogonal functions like the Legendre polynomials, require a larger number of terms in the expansion to obtain the same accuracy as the Chebyshev expansion of fewer terms. However, for the Chebyshev expansion the coefficients have to be evaluated numerically, using a fairly complicated series which lengthens the computation time. If a polynomial expansion can be devised with coefficients that have a closed form expression, it could be computationally more efficient than using Chebyshev polynomials. In fact, for the quadratic focal length function [D(s) = a + ks2] the Legendre coefficients have closed-form expressions but are fairly complicated [20]. A comparison of reconstruction times between an expansion using Chebyshev and one using Legendre polynomials for the quadratic case has not yet been evaluated. Another possibility is to use the Taylor series expansion which has an efficient closed-form expression to evaluate the coefficients. The disadvantage of the Taylor series expansion is that its range of convergence changes from point to point and usually does not cover the whole range of the projection data.

8 582 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 12, NO. 3, SEPTEMBER 1993 In this paper, we use a truncated polynomial expansion of 1/g2 to approximate l/g2. The expansion decouples the variable s from other parameters so that the algorithm can be written as a summed convolution backprojection. The rate of convergence does not depend upon the projection data and the projection data array size, but depends upon the smoothness of the function l/g2, and, in tum, depends upon the smoothness of the focal length function D(s). If the function D(s) is smooth for every s, only a small number of terms are required to approximate l/g2. On the other hand, if D(s) is not smooth at a particular value s, the expansion will not converge and artifacts will appear in the reconstructed image. A drawback of our algorithm is the presence of a reconstruction artifact at the center of rotation for geometries with nonsmooth focal length functions D(s). Comparing the two examples in the text, the quadratic varying focal length function is smooth for all s giving a smooth 1/g2. the other hand, the linear varying focal length function is not smooth at s = 0 and neither is 1/g2. In this case, the derivative of l/g2 does not exist at s = 0. Therefore, the expansion of 1/g2 which requires derivatives does not converge at s = 0, while for the quadratic case it does. For geometries with smooth focal length function D(s), the algorithm gives good results if Chebyshev polynomials are used to expand 1/g2, but the convergence rate depends upon the rate of change of the focal length function. Computer simulations were performed for the quadratic function [D(s) = a + ks2] with larger values for IC than those used in simulations presented in this paper. Keeping the variable a the same, larger values of IC cause the focal length function to vary more rapidly over the imaging area. When the parameter k is increased by ten times, the number of convolution terms N had to be doubled to obtain the same image quality in the reconstruction. The question conceming the variable focal length fan-beam geometry is whether a traditional convolution backprojection algorithm can be derived. Other geometries, like that found with image intensifier CT [21], have come across this same problem. It may be that other forms of the filter function will allow the reconstruction algorithm to reduce to the conventional algorithm of one convolution per projection followed by a backprojection operation. It is interesting to observe that other geometries such as a cone-beam geometry that is sampled on a sphere also does not reduce to a convolution backprojection type algorithm [22]. Even if convolution backprojection algorithms do not exists, we are finding more often than not that backprojection filtering algorithms do apply for several of these geometries. For the cone-beam sampling on a sphere, a backprojection filtering algorithm has been derived [23]. We have shown that for variable focal length fan-beam geometries a backprojection filtering algorithm also applies [14]. This is an area of future research that may develop more accurate and more efficient tomographic reconstruction algorithms for some very interesting tomographic geometries. ACKNOWLEDGMENT We would like to thank Biodynamics Research Unit, Mayo Foundation, for use of the Analyze software package. REFERENCES R. J. Jaszczak, L.-T. Chang, and P. H. Murphy, Single photon emission computed tomography using multi-slice fan-beam collimator, IEEE Trans. Nucl. Sci., vol. NS-26, pp , B. M. W. Tsui, G. T. Gullberg, E. R. Edgerton, D. R. Gilland, J.R. Perry, and W.H. McCarmey, Design and clinical utility of a fan beam collimator for SPECT imagine -- of the head. J. Nucl. Med, vol. 27. DD , G. T. Gullbere. -. G. L. %ne. F. L. Datz. P. E. Christian, C.-H. Tung. I, and H. T. Morgan, Review of convergent beam tomography in single photon emission computed tomography, Phys. Med. Biol., vol. 37, pp , J. Hsieh, Scintillation camera and multifocal fan-beam collimator used therein, United States Patent 4,823,017, Apr. 18, R. J. Jaszczak, J. Li, H. Wang, and R. E. Coleman, Three-dimensional SPECT reconstruction of combined cone beam and parallel beam data, Phys. Med. Biol., vol. 37, pp , L. A. Shepp and B. F. Logan, The Fourier reconstruction of a head section, IEEE Trans. Nucl. Sci., vol. NS-21, pp. 2143, G. T. Herman and A. Naparstek, Fast image reconstruction based on a Radon inversion formula appropriate for rapidly collected data, SIAM J. Appl. Math, vol. 33, pp , B. K. P. Hom, Fan beam reconstruction methods, IEEE Proc., vol. 67. pp G. T. Gullberg, C. R. Crawford, and B. M. W. Tsui, Reconstruction algorithm for fan beam with a displaced center-of-rotation, IEEE Trans. Med. Imag., vol. MI-5, pp , C. R. Crawford and G. T. Gullberg, Reconstruction for fan beam with an angular-dependent displaced center-of rotation, Med. Phys., vol. 15, pp , R. J. Jaszczak, J. Li, H. Wang, and R. E. Coleman, SPECT collimation having spatially variant focusing (SVF), J. Nucl. Med., vol. 33, p. 891, (abstract) P. bike and D. P. Boyd, Convolution reconstruction of fan-beam projections, Comp. Graph. and Imag. Proc., vol. 5, pp , G. T. Gullberg, The reconstruction of fan-beam data by filtering the back-projection, Comp. Graph. and hag. Proc., vol. 10, pp , G. L. Zeng and G. T. Gullberg, Fan-beam backprojection filtering algorithm for a spatially varying focal length collimator, submitted to IEEE Trans. Med Imag. H. Hui, G. T. Gullberg, and R. A. Kruger, Convolutional reconstruction algorithm for fan beam concave and convex circular detectors, IEEE Trans. Med. Imag., vol. MI-7, pp , P. C. Hawman, J. Hsieh, and B. E. Hasselquist, The cardiofocal collimator: a novel focussing collimator for cardiac SPECT, J. Nucl. Med., vol. 33, p. 852, (abstract) G. Men, Mathematical Methods for Physicists, 3rd ed. San Diego, CA: Academic Press, V. I. Krylov (translated by A. H. Stroud), Approximate Calculation of Integrals New York: Macmillan, R. H. Huesman, G. T. Gullberg, W. L. Greenberg, and T. F. Budinger, Users Manual: Donner Algorithms for Reconstruction Tomography, Lawrence Berkeley Laboratory, University of California, C. D. Hodgman, C.R.C. Standard Mathematical Tables (12th ed. Cleveland, OH Chemical Rubber Pub., R. N. hot, R. J. Willeters, J. R. Battem, and J. S. Orr, Investigations using an x-ray image intensifier and a TV camera for imaging transverse sections in humans, Er. J. Rudiol., vol. 57, pp , R. V. Denton, B. Friedlander, and A. J. Rockmore, Direct threedimensional image reconstruction from divergent rays, IEEE Trans. Nucl. Sei., vol. 26, pp , F. C. Peyrin, The generalized back projection theorem for cone beam reconstruction, IEEE Trans. Nucl. Sci., vol. 32, no. 4, pp , 1985.