Cone-beam mammo-computed tomography from data along two tilting arcs

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1 Cone-beam mammo-computed tomography from data along two tilting arcs Kai Zeng, a Hengyong Yu, b Laurie L. Fajardo, c and Ge Wang d CT/Micro-CT Laboratory, Department of Radiology, University of Iowa, Iowa City, Iowa Received 23 December 2005; revised 19 July 2006; accepted for publication 20 July 2006; published 13 September 2006 Over the past several years there has been an increasing interest in cone-beam computed tomography CT for breast imaging. In this article, we propose a new scheme for theoretically exact cone-beam mammo-ct and develop a corresponding Katsevich-type reconstruction algorithm. In our scheme, cone-beam scans are performed along two tilting arcs to collect a sufficient amount of information for exact reconstruction. In our algorithm, cone-beam data are filtered in a shiftinvariant fashion and then weighted backprojected into the three-dimensional space for the final reconstruction. Our approach has several desirable features, including tolerance of axial data truncation, efficiency in sequential/parallel implementation, and accuracy for quantitative analysis. We also demonstrate the system performance and clinical utility of the proposed technique in numerical simulations American Association of Physicists in Medicine. DOI: / Key words: cone-beam CT, mammography, exact reconstruction, Katsevich algorithm I. INTRODUCTION Breast cancer is ranked as the second leading cause of cancer death in women in the United States. It has been recognized that mass screening and early treatment are extremely important to reduce the mortality of breast cancer. Due to its specificity and sensitivity, x-ray mammography has been the method of choice for screening and diagnosis. 1,2 However, x-ray mammography is far from being perfect because up to 17% of breast cancers are not identified with mammography, and normal breasts are associated with 70% 90% of mammograms suspicious of cancers. 3 A major limitation of x-ray mammography is its projective nature, while the real anatomy and pathology is really in three dimensions 3D.To address this problem, x-ray tomosynthesis and cone-beam computed tomography CT are two compelling solutions. Tomosynthesis is a three-dimensional 3D imaging technique to reconstruct a series of images from a limited number of projections. 4 Since its introduction in 1972, the area of tomosynthesis has been significantly advanced largely due to the development of the area detectors. 5 A primary application of tomosynthesis is for breast imaging. 6 8 The tomosynthetic algorithms are either analytic or iterative. The analytic algorithms are straightforward and efficient, such as self-masking, 9 selective plane removal, 10 and matrix inversion tomosynthesis. 11 While the iterative algorithms are robust against noisy data and flexible to integrate prior knowledge, as it is done using algebraic reconstruction techniques, 12,13 expectation-maximization, 14 etc. none of these algorithms can avoid the inherent drawback of tomosynthesis due to the data incompleteness. Technically speaking, a breast volume should be imaged very well by cone-beam CT. Since more information of the object is acquired, the image quality of CT is much better than tomosynthesis, in terms of contrast resolution, geometrical distortion, etc. The concept of breast CT was proposed two decades ago, 15 but little progress had been made initially because of compromised image quality and involved radiation exposure. Again, thanks to the advancement in the digital detector technology, a number of groups investigated the feasibility and prototypes of cone-beam mammo-ct. 16,17 Nevertheless, the algorithms for breast CT are still based on the traditional Feldkamp-type algorithms, 18 and reconstruct images approximately with various artifacts. The fundamental classic results on exact cone-beam CT reconstruction were achieved by Grangeat, 19 Smith, 20 and Tuy. 21 The recent breakthroughs on exact cone-beam CT algorithms were reviewed by Zhao et al. 22 Up to now, there are a number of accurate and efficient cone-beam CT algorithms for various scanning trajectories, such as a helix, an arc-plus-line, 26,27 a circle-plus-arc, 28,29 and a saddle curve. 30,31 Also, there are several algorithms which allow exact image reconstruction in the case of general trajectories To improve image quality with cone-beam mammo-ct, we are motivated to design a cone-beam scanning mode that allows theoretically exact image reconstruction. In this article, we propose a novel scheme for cone-beam mammo-ct, which is theoretically exact, and develop a corresponding Katsevich-type reconstruction algorithm. In our scheme, cone-beam scans are performed along two tilting arcs to collect a sufficient amount of information for exact reconstruction. We derive our corresponding algorithm in the framework established by Katsevich, 29,33 which may handle axial truncation of cone beam data. In what follows, we describe our system setup and derive the algorithm in Sec. II, describe numerical simulation results in Sec. III, and discuss relevant issues and conclude the article in Sec. IV. II. METHODS AND MATERIALS A. Cone-beam mammo-ct system In the proposed cone-beam mammo-ct system Fig. 1, a patient lays down on a table with one breast hanging through a hole. The x-ray tube and a flat-panel camera are fixed to a 3621 Med. Phys , October /2006/33 10 /3621/13/$ Am. Assoc. Phys. Med. 3621

2 3622 Zeng et al.: Cone-beam mammo-ct 3622 B. Cone-beam reconstruction method 1. Review of general cone-beam reconstruction formula First, we need to review Katsevich s general cone beam reconstruction framework briefly. 33 Let the scanning locus be a finite union of smooth curves defined in R 3 : I ª a l,b l R 3, l s I ys R 3,ẏs 0onI, 1 where a l b l and ẏsªdy/ds. Assume f is smooth, compactly supported and identically equals zeros in a neighborhood of the locus, the cone beam transform of f along is defined as FIG. 1. Proposed 3D mammography cone-beam CT system. a System configuration and b the cross section along the dashed line in a. rigid frame such as a C-arm to produce cone-beam projections. Then, a beast can be scanned twice along two tilting arcs respectively Figs. 1b and 2a by rotating the C-arm in different scanning planes. The other breast can be scanned in a similar fashion. Compared with the existing breast CT systems that only perform approximate reconstruction, our system for the first time aims at theoretically exact conebeam mammo-ct with fewer artifacts and better accuracy. Unlike other C-arm based algorithms, 27,29 our scanning trajectory is more symmetric, which helps improve image quality in general. D f y, ª0 fy + tdt, y, S 2, 2 where S 2 is the unit sphere in R 3. For given xr 3 and R 3 0, we also define s,x ª x ys x ys, x R3 \, s I, x, ª z R 3 :z x =0. With ys j denoting points of intersection of x, and and denoting the great circle S 2 : =0 which consists of unit vectors perpendicular to, we introduce the sets Crits,x ª s,x:x, is tangent to or contains an end point of 3 I reg x ª s I:Crits,x s,x Critx ª Crits,x. si 4 FIG. 2. Scanning geometry for exact cone-beam reconstruction. a 3D view of the tilting arcs and the breast and b the exact reconstruction region U. To reconstruct f at any fixed xr 3, Katsevich suggested the following main assumptions about the locus. 33 Property C1. (Tuy s condition Ref. 21). Any plane through x intersects at least at one point. Property C2. There exists a constant N 1 such that the number of directions in Crits,x does not exceed N 1 for any si reg x. Property C3. There exists a constant N 2 such that the number of points in x, does not exceed N 2 for any S 2 Critx. If we only use a subset x for image reconstruction at x, and I are replaced by x and Ix for the earlier mentioned formulas, respectively. Let us consider a weighting function ns,x,, s I reg x, s,x Crits,x. Ifns,x, is the piecewise constant and j:ysj x,ns j,x,=1 for almost all S 2, we arrive at Katsevich s general inversion formula 33 as follows:

3 3623 Zeng et al.: Cone-beam mammo-ct 3623 fx = Ix m 0 2 c m s,x x ys q D fyq,cos s,x + sin d s,x, m sin q=s ds, 5 s,x, ª s,x s,x,,s,x, s,x, 6 where is a polar angle in the plane perpendicular to s,x and m 0, are the points where s,x, are discontinuous, and c m s,x are values of the jumps c m s,x ª lim s,x, m + s,x, m, s,x, ª sgn ẏsns,x,, = s,x, s,x. 8 FIG. 3. Illustration of Pl lines for a xū 2 and b xū Existence of PI segments For our two-tilting-arcs scanning locus Fig. 2a, wedescribe our geometry first, and then validate every property required by Katsevich s general reconstruction framework. Let the three components of xr 3 are x 1,x 2,x 3, respectively. fxc 2 0 is a smooth function inside the cylinder x 1 +x 2 2 r 2. Our scanning orbit =C a C b defined on the real interval I=I 1 I 2 consists two tilting arcs see Fig. 2a is parameterized by C a ª ys:y 1 = R coss + s mx cos t,y 2 = R sins + s mx,y 3 = R coss + s mx sin t,s I 1, C b ª ys:y 1 = R coss s mx cos t,y 2 = R sins s mx,y 3 = R coss s mx sin t,s I 2 where I 1 = /2 2s mx,/2, I 2 =/2,3/2+2s mx, Rr is the radius of the arcs, t is the tilting angle satisfying R cos tr, and s mx =arcsinr/r cos t. s mx is the fan angle. The cone beam projection data of f is defined as in Eq. 2. As defined in Appendix A, the reconstruction region Ū can be decomposed into two kinds of reconstruction zones Ū 1 and Ū 2 Figs. 2b and 14. There are exactly two PI segments within Ū 2 and there is only one PI segment within Ū 1. Here PI means, and a PI segment of x is the line segment containing x and its two end points on the scanning orbit. Our problem is to reconstruct f inside a region U which is the intersection of Ū and the cylindrical support Fig. 2b. In practical clinical applications, the region in the chest should be avoided and only the breast part is concerned. Based on the definition of Ū, for any fixed xū, there exist at least one and at most two PI segments. For each PI segment, one end point must be on arc A while the other 9 must be on arc B if we ignore those points on the scanning arcs planes of zero measure. We denote the corresponding angular variable as s a =s a x for arc A and s b =s b x for arc B. The existence and uniqueness of these PI segments can be divided in the two cases: 1 R sins a +s mx x 2 R sins b s mx and 2 R sins a +s mx x 2 R sins b s mx. Moreover, for any xū 2 Fig. 3a, there exist exactly two PI segments with our symmetric imaging geometry. While, for any xū 1 Figs. 3b and 14d, there exists exactly one PI segment, either in case 1 or case 2. With our scanning geometry, for a pair of given s a x and s b x for xū 2 Fig. 3a, the PI segment may correspond to two PI intervals Ī 1 x= /2 s mx,s a xs b x,3/2+s mx and Ī 2 x=s a x,/2 s mx /2+s mx,s b x yielding two PI arcs 1 = 1 x=ī 1 x and 2 = 2 x=ī 2 x Fig. 3a. 3. Reconstruction method To derive a theoretically exact image reconstruction algorithm in Katsevich s framework, 33 we need to check the properties of the locus and the weighting function in our TABLE I. Weighting functions ns,x, used for our Katsevich-type reconstruction. Case Value of ns,x, 1IP,s 1 I 1 1 1IP,s 1 I 2 1 3IPs,s 1,s 2 I 1 ;s 3 I 2 ns 1,x,=ns 2,x,=1, ns 3,x,= 1 3 IPs, s 1 I 1 ;s 2,s 3 I 2,s 2 s 3 ; x is above the plane in Ū 2, Fig. 2b ns 1,x,=ns 3,x,=1, ns 2,x,= 1 3 IPs, s 1 I 1 ;s 2,s 3 I 2,s 2 s 3 ; x is below the plane in Ū 1, Fig. 2b ns 1,x,=ns 2,x,=1, ns 3,x,= 1

4 3624 Zeng et al.: Cone-beam mammo-ct 3624 particular geometry. Let us only consider the PI-arcs 1 to reconstruct xū 2 in the following, while 2 can be similarly utilized. As there exists at least one PI segment for any xū, property C1, Tuy s condition, is satisfied. Also, for ys 1 x there are a limited number of planes x,, s,x, which satisfies x, contains an endpoint of 1 or is tangent to 1. Thus property C2 is satisfied. Moreover, apart from a set of zero measure, any plane x,, S 2, through x intersects 1 x at an odd number of intersection points IP. And the number of IP is either one or three which validates the property C3. Overall, 1 satisfies the three properties C1 C3 required in Ref. 33. Following the work by Katsevich, 29 we define a similar weighting function ns,x, in Table I, where s j, j=1,2,3 are the angular parameters satisfying ys j 1 x,. Thus, we have j:ysj 1 x,ns j,x,=1 for almost all S 2, and ns,x, is the piece-wise constant. Therefore, our scanning arcs and weighting function are qualified to be fitted into Katsevich s general reconstruction framework. Then, we need to find the discontinuities in of 2.8 for each sī 1 x, and determine the filtering directions. Similar to Katsevich s treatment, 29 we can analyze the discontinuities by projecting the source trajectory =C a C b onto the detector plane DPs 0 Fig. 4, which is defined as the plane through the origin O and perpendicular to the line through ys 0 and O. In this article, all the variables are denoted with hat signify the objects on a projection plane. The detector coordinates are defined along the directional vector u and v. For the sources on the arc A, that is, s 0 /2 +2s mx,/2, u = sins 0 + s mx cos t, coss 0 + s mx,sins 0 + s mx sin t, v = sin t,0,cos t. 10 And for the source on the arc B, that is, s 0 /2,3/2 +2s mx FIG. 4. Projections of a Pl line on a detector plane for the configuration of Fig. 3a. a The projected trajectory with the source on C a, b the projected trajectory with the source on C b,andc an impossible trajectory projection. u = sins 0 + s mx cos t, coss 0 s mx, sins 0 s mx sin t, v = sin t,0,cos t. 11 With the source on C a, we can describe the projection of C b on the detector plane as us = R sins 0 + s mx coss s mx cos 2t coss 0 + s mx sins s mx coss s mx cos2tcoss 0 + s mx + sins s mx sins 0 + s mx 1, coss s mx sin 2t vs = R coss s mx cos2tcoss 0 + s mx + sins s mx sins 0 + s mx 1, 12 where s 0 /2 2s mx,/2 indicates the source position on the arc A, s/2,3/2+2s mx and us and vs are the detector coordinates. With the source on C b, by geometric symmetry we can immediately obtain the same formula except for a changed range of s. When the source is on C a Fig. 4a, the discontinuities may occur at polar angle 1, 2, and 3 which corresponding to the plane P 1, P 2, and P 3. It is easy to compute that the magnitudes of the jumps at these three positions are 2, 0, and 0, respectively. Thus, we only have a jump along direction parameterized by 1, which is the same filtering direction used in the Feldkamp algorithm. 18 When the source is on C b Fig. 4b, the projection of a PI segment can only have one IP with the projection of 1 x, giving the tangential direction 1 with a jump of magnitude 2. The last case Fig. 4c is simply impossible, as shown in Appendix B. Thus, the filtering directions should be along horizontal

5 3625 Zeng et al.: Cone-beam mammo-ct 3625 TABLE II. Linear attenuation coefficients at 38 kev. Material Attenuation coefficient cm 1 Skin Breast Calcifications Mass Fibrous f Ī2 x = Ī 2 x m=1, x ys q D fyq,cos s,x + sin d s,x, m sin q=s ds. Combining these two formulas, we have fx = f Ī1 x + f Ī2 x/ FIG. 5. Uncompressed breast phantom which contains masses, calcifications, and fibroses of different sizes and densities. lines Fig. 4a or tangential lines Fig. 4b, depending on m determined based on our above analysis. This is exact the basic same result obtained by Katsevich. 29 Therefore, the reconstruction formula can be expressed as f Ī1 x = 1 1 x ys 2 2 Ī 1 x m=1,2 0 2 q D fyq,cos s,x + sin d s,x, m sin q=s ds. 13 On the other hand, we can also reconstruct x from 2 x. Similarly, we have Consequently, our reconstruction algorithm can be divided into four steps: 1 differentiate the projection data collected along PI-arc 1 x to obtain /qd f ; 2 filter every projection along the direction defined by m. Projection data on arc A are horizontally filtered, and projection data on arc B are tangentially filtered; 3 backproject the filtered data to reconstruct an image; and 4 repeat the above three steps for the projection data collected along the other PI-arc 2 x, and average the two images for the final reconstruction. For xū 1 Fig. 3b, we can derive the formula and steps in a similar fashion as that for xū 2 Fig. 3a. Since our reconstruction formula is derived in Katsevich s framework, 29,33, it can be regarded as a variant of the conebeam reconstruction algorithm in the case of circle-and-arc. 29 C. Uncompressed breast phantom A 3D mathematical mammography phantom Fig. 5 was designed in reference to several commercially available mammography phantoms, such as the American College of Radiology ACR accreditation phantom, Mammography Imaging Screening Trial phantom and Uniform phantom. 37,38 TABLE III. Dimensions and linear attenuation coefficients of the fibroses. Fibrous No Diameter mm Height mm AC cm

6 3626 Zeng et al.: Cone-beam mammo-ct 3626 TABLE IV. Dimensions and linear attenuation coefficients of masses. Mass No Diameter mm AC cm The breast was modeled as half an ellipsoid. While the existing phantoms contain important structures simulating mass, fibrous and calcification, they mimic a compressed breast for x-ray mammography, being suboptimal for our 3D CT simulation. To address this limitation, our phantom targeted an uncompressed breast with representative anatomical and pathological features, including structures of different sizes and contrasts. A detailed description of the phantom is as follows. The three ellipsoidal semiaxes were set to 50, 50, and 100 mm. The skin thickness was set to 2.5 mm. While the fibroses were modeled as cylinders, the calcifications and mass as balls Fig. 5. The phantom was positioned in the nonnegative space, attached to the chest wall defined on z =0 mm. Fibroses were placed on the planes of z=22.5 mm and z=62.5 mm. Masses were centered on the planes of z =35 mm and z=75 mm. Calcifications were scattered on the plane of z=42.5 mm. The linear attenuation coefficients AC of the breast structures were specified for 38 kev x-rays 16,39 as listed in Table II. The dimensions of the features in the phantom were specified in reference to those in the existing mammography phantoms, especially the ACR phantom. Tables III V enlist the sizes and attenuation properties of the fibroses in the planes z=22.5 and 62.5 mm, the parameters of the masses in the planes z=35.0 and 75.0 mm, and the characteristics of the calcifications at z = 42.5 mm, respectively. III. NUMERICAL SIMULATION To corroborate the correctness of our proposed algorithm and demonstrate its clinical utility, we implemented it in C for numerical tests. These results were also compared with a circular cone-beam CT scan using the FDK algorithm. The system setup and geometric parameters were summarized in Tables VI and VII, respectively. In our simulation, the reconstruction region U was delimited by the chest wall Fig. 6a. The FDK algorithm can only approximately reconstruct the volume above the scanning plane Fig. 6b, while our proposed method can exactly reconstruct a larger volume. In all the cases, the reconstruction results obtained using our algorithm were excellent except for the streak artifacts primarily caused by the high-density calcifications ten times higher than the background Figs. 6 8, which were also observed in the FDK results. The other structures, such as fibrous and mass, were reconstructed very well. Although the detector element size was comparable or even larger than the fine structures in our phantom, these features were clearly visible in the reconstructed images. As far as the calcifications are concerned, despite that they were substantially smaller than one pixel, they were revealed in the reconstruction. The low-contrast structures were well revealed Figs. 7 and 8 using our algorithm. Our algorithm generally produced more accurate results than the FDK algorithm, because there were significant density drops in the FDK results, especially far away from the central plane Figs Also, due to the approximate nature of the FDK algorithm, significant streak artifacts were observed Figs. 7 and 8 in the FDK results. Besides, our algorithm preserved the shape of the breast phantom, but there were substantial shape distortions in the FDK results especially near the top of the phantom Figs. 7 and 8. A number of numerical tests with noisy projection data were performed to simulate the practical imaging process. As the image quality is closely related to radiation dose, which is proportion to the total number of involved x-ray photons given other conditions being equal, the same total number of photons was used in each test for image quality comparison. In our study, the same radiation dose was uniformly distributed to every detector cell in each projection view for simplicity. Then, the Poisson noise was added to the projection data. The image quality difference between our algorithm and the FDK algorithm was similar to that in the noise-free case, except that in the noisy case our algorithm showed a better noise tolerance than the FDK algorithm. 40 Specifically, the images obtained using our algorithm were smoother than the FDK results, and had no geometric distortion that was shown in the FDK counterparts Figs In Figs. 10 and 11, the image noise associated with the proposed algorithm appeared to be larger than that with the FDK algorithm. The reason was that the density drop inherent in the FDK reconstruction, made many pixel values go outside the fixed display window. As a result, a large portion of noise became invisible in the displayed image. The aforementioned image artifacts in our simulation were around the high contrast structures. The profiles across their boundaries can be modeled as piece-wise constant func- TABLE V. Dimensions and linear attenuation coefficients of calcifications. Calcification No Diameter mm AC cm

7 3627 Zeng et al.: Cone-beam mammo-ct 3627 TABLE VI. Simulated imaging parameters for tilting arc based mammo-ct scan. Number of projections 5002 Detector array size mm 2 Number of detector elements Detector element size mm 2 Number of photons per 1e view for noisy projections Arc radius 330 mm Arc tilting angle 20 Arc angular range 226 Reconstruction grid Voxel size mm 3 tions, which do not satisfy the assumption f C 0 R 3 that was made for the formulation of Katsevich s general exact cone-beam reconstruction scheme. 33 Therefore, it should not be surprising that such high-contrast nondifferentiable features had caused intensity fluctuations in the reconstructed images, as we previously discussed. 41 To confirm the correctness of the implementation of our proposed reconstruction formula, we also applied it to reconstruct the traditional 3D Shepp-Logan phantom. The reconstruction results looked TABLE VII. Simulated system setup parameters of a circular scan for FDK algorithm. Number of projections 1000 Two-dimensional detector array size mm 2 Detector elements Detector element size mm 2 Radius of circle 330 mm Total photon number per 1e view for noisy projections Reconstruction grid Voxel size 0.43 mm 3 flawless, precisely as we expected, but the results using the FDK algorithm contained significant density dropping artifacts Figs. 12. IV. DISCUSSIONS AND CONCLUSION Our reconstruction algorithm was derived from Katsevich s general scheme and Katsevich s circle-and-arc algorithm, which can survive the longitude data truncation. As a result, no x-rays need to be sent into the chest wall. Also, this algorithm can be efficiently implemented in parallel similar to what was recently reported of the parallel implementation of the Katsevich algorithm. 42 Additionally, our scanning or- FIG. 6. Representative phantom slices. a The ideal phantom slice at x 2 =0.0 mm, b x 1 =0.0 mm, c x 3 =35 mm, and d x 3 =75.0 mm. The dotted lines shows the exact reconstruction region. The display window is 0.244,

8 3628 Zeng et al.: Cone-beam mammo-ct 3628 FIG. 7. Reconstructed images at x 2 =0.0 mm using the proposed algorithm and the FDK algorithm. The images in the left column are from the proposed algorithm, and that in the right column are from the FDK algorithm. The images in the top row are from noise free projection data, and that in the bottom row from noisy data with the same number of photons. The display window is 0.244, FIG. 8. Reconstructed images at x 1 =0.0 mm using the proposed algorithm and the FDK algorithm. The images in the left column are from the proposed algorithm, and that in the right column are from the FDK algorithm. The images in the top row are from noise free projection data, and that in the bottom row from noisy data with the same number of photons. The display window is 0.244, bit is symmetric, and may be numerically more stable than other less symmetric scanning trajectories, such as circleplus-line and circle-plus-arc. 27,29 Although our algorithm requires a longer scanning trajectory than a circular scan, it does not mean that our algorithm require more radiation dose. In our imaging protocol, we can distribute the same dose as that with the circular scan to each view and reconstruct 3D images from these projections. As our algorithm can utilize all the information from two arcs, the final image shall not be inherently noisier than the circular scan reconstruction. Our numeric experiments have indicated that our results were actually less noisy than that from the circular scan using the FDK algorithm. To compare the noise levels in the images reconstructed using our algorithm and the FDK algorithm respectively, the standard deviations of pixel values within a homogeneous region-of-interest were computed Fig. 11. Generally speaking, the variance of a stochastic process cannot be computed from a single realization of that process. To compare the noise properties of the reconstructed images, the ensemble averages should be used, instead of the averages based on a single image. These averages are equivalent only in the case of an ergodic stochastic process. However, it is not common in either the quality-control practice or the reconstruction literature to do the scan and reconstruction many times for evaluation of the noise level. 40,43 45 Hence, in this project the noise image has been approximately considered as being ergodic. In conclusion, we have proposed a novel cone-beam mammo-ct scheme based on two tilting arcs and formulated a Katsevich-type reconstruction algorithm. The numerical simulation results are consistent with the ideal 3D image volume within the numerical error. Compared with other existing mammo-ct algorithms, our work promises better diagnostic performance for breast imaging, and may have a FIG. 9. Reconstructed images at x 3 =35.0 mm using the proposed algorithm and the FDK algorithm. The images in the left column are from the proposed algorithm, and that in the right column are from the FDK algorithm. The images in the top row are from noise free projection data, and that in the bottom row from noisy data with the same number of photons. The display window is 0.244,

9 3629 Zeng et al.: Cone-beam mammo-ct 3629 x 1 = R cos s a cos t + 1 R cos s b cos t, x 2 = R sin s a + 1 R sin s b, x 3 = R cos s a sin t + 1 R cos s b sin t. A1a A1b A1c We can rewrite A1a A1c as x 1 R cos t = cos s a + 1 cos s b, A2a x 2 R = sin s a + 1 sin s b, A2b FIG. 10. Reconstructed images at x 3 =75.0 mm using the proposed algorithm and the FDK algorithm. The images in the left column are from the proposed algorithm, and that in the right column are from the FDK algorithm. The images in the top row are from noise free projection data, and that in the bottom row from noisy data with the same number of photons. The display window is 0.244, commercial potential. Our scheme also can be used to other similar tomographic imaging applications, such as singlephoton emission computed tomography. ACKNOWLEDGMENTS This work is partially supported by the NIH/NIBIB Grant Nos. EB and EB APPENDIX A: DEFINITION OF REGION Ū, Ū 1, and Ū 2 1. Monotony of x 3 with respect to s a and s b First, we prove that x 3 is monotonic for the first case R sins a +s mx x 2 R sins b s mx with respect to s b as what Katsevich did. 29 Without loss of generality, let us parameterize x by 01 with s b=s b s mx and s a=s a +s mx. Hence, we have x 3 R sin t = cos s a 1 cos s b. A2c Then, let us fix x 1 and x 2, and let, s a be functions of s b, i.e., s b and s as b. Differentiating A2a and A2b, we have cos s a cos s b d ds b ds a sin s a = 1 sin s b, A3a ds b sin s a sin s b d ds a + cos s a = 1 cos s b. A3b ds b ds b Solving A3 for d/ds b and ds a/ds b, we obtain d ds b = 1 sins b s a, A4a 1 coss a s b ds a = 1. A4b ds b Also, differentiating A2c with respect to s b, we have 1 dx 3 = cos s a + cos s b d R sin t ds b ds b Noting ds a sin s a + 1 sin s b. A5 ds b FIG. 11. Reconstructed images at x 3 =46.8 mm from the proposed algorithm and the FDK algorithm with the same number of photons. The noise level is measured as the standard deviation from the homogeneous region indicated by the dotted rectangle. a The noise from the proposed algorithm is 2.03e-0.3, and b the noise from the FDK algorithm is 2.92e-03. The display window is 0.244,

10 3630 Zeng et al.: Cone-beam mammo-ct 3630 FIG. 12. Reconstructed Shepp-Logan phantom images at x 2 = 0.25 and profiles along the dash lines. a is reconstructed from the proposed algorithm and b from the FDK algorithm. The images are displayed with the window 1.0, sins b s acos s a + cos s b sin s b sin s acoss b s a = sin s b cos s a cos s b sin s acos s a + cos s b sin s b sin s acos s a cos s b + sin s a sin s b = cos 2 s a sin s b sin s a cos 2 s b sin s a sin 2 s b + sin 2 s a sin s b = sin s b sin s a A6 and substituting A4 and A5, we have dx 3 = 1 R sin t cos s a + cos s bsins b s a ds b 1 coss b s a + sin s b sin s a =21 R sin t sin s b sin s a 1 coss b s a. A7 Because 01 and R sins ax 2 R sins b, dx 3 /ds b does not change its sign. Hence, x 3 is monotonic decreasing with s b or s b.ass a or s a rotates together with s b or s b in the same direction, x 3 is also monotonic decreasing with s a or s a. Similarly, for the second case R sins a +s mx x 2 R sins b s mx, we can prove that x 3 is monotonous increasing with s b or s b.ass a or s a rotates together with s b or s b in the same direction, x 3 is also monotonic increasing with s a or s a. 2. Upper and lower limits of x 3 given x 1,x 2 to admits a PI line It is easy to see that if we project all the x that admit at least one PI line onto the plane x 3 =0, the results must be within an ellipse x 1,x 2 x 1 2 /cos 2 t+x 2 2 =R 2. Moreover, given a x 1,x 2 within this ellipse, we can readily get the limit of x 3 with respect to the assumption of cases 1 and 2 based on the aforementioned monotony property. If we do so for every point x 1,x 2 within the ellipse, we can obtain an image showing the appearance of the region. Based on assumption for the first case, we have x 2 ys a x 2 x 2 ys b where x 2 ys b represents the second component of ys b. For any fixed x 1,x 2 within the ellipse and all the points admitting a PI line Fig. 13, we have the monotony of x 3 with respect to s a or s b. Therefore, we can determine the upper and lower limits of x 3 for any given x 1,x 2 as follows: Let s a x 2,s a+ x 2 be two particular s a± on arc A Fig. 13a, at which x 2 y ArcA s a± =x 2, x 1 y ArcA s a+ 0 and x 1 y ArcA s a 0. Note that for x 2 R coss mx there does not exist s a. Let s b x 2, s b+ x 2 be two particular s b± on arc B Fig. 13b, at which x 2 y ArcB s b± =x 2, x 1 y ArcB s b+ 0 and x 1 y ArcB s b 0. Note that for some x 2 R coss mx there dose not exist s b+. According to the monotony, i.e., x 3 is monotonous decreasing with s a, the upper limit of x 3 can be reached at the FIG. 13. Projection of the scanning arcs, the Pl line case 1 and X onto the traverse plane. a The projection of ArcA, and b the projection of ArcB. The dotted curve shows the range of s a and s b.

11 3631 Zeng et al.: Cone-beam mammo-ct 3631 minimum s a, i.e., s a =s a+. As far as the lower limit of x 3 case1 x 1,x 2 is concerned, it is more complicated to calculate the maximum s a for a particular x 1,x 2. We must check all the possible boundaries from arc A the dotted line in Fig. 14a, such as s a and s a max. Besides, as ys a, x, and ys b are on the same line, the maximum s a is also limited by the boundaries on arc B the dotted line in Fig. 14b, such as s b+ and s b max. Hence, the lower limit of x 3 is the maximum one of those corresponding x 3 s at those four critical positions. For example, in Fig. 14, the maximum s a is achieved when the PI line intersects arc B on ys b max. Finally, the strategy to determine the limits can be described as follows: Upx 3 case 1 x 1,x 2 = x 3 s a+ = x 3 s b ifsa exists x 3s a max = Lowx 3 case 1 x 1,x 2 2 = maxx3sa, +2sm A8 x 3 s b+,ifs b+ exists x 3s b max = 3 2 where Upx 3 case1 x 1,x 2 is the upper limit of x 3, Lowx 3 case1 x 1,x 2 is the lower limit of x 3. And for the second case s assumption, we can analysis it similarly. 3. Definition of Ū, Ū 1, and Ū 2 Now, we can define the region Ū as A9a, which is the disjunction of the regions determined in cases 1 and 2. Based on the earlier results, it is easy to see that for any xū, it admits at least one PI line and at most two PI lines. The upper and lower surfaces of the region Ū are shown in Fig. 14. Furthermore, the two PI-segments region Ū 2 is defined as in A9b, which admits two PI segments and has the same upper surface as region Ū. Ū 2 s lower surface is shown in Fig. 14c. Ū 1 is defined as A9c, which admits only one PI line. Figure 14d shows the relative position between Ū 2 and Ū 1, FIG. 14. Geometry of the exact reconstruction region Ū. a The upper surface of the region Ū with respect to two tilting arcs, b the lower surface of the region Ū, c the lower surface of the region that allows 2 Pl lines, and d the 2 Pl lines region vs the 1 Pl line region the 1 Pl line region in green, and the 2 Pl line region in red.

12 3632 Zeng et al.: Cone-beam mammo-ct 3632 = x 2 1 Ū x cos 2 t + x 2 2 R 2, and given a x 1,x 2, x 3 minlow case 1,Low case 2,maxUp case 1,Up case 2, Ū 2 = x 2 1 x cos 2 t + x 2 2 R 2, and given a x 1,x 2, x 3 maxlow case 1,Low case 2,minUp case 1,Up case 2, Ū 1 = Ū Ū 2. A9a A9b A9c APPENDIX B: PROOF OF THE IMPOSSIBLE CASE FIG. 4 c Again, let us denote s b=s b s mx, s =s s mx, and s t=s t +s mx. If the case exists Fig. 4c, as pointed out by Katsevich 29 for the circle-and-arc trajectory, there must be a plane tangent to C a at s t. Thus, we must have sin s t cos t cos s t sin s t sin t t cos s cos s tcos t sin s sin s t cos s t + cos s sin t =0, B1 cos s b cos s tcos t sin s b sin s t cos s t + cos s bsin where 3/2s s b, sin s bsin s t. Simplifying B1, we have 2 cos t sin tsin s t sins s b + cos s cos s b =0. B2 Since 0t/2, cos t sin t0, we have sin s t sins s b + cos s cos s b =0. On the other hand, we obtain that cos s cos s b + sin s t sins s b =cos s cos s b sin s b sins b s + sin s b sin s tsins b s =cos s cos s b sin 2 s b cos s + sin s b cos s b sin s + sin s b sin s tsins b s = cos s b1 sin s b sin s + cos 2 s b cos s + sin s b sin s tsins b s B3 = cos s b1 coss b s + sin s b sin s tsins b s 0, B4 where we have used the fact that for 3/2s s b, sin s bsin s t. Therefore, we arrive at a contradiction. a Electronic mail: kai-zeng@uiowa.edu b Electronic mail: hengyong-yu@ieee.org c Electronic mail: L-Fajardo@uiowa.edu d Corresponding author. Electronic mail: ge-wang@ieee.org 1 L. T. Niklason, Current and future developments in digital mammography, Eur. J. Cancer 38, S14 S J. M. Lewin, C. J. D Orsi, and R. E. Hendrick, Digital mammography, Radiol. Clin. North Am. 42, A. I. Mushlin, R. W. Kouides, and D. E. Shapiro, Estimating the accuracy of screening mammography: A meta-analysis, Am. J. Prev. Med. 14, D. G. Grant, Tomosynthesis: A three-dimensional radiographic imaging technique, IEEE Trans. Biomed. Eng. 19, J. T. Dobbins III and D. J. Godfrey, Digital x-ray tomosynthesis: Current state of the art and clinical potential, Phys. Med. Biol. 48, R65 R E. D. Pisano and C. A. Parham, Digital mammography, sestamibi breast scintigraphy, and positron emission tomography breast imaging, Radiol. Clin. North Am. 38, x. 7 L. T. Niklason et al., Digital tomosynthesis in breast imaging, Radiology 205, G. M. Stevens et al., Circular tomosynthesis: Potential in imaging of breast and upper cervical spin Preliminary phantom and in vitro study, Radiology 228, D. P. Chakraborty et al., Self-masking subtraction tomosynthesis, Radiology 150, D. N. Ghosh Roy et al., Selective plane removal in limited angle tomographic imaging, Med. Phys. 12, D. J. Godfrey, A. Rader, and J. T. Dobbins III, Practical strategies for the clinical implementation of matrix inversion tomosynthesis MITS, Medical Imaging 2003: Physics of Medical Imaging, Feb The International Society for Optical Engineering, San Diego, CA, R. Gordon, R. Bender, and G. T. Herman, Algebraic reconstruction techniques ART for three-dimensional electron microscopy and x-ray photography, J. Theor. Biol. 29, P. Bleuet et al., An adapted fan volume sampling scheme for 3-D algebraic reconstruction in linear tomosynthesis, IEEE Trans. Nucl. Sci. 49, T. Wu et al., A comparison of reconstruction algorithms for breast tomosynthesis, Med. Phys. 31, C. H. Chang et al., Computed tomographic evaluation of the breast, AJR, Am. J. Roentgenol. 131, B. Chen and R. Ning, Cone-beam volume CT breast imaging: Feasibility study, Med. Phys. 29, J. Zhong, R. Ning, and D. Conover, Image denoising based on multiscale singularity detection for cone beam CT breast imaging, IEEE Trans. Med. Imaging 23, L. A. Feldkamp, L. C. Davis, and J. W. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A 1, P. Grangeat, Mathematical frame work of cone beam 3D reconstruction via the first derivative of the radon transform, Lect. Notes Math. 1497,

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