THE FAN-BEAM scan for rapid data acquisition has

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1 190 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 2, FEBRUARY 2007 Hilbert Transform Based FBP Algorithm for Fan-Beam CT Full Partial Scans Jiangsheng You*, Member, IEEE, Gengsheng L. Zeng, Senior Member, IEEE Abstract This paper presents a new type of filtered backprojection (FBP) algorithm for fan-beam full- partial-scans. The filtering is shift-invariant with respect to the angular variable. The backprojection does not include position-dependent weights through the Hilbert transform the one-dimensional transformation between the fan- parallel-beam coordinates. The strong symmetry of the filtered projections directly leads to an exact reconstruction for partial data. The use of the Hilbert transform avoids the approximation introduced by the nonuniform cutoff frequency required in the ramp filter-based FBP algorithm. Variance analysis indicates that the algorithm might lead to a better uniformity of resolution noise in the reconstructed image. Numerical simulations are provided to evaluate the algorithm with noise-free noisy projections. Our simulation results indicate that the algorithm does have better stability over the ramp-filter-based FBP circular harmonic reconstruction algorithms. This may help improve the image quality for in place computed tomography scanners with single-row detectors. Index Terms Filtered backprojection (FBP) algorithm, Hilbert transform, partial scan. I. INTRODUCTION THE FAN-BEAM scan for rapid data acquisition has been widely adopted in commercial computed tomography (CT) for years. The most common image reconstruction method is the ramp filter-based filtered backprojection (rfbp) algorithm from [1], [2]. The rfbp algorithm is currently the stard method in CT scanners with single-row X-ray detectors. In all FBP-type algorithms of [1] [5], position-dependent weights are required in the backprojection the region close to the X-ray source trajectory is observed to suffer from intolerable errors due to the reduced cutoff frequency [6]. The position-dependent weights also cause nonuniformity of resolution noise in the reconstructed image [7]. In order to address these issues, two methods were suggested in [6], in which one is the circular harmonic reconstruction (CHR) algorithm that was evaluated earlier in [8] the other is the exact rebinning algorithm [6]. Recently, more investigations on the analysis treatment of nonuniform resolution Manuscript received August 31, 2006; revised November 2, 2006 The work of J. You was supported in part by the Chinese National Science Foundation under Grant in part by the Chinese government 973 program under Grant 2006CB during his visit to Beijing University. The work of G. L. Zeng was supported by the National Institutes of Health under Grant CA Grant EB Asterisk indicates corresponding author. *J. You is with Cubic Imaging, Auburndale, MA USA. He was with the Department of Radiology, State University of New York, Stony Brook, NY USA ( jyou@cubic-imaging.com). G. L. Zeng is with the Department of Radiology, University of Utah, Salt Lake City, UT USA ( larry@ucair.med.utah.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TMI noise were reported in [9] [10] using the rfbp algorithm, the shift-variant filtering methods were then suggested. For the fan-beam short-scan, one of the major tasks in [11] [15] was to find smooth weights for the redundant projection data in order to reduce the streak artifacts that result from using the rfbp algorithm, in which position-dependent weights in the backprojection always cause nonuniformity of resolution noise. The position-dependent weights also exist in [16]. The main goal of this paper is to derive a new type of FBP algorithm to avoid position-dependent weights in the backprojection by using the Hilbert transform, which was previously used for fan-beam data reconstruction in [4] [17] [19]. In this paper, by using the Hilbert transform one-dimensional (1-D) coordinate transformation between the parallel-beam fan-beam coordinates, we have derived a new type of FBP algorithm (15), which is expressed as (21) for the full scan (26) for the partial scan. The new algorithm does not need position-dependent weights in the backprojection so that the cutoff frequency related nonuniformity issue in [7] is avoided. Actually, under the fan-beam coordinates, the cutoff frequency is uniform in the new algorithm the filtering is shift-invariant with respect to the equiangular projection rays. The new algorithm has the symmetry property (25) similar to that in the parallel-beam FBP algorithm, which easily leads to a mathematically exact stable reconstruction formula (26) for fan-beam partial scans. Variance analysis (39) suggests that the new algorithm could generate a better uniformity of resolution noise compared with the rfbp algorithm. Simulation results have revealed noticeable advantages of the new algorithm over the rfbp CHR algorithms when the projection noise becomes severe the object is close to the X-ray source. II. REVIEW OF PARALLEL-BEAM FBP ALGORITHM Let denote the two-dimensional (2-D) Euclidean space with point representation in the Cartesian coordinates in the polar coordinates. Define as the Cartesian coordinates after rotating by an angle counterclockwise (see Fig. 1 for a typical parallel-beam CT scanning geometry). These two coordinates satisfy the following relations: In CT imaging, function of represents the distribution of X-ray linear attenuation coefficients. Let denote the same function in (1) /$ IEEE

2 YOU AND ZENG: HILBERT TRANSFORM BASED FBP ALGORITHM FOR FAN-BEAM CT FULL AND PARTIAL SCANS 191 Fig. 1. Illustration of the parallel-beam coordinates projections. the coordinates. The Radon transform of is defined as Using the Fourier slice theorem, the FBP algorithm of [20] for the parallel-beam scan can be expressed as (2) Fig. 2. Each projection ray is uniquely determined by (; ). Points O S st for the coordinate origin one X-ray point source, respectively. Point P is an arbitrary reconstruction point. Here denotes the angle between OS the projection ray, denotes the angle between OS PS, denotes the angle between OS the y-axis, D is the radius of the circular X-ray source orbit. fan-beam coordinates, respectively. The coordinate transformation between is given by (5) Hereafter, denotes the Fourier transform of with respect to the first variable. There have been many convolution-style expressions of (3) in the literature. In this paper, we will use the following expression: (3) The transformation Jacobian from to is. Let, hereafter denotes the maximum fan-subtending angle. The difference of (4) in fan-beam coordinates can be calculated by (6) where (7) (4) Here, is the ramp filter pv sts for the principalvalued integral. Notice that the first line of (4) corresponds to the rfbp algorithm the second line corresponds to the Hilbert transform based FBP algorithm. Our fan-beam reconstruction formula (15) will be derived from the second line of (4). III. DERIVATION OF THE HILBERT TRANSFORM BASED FBP ALGORITHM Let denote the projections from the fan-beam scan as illustrated in Fig. 2 with a circular X-ray source orbit equiangular detector array. In the following, the convention in [21] for fan-beam coordinates geometry parameters will be used throughout this paper, the equiangular sampling is always assumed. As illustrated in Fig. 2, a reconstruction point P is expressed as in the polar coordinates, represent the same line PS in the parallel-beam Notice that is the length of line PS. For convenience, will be simply written as depending on the context. The conventional Hilbert transform of the parallel-beam projection is defined as In the polar coordinate system, we define the Hilbert transform as follows: (8) (9) (10)

3 192 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 2, FEBRUARY 2007 It is known that for satisfy the equation (11) To the authors knowledge, (11) was first used for fan-beam image reconstruction from partial data in [4]. The counterparts of (9) (10) for equally spaced sampling were described in [18]. In this paper, the authors intend to investigate the use of (11) for avoiding the weighting factor in the rfbp algorithm toward mitigating the nonuniformity of resolution noise. Using the chain rule of derivative (11), we obtain the following relations: Equation (18) is equivalent to the algorithm in [4] [5] because the order of partial derivatives the Hilbert transform can be exchanged, see (37) for details. Equation (17) indicates that the weighting factor comes from the coordinate transformation between. By relations (5), (8), (13), we derive the following equality: Using (23), we have (19) Combine (12) (13), define (12) (13) Thus, (15) can be further simplified to (20) By 1-D transformation, be rewritten as (14) (10), (14), (4) can (15) Notice that the backprojection of (15) is calculated under the parallel-beam coordinates so that the position-dependent factor is avoided. Actually, the parallel-beam coordinates were used to design the orbit-independent redundancy weights in [13]. We call (15) the Hilbert transform based FBP algorithm denote it by hfbp. Through simple calculations, see Appendix for details, from (8), we can derive the following equalities: (21) Equation (21) makes sense for image reconstruction in nuclear medicine in which the sampling interval for is so coarse that the finite difference introduces severe numerical errors. During the review of this paper, the authors learned that equalities similar to (20) (21) were obtained earlier in [24]. In summary, we restate the preceding discussion as the following theorem. Theorem 1: Assume for, projection is known within, then function can be reconstructed with three steps of calculations: 1) the Hilbert transform (10), 2) the partial derivative (14), 3) the backprojection (15). The partial derivative with respect to is not required in (14) when using (21). IV. SYMMETRY OF FILTERED PROJECTIONS AND PARTIAL SCAN It is well known that the parallel-beam projection satisfies the following symmetry: (22) (16) From the definition of following equation: by (9), it is easy to derive the Then, (15) can be rewritten as (17) (18) (23) Thus, it is easy to see that 180 projections in a parallel-beam scan are enough for an exact reconstruction. For fan-beam data, it is generally known that the scanning range of needs to be for an exact reconstruction. In this paper, we call the data acquisition a partial scan if the scanning range of is a true subset of. In the literature, the scanning range

4 YOU AND ZENG: HILBERT TRANSFORM BASED FBP ALGORITHM FOR FAN-BEAM CT FULL AND PARTIAL SCANS 193 is usually called a short-scan the scanning range larger than is called an over-scan. From Theorem 2 the results in [4] [5], a certain region could be accurately reconstructed for the scanning range with total sum less than. Reconstruction for the short-scan was initially investigated in [11] [12]. Some simulation results were reported first in [13] recently [4] for less than short-scan. In early works, redundancy weights for projections were directly multiplied to in [11], [12], [14], [15]. As pointed out in [11] [12], the projection redundancy weights have to be smooth, otherwise severe streak artifacts are observed. However, if the redundancy weight is applied to the filtered projection, the streak artifacts do not appear at all or can be reduced to a negligible level [4], [13]. In the fan-beam coordinates, we define for. From (5), correspond to in the parallel-beam coordinates, respectively. By (11), in fan-beam coordinates (23) becomes the following equation: By (14), it follows that (24) (25) Let be a subset of to represent the scanning range, be its characteristic function. From the symmetry (25), we derive the following theorem. Theorem 2: Assume for projection is known within. Denote.If, then can be accurately reconstructed using the following formula: (26) Here, is defined as (28) Formula (27) is the same algorithm as in [4] [5]. The rfbp algorithm in [1], [2] can be expressed as Here, is defined as (29) (30) As discussed in [6] [7], the weighting factor is related to the cutoff frequency in the parallel-beam coordinates tends to zero when. The fact that tends to zero eventually leads to severe numerical errors in the backprojection when the object is very close to the X-ray source. The authors would like to point out that these kinds of numerical errors do not go away no matter how smooth the object function is. The same problem exists in the algorithms of [4] [5]. The numerical error by is due to an approximation in the derivation of the FBP-type algorithm as detailed in [6] [7]. As a result, the weighted version of (29) for partial scans cannot be mathematically accurate. In the following, we give a brief overview of the approximation. Define the b-limited ramp filter as (31) Certainly, is not homogeneous with respect to. From the first line of (4) (6), it is easy to deduce the following equation: Because of the symmetry (25), one can replace 0.5 of (26) by any (maybe negative) function, with along the redundant projection lines. The choice of 0.5 is because the sum of two weighted rom variables with the same variance takes the minimum variance with weight 0.5. For the short-scan over-scan, (26) degenerates to the 180 parallel-beam reconstruction. V. COMPARISON WITH OTHER EXISTING FBP-TYPE ALGORITHMS In this section, we give a comparison between (15) the algorithms in [1] [5]. Since the order of the differential Hilbert transform in (14) can be exchanged, we rewrite (18) as (27) (32) Here,. The filtering in the second line of (32) is shift-variant. In order to derive the shift-invariant FBP-type algorithm (29), one has to assume that is a constant. By doing so, the inner integral of the last line of (32) appears to be a linear shift-invariant operation. This practice actually results in a nonstationary cutoff

5 194 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 2, FEBRUARY 2007 frequency in the image [7]. As pointed out in [6], that kind of approximation should not be a concern when,but becomes unacceptable when. To be precise, the exactness proof presented in [15] is for the algorithm (32) instead of the conventional FBP-type algorithm (29) (30). The main contribution in this paper includes two aspects: one is usingthefbp-typeformulas(15) or(21) toavoid inthe backprojection, the other is using the strong symmetry condition (25) to hle the redundancy for partial scans. Although tends to zero when, it does not introduce singularities in the fan-beam coordinates since is equivalent to in the parallel-beam coordinates. Here, keep in mind that is bounded if is smooth. Our simulation results confirmed this observation. It may be concluded that the nonuniformity of cutoff frequency issue in [6] [7] can be resolved by (15). For partial scans, the exactness of formula (26) is obvious due to the strong symmetry equality (25). Next, we comment on the partial data reconstruction. The weighting in (26) the algorithms of [4] [5] are performed on the filtered projections in the backprojection, respectively. As discussed after (26), no smoothness requirement is needed for if along the redundant lines. Our simulation study indicated that no streak artifacts were present from the method in [4] with the weight 0.5. On the other h, the method in [11], [12], [14], [15] was weighting the original projection before the filtering (30). The ramp filter is a combination of the differential the Hilbert transform, it could generate severe numerical errors if weighted projection is not smooth. This may explain why smoothness was required on the redundancy weights of [12] in order to reduce the streak artifacts [11]. Notice that the exactness of (26) is straightforward due to the symmetry condition (25). No symmetry seems to be derived in the literature for the filtered projections in (28) (30). For a fixed point in (28), through coordinate transformation, it is easy to deduce the following weak symmetry condition for 2) Differential by central difference (35) 3) Backprojection by bilinear interpolation For each point, let be its polar coordinate expression, i.e.,, we compute by bilinear interpolation of. Then function is reconstructed by the following equation: (36) The central difference with respect to in (35) is not needed for (21), the redundancy weights need to be considered when implementing (26). Equations (34) (35) are discretized in the fan-beam coordinates. Such numerical discretization would ensure that (35) does not generate singular values on the points that are close to the X-ray source, as demonstrated in our numerical simulations. More analysis of the fan-beam data sampling can be found from [22]. The filtering of (34) is shift-invariant the backprojection (36) only needs the bilinear interpolation. From a computational perspective, this may reveal certain advantages of the hfbp algorithm over the algorithms in [9], [10], [16]. The noise propagation of the algorithm (34) (36) is studied by calculating the variance of the reconstructed image. Here, for simplicity, we only consider formula (21). Using the equality, we derive the following equations: (33) Here,. However, the authors have not found any kind of symmetry for the filtered projection in (30). VI. DISCRETIZATION AND VARIANCE ANALYSIS Denote by, the sampling grid intervals with respect to the variables,, respectively. The numerical implementation of (15), (21), (26) follows three steps. 1) Discrete Hilbert transform Then (21) becomes (37) (38) (34) From [23], if is a stationary process, its partial derivative remains a stationary process. Let de-

6 YOU AND ZENG: HILBERT TRANSFORM BASED FBP ALGORITHM FOR FAN-BEAM CT FULL AND PARTIAL SCANS 195 note the variance of reconstructed image by (38) is, then the variance of the (39) The constant does not affect the uniformity. Compared with the rfbp algorithm that needs in the backprojection, the factor only depends on. Intuitively, this might suggest that (34) (36) could give better uniformity, which was confirmed by numerical simulations in recent work [16], [24]. However, the projection noise never becomes stationary in practice. By our recent results in [25], the noise can be decomposed into, where denotes the null space component denotes the range space component. The reconstruction of using the FBP algorithm is perfectly zero. Even when the variance of is large, should not affect the reconstructed image. Thus the common variance analysis for stationary noise alone may not reveal all aspects of the noise propagation from the projection to the reconstructed image. Let st for the noisy projection the corresponding reconstructed image, respectively. The signal-to-noise ratios (SNRs) of the discrete projection reconstructed image are calculated by (40) The SNR is also closely related to the variance when the signal is understood as a rom process. In this paper, we use the propagation of noise from the projection to the reconstructed image to measure the noise effect of different algorithms. Our simulation results revealed that the hfbp algorithm gave a better SNR compared with the CHR rfbp algorithms. VII. SIMULATION RESULTS The geometry of a typical third-generation CT scanner with a circular X-ray source orbit was used in the computer simulations. The radius of the circular orbit was 50 cm, i.e., cm. The maximum fan-subtending angle was. The variable was evenly sampled in with a sampling angle of was equally sampled in with a total of 360 ray sums, i.e.,. The Shepp Logan phantom was used to simulate the object function, the line integral was analytically calculated so that it is mathematically exact. The largest ellipse in Shepp Logan phantom has a major axis of 50 cm so one only needs to reconstruct the image at points with cm. The linear attenuation range was 0.06 cm to 0.18 cm. In simulations with noise consideration, the line integral was then converted to the mean number of photons with two effective source influences of photons for each X-ray tube. The measurement was drawn from a Poisson deviate with the corresponding mean, the noisy projection was then calculated by. The SNRs of the generated noisy projection data were for, respectively. The size of all the images reconstructed was In all simulations, we took, which is the same as. First, we reconstructed the images from the full-scan noisefree noisy projection data by hfbp, CHR, rfbp algorithms. Formula (21) was implemented in the hfbp algorithm. The implementation of [8] was used for the CHR algorithm (30) was implemented in the rfbp algorithm. All reconstructed images are shown in Fig. 3. For the noise-free data, all algorithms seem to give perfect results, which is consistent with the analysis in [6] that does not affect the uniformity when. However, the difference becomes more visible when the noise is present. The SNRs of the reconstructed images from noisy projection data were calculated. Table I lists all the SNRs for three reconstruction algorithms. Notice that the hfbp algorithm (21) gave better SNRs compared with the CHR rfbp algorithms. We also used (27) (28) to reconstruct the image from same projection data, the results were observed to be very similar to the reconstruction by (21). Second, to evaluate the numerical stability of (21) when the object gets close to the X-ray source, we changed the subtending angle to, reduced the radius of X-ray source orbit to cm, then reconstructed the image for. With that change, we had. The hfbp algorithm did not reveal any noticeable numerical errors when. The CHR algorithm suffered from some artifacts even there was no singularity near the boundary. The rfbp algorithm generated severe singularity when. The simulation results by (27) (28) revealed similar behavior to the rfbp algorithm (29) (30). After clipping extremely large values in the reconstructed image by the rfbp algorithm, all the reconstructed images are shown in Fig. 4. The authors would like to point out that some reconstructed values in Fig. 4(c) were 1000 times larger than the actual maximum value. In Fig. 5, the profiles are drawn across the vertical central line of the reconstructed images by the hfbp, CHR, rfbp algorithms to show the differences. Please keep in mind that singular values for in the reconstructed image by the rfbp algorithm were clipped in Fig. 4 for display. It may be concluded that (21) could hle the nonunifomity introduced by. Next, we use (26) to reconstruct images from over-, short ROI-scan projection data. In this simulation, we used scanning angle sets of,, a subtending angle of. All the reconstructed images are shown in Fig. 6. For over- half-scans, the reconstructed images from noise-free data are virtually identical to the reconstructed image with the full-scan data. For the ROI-scan, only the area inside the black triangle in Fig. 4 should have the accurate reconstruction. The simulation results were consistent with the prediction by Theorem 2. The

7 196 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 2, FEBRUARY 2007 Fig. 3. The reconstructed images from full-scan data: from left to right by hfbp, CHR, rfbp, from top to bottom for noise-free noisy data with N = TABLE I SNRS FOR THREE ALGORITHMS USING TWO SETS OF NOISY PROJECTIONS noise effect of (26) for the over- half-scans is very similar to the case for the full-scan by (21). The calculated SNRs are shown in Table II. Here, we noticed that the SNRs for partial scans were reduced compared with the full-scan. This is due to two reasons. One is that less data is used in (26), the other may introduce numerical is that the extra factor errors compared with (21) in which that factor is not needed. This indicates that (21) not only improves the computation efficiency but also helps reduce numerical errors. VIII. DISCUSSION AND CONCLUSION The nonuniformity of resolution noise became an issue in recent research [6], [7], [9], [10]. By a relationship of the Hilbert transforms of the parallel-beam fan-beam data using the 1-D transformation between parallel-beam fan-beam coordinates, we derived a new type of FBP algorithm (15) that does in the backnot need the position-dependent weight projection so that intolerable numerical errors around the region that is close to the X-ray source can be avoided. Compared with in the backprothe rfbp algorithm, the weight in (14). Intuitively, this jection of (29) was reduced to might suggest that the nonuniformity discussed in [6], [7], [9], [10] can be mitigated to some extent. Moreover, an exact reconstruction formula (26) was derived for fan-beam partial scans. The formula (26) is applicable to full-, over-, short-, ROI-scans without smoothness requirements on the redundancy

8 YOU AND ZENG: HILBERT TRANSFORM BASED FBP ALGORITHM FOR FAN-BEAM CT FULL AND PARTIAL SCANS 197 Fig. 4. The reconstructed images for r D with =90 by: (a) hfbp, (b) CHR, (c) rfbp algorithms, respectively. Fig. 5. The profiles across the vertical central line of the images reconstructed from noise-free data with the hfbp, CHR rfbp algorithms. weights. The filtering in (10) is shift-invariant. This may have certain computational advantages over the methods in [9], [10], [16]. Numerical simulations revealed the better stability of the hfbp algorithm over the rfbp CHR algorithms. It may be concluded that (21) (26) are recommended for some situations in which the X-ray source is very close to the object the projection noise is severe. APPENDIX DERIVATION OF (16) AND (17) The first equation of (8) is equivalent to From the first equation of (8), we also derive the following relation: (A3) The calculation of the right-h side of (A3) is rather straightforward (A4) The left-h side of (A3) is calculated as follows: (A1) Then, from following equality: we derive the (A2) Calculating the square root of (A2), we derive (16). (A5)

9 198 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 26, NO. 2, FEBRUARY 2007 Fig. 6. The reconstructed images: from left to right for the over-, half- ROI-scans, from top to bottom for noise-free, N = data. ACKNOWLEDGMENT TABLE II SNRS OF RECONSTRUCTED IMAGES FROM NOISY DATA The authors would like to express their thanks to the C++ numerical library for computer simulations from website REFERENCES Then, (17) can be derived by the following calculations: (A6) [1] G. T. Herman A. V. Naparstek, Fast image reconstruction based on a Radon inversion formula appropriate for rapidly collected data, SIAM J. Appl. Math., vol. 33, pp , [2] B. K. P. Horn, Fan-beam reconstruction methods, Proc. IEEE, vol. 67, no. 12, pp , Dec [3] G. Besson, CT fan-beam parameterizations leading to shift-invariant filtering, Inverse Problems, vol. 12, pp , [4] F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, Image reconstruction from fan-beam projections on less than a short scan, Phys. Med. Biol., vol. 47, pp , [5] G. H. Chen, A new framework of image reconstruction from fan beam projections, Med. Phys., vol. 30, pp , [6] F. Natterer F. Wübbeling, Mathematical Methods in Image Reconstruction. Philadelphia,, PA: SIAM, 2001.

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