Planar tomosynthesis reconstruction in a parallel-beam framework via virtual object reconstruction

Planar tomosynthesis reconstruction in a parallel-beam framework via virtual object reconstruction Brian E. Nett a,shuaileng a and Guang-Hong Chen a,b a Department of Medical Physics, University of Wisconsin-Madison, WI 53704; b Department of Radiology, University of Wisconsin-Madison, WI 53792 ABSTRACT A framework for image reconstruction from planar tomosynthesis trajectories (i.e. all source positions reside in a single plane) is presented. The parallel beam geometry is a convenient starting point in deriving reconstruction algorithms, both analytic and iterative, as the relation between frequency space and image space is well known. We present a method for utilizing parallel beam reconstruction algorithms in an internally consistent manner. The concept of a virtual image object is utilized. This virtual image object has the property that cone-beam projections through the real object are directly related to parallel-beam projections of the virtual object. The virtual object may then be reconstructed using any algorithm derived for parallel beam projections. Finally, an affine transform may be applied to the virtual image object in order to generate the reconstruction result. In the implementation described here the backprojection operation is performed such that the real image object is reconstructed without introducing a rebinning in image space. Image reconstruction comparisons are given for a standard filtered backprojection (FBP) type algorithm where parallel projections were assumed in the algorithm derivation. Numerical simulations were performed for a C-arm type geometry and parallel beam FBP reconstructions using the virtual object are compared with the standard backprojection algorithm. Finally, a comparison was made between the new parallel beam reconstruction and the standard approximation where the cone-beams are assumed to be approximately parallel beams and a cone-beam backprojection is employed. A reduction in streaking artifacts was observed using the new algorithm compared with the standard approximation. 1. INTRODUCTION In a tomosynthesis acquisition typically several cone-beam projections of the object are taken from different source and detector positions. In the literature on tomosynthesis parallel beam geometry is often assumed when deriving reconstruction algorithms. 1 4 These algorithms which have been derived in the parallel beam limit are then applied directly to the cone-beam case, without a strong theoretical foundation. There are significant advantages of assuming parallel beam geometry as the relationship between the projection data and the Fourier transform of the image object is well known. Thus, there is a motivation to utilize a parallel-beam geometry. Since it is not feasible to acquire parallel beam data from the tomosynthesis trajectories in practice. In this paper, we utilize the concept of a virtual object to convert cone-beam projections of a real object into parallel beam projections of a virtual object. Although methods have been proposed to populate frequency space 5 from divergent beam projections via summing many shifted local Fourier transforms, the projection from a single view no longer corresponds to a single localized slice in frequency space. In the framework presentedherewedemonstrate, for planar tomosynthetic trajectories, a local transform that preserves the one-to-one correspondence between a measured projection and a slice in frequency space, as this correspondence is crucial in algorithm development. This framework for tomosynthesis reconstruction may be utilized for any planar tomosynthesis acquisition (i.e. the source moves in a single plane during the acquisition). When the detector is oriented parallel to the source plane the conversion algorithm may be applied directly to the acquired projection data. Another possible acquisition for planar tomosynthesis is acquiring cone-beam projections from a circular tomosynthesis acquisition using an interventional C-arm system. 6 This type of acquisition may be of interest for providing interventional fluoroscopy and/or interventional perfusion measurements. In order to utilize the virtual object concept to perform the reconstruction rebinning of the projection data is utilized. After the data has been rebinned to a All scientific correspondence should be addressed to G.-H. Chen via email: gchen7@wisc.edu 6510-79 V. 7 (p.1 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

virtual detector that is parallel to the source trajectory the reconstruction may be carried out using the virtual object concept. Simulation results included here demonstrate: the concept of the virtual object, the dependence of the virtual object scaling on acquisition geometry, and the correspondence between cone-beam projections of the real object and parallel beam projections virtual object. Additionally, one of the reconstruction algorithms derived in the parallel beam framework is implemented here using the virtual object concept. The filtered backprojection based algorithm proposed by Pelc 1 that compensates for the density of frequency space sampling of the tomosynthesis trajectory was implemented here. The filter function in this algorithm was derived in a parallel beam framework. A comparison between the implementation using parallel projections through the virtual object and cone-beam projections through the real object is presented. A reduction in streaking artifacts was observed using the new algorithm compared with the standard approximation. 2. METHODS In this work we propose an internally consistent reconstruction algorithm for planar tomosynthesis (i.e. all source positions reside in a single plane that is parallel to the reconstructed image planes). This will be accomplished via reconstruction of a virtual image object which has the property that for our proposed acquisition geometry the cone-beam projections of the real image object are directly related to parallel beam projections of the virtual image object. In this framework the virtual image object need not be directly reconstructed as the conversion from real object to virtual object as well as the scaling operation may be taken account in the backprojection step. Note that the virtual object must be invariant as the x-ray source moves to different positions. In planar tomosynthesis the x-ray source is constrained to a given plane (e.g. linear tomosynthesis, circular tomosynthesis, etc.). Thus, the mapping relating the real object to the virtual object may be dependent on the distance between the source and detector plane, but must be independent of the source s position within the source plane. The affine transform which we have utilized in this work to map between the real and virtual object satisfies this criteria. Additionally, we note that in some acquisition geometries the projection data is acquired on a detector that is not parallel to the source-plane. In this case the projection data may be re-binned to a corresponding virtual detector that is parallel to the source plane. The virtual object framework may then be applied directly using this virtual detector. We stress that for planar tomosynthesis we will use a source position independent (within a given plane) mapping between the real object space and the virtual object space. 2.1. Three Dimensional Virtual Object The mapping from the real object to the virtual object is presented below. This mapping is derived so that the measurements on the detector of cone-beam projections may be directly related to parallel beam projections of a virtual object. The concept of treating divergent X-ray projections as parallel projections was previously proposed by Edhom and Danielsson. 7 Here we follow a similar formalism but introduce an additional parameter in our system parameterization and the virtual object definition utilized here is slightly different than that previously proposed. The direction chosen here for the parallel rays is parallel to the iso-ray which passes from the source point to the origin of the object space. The geometry utilized in this derivation is given in Figure 1. Where a given point source of divergent x-rays is denoted by S(x s,y s,r) and resides in a given plane a height R above the x y axis. In the 3D case the source is free to move along any planar trajectory. The measurements of the divergent rays are then made on a detector which is separated from the planar source by a distance D and lies parallel to the source plane. For a sample point in the object, P (x p,y p,z p ), the divergent ray that originates at the source point intersects the detector plane at point A(x A,y A, (R D)). The first step in generating a virtual object is to establish a mapping with one-to-one correspondence between the real image object and the virtual image object. The iso-ray is defined here as the ray which originates from the source point and passes through the origin of the real image space. The attenuation along ray SA provides the divergent beam measurement detected at a detector element centered at point A. Thismeasurementatpoint A may also be associated with the attenuation of the virtual object along a parallel ray. The direction of choice here is rays which are parallel to the iso-ray. In this case the ray VAis parallel to the ray S0 and thus we would like to relate the attenuation along the ray VAin the virtual object to the attenuation along SA. Therefore, we 6510-79 V. 7 (p.2 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

need a one-to-one mapping which relates each point along the ray SA to a point along the ray VA. For simplicity the mapping we choose here is to draw a line from the origin though the image point and the point at which this line intersects the virtual ray is the virtual point associated with this real point (i.e. P (x, y, z) V (x v,y v,z v )). Thus, in order to derive the relationship between the real and virtual object we must simply relate (x, y, z) to (x v, y v,z v ). This mapping is derived using the similar triangles of Figure 1. z z v = PP 0 VV 0 = OP OV = z v = 1+ PV z OP VA SO = VV D SS 0 VA SO = PV OP z v z 1 = z v + D R R z v = = z v + D R R OP OP + PV = 1 1+ PV OP D R z z (6) z v = Q z, where Q = D R z. (7) Based on our mapping choice it is clear that the scaling factor is identical for all three axes. Therefore, the virtual object coordinates are related to the real object coordinates as given below. x v = Q x, y v = Q y, z v = Q z, where Q = D R z. (8) The second step in establishing the virtual object concept is to account for the difference in projection value for the rays traversing the real object and the rays traversing the virtual object. Namely, we examine the path lengths in the real and virtual object spaces: Proj = dlf(x, y, z), and Proj v = dl f v (x v,y v,z v ), (9) PA where dl = VA dz, and dl = dz v cos θ 0 cos θ = DRdz (R z) 2 cos θ, where each geometric parameter is demonstrated in Figure 1, and the parameters of Proj and Proj v have been suppressed. After substituting the relationships for dl and dl back into the integral equation one obtains the following relationships dz Q 2 Rdz Proj = f(x, y, z), and Proj v = cos θ 0 D cos θ f v(x v,y v,z v ). (10) By examining the above equations one may make an informed choice of the virtual object, f v (x v,y v,z v )= 1 f(x, y, z). (11) Q2 (1) (2) (3) (4) (5) Given this choice of the virtual object we may substitute this definition back into our expression for a given virtual projection. 6510-79 V. 7 (p.3 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

z S( x, y, R) S S V( xv, yv, zv) R 0 S0 ( xs, ys,0) y P( x, y, z) V0 ( xv, yv,0) D O P 0 ( x, y,0) x V ( x, y, R D) D v v (D-R) A( x, y, R D) A A (a) Figure 1. The geometrical configuration assumed here in deriving the virtual object mapping. Proj v = Rdz f(x, y, z) (12) D cos θ By comparing the expression for Proj in Eq. 10 to the expression for Proj v in Eq. 12 we obtain the relationship between the projections of the real object and the projections of the virtual object. Proj v = R D cos θ 0 Proj (13) cos θ We may then incorporate the definitions of the cos θ and cos θ 0 into this relationship, as provided below cos θ = R D and cos θ 0 = x 2 s + ys 2 + R 2 (xa x s ) 2 +(y A y s ) 2 + D, (14) 2 Proj v = x 2 s + y 2 s + R2 (xa x s ) 2 +(y A y s ) 2 + D 2 Proj. (15) This multiplicative factor that relates the parallel beam projections of the virtual object and the cone-beam projections of the real object will be referred to here as the virtual projection scaling. 2.2. Geometric Detector Description of C-arm Type System First, we will introduce the geometry of consideration here as shown in Figure 2. The angle t is measured with respect to the x axis and positive angles are measured from x in the direction of y. The angle θ is measured 6510-79 V. 7 (p.4 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

t z x R y D û SSD û vˆ ŵ vˆ ŵ Figure 2. Demonstration of the three separate coordinate systems which will be employed here. The geometrical parameters of interest in this geometry are: R the distance from the source plane to the origin, D the distance from the source plane to the virtual detector plane, û, ˆv, ŵ the unprimed local unit vectors of the real detector, û, ˆv, ŵ the primed local unit vectors of the virtual detector, ˆx, ŷ, ẑ are the lab unit vectors, SSD distance from the source to the detector surface, SOD distance from the source to the origin, t the view angle parameter, and θ the half tomo angle for this geometry. with respect to the z axis. In the cone-beam geometry the iso-ray originates from the focal point and passes through the origin and intersects the center of the detector. The source to surface distance of the detector will be referred to as the SSD, and likewise the source to origin distance is referred to as the SOD. These distances are measured along the iso-ray. When utilizing a standard interventional C-arm system each projection is taken in the unprimed coordinate system (Figure 2). The virtual object concept derived above was done under the condition that the detector plane is parallel to the source plane. Therefore, in order to apply the virtual projection scaling we introduce the primed coordinate system which is simply a translation of the world coordinate system (Figure 2). The relation between the unprimed and the primed coordinates is given below. We introduce several coordinate systems in order that we may easily derive associated relations to connect the primed and unprimed systems. The global three dimensional Cartesian coordinate system enables one to express a given position, r, in terms of its three components (x, y, z) r = xî + yĵ + zˆk, (16) where î, ĵ, ˆk are the unit vectors along the x, y, z axes respectively. Equally, well we may express r in terms of a coordinate system which rotates along with a detector that is perpendicular to the iso-ray (where the iso-ray originates at the focal point and passes through the origin). Note that in general these two coordinate systems do not share a common origin. Thus, we may represent r in this rotating coordinate system as, r = uû + vˆv + wŵ + O, (17) r = uû + vˆv + wŵ (SSD SOD)ŵ, (18) r = uû + vˆv +(w SSD + SOD)ŵ, (19) 6510-79 V. 7 (p.5 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

where if one stands upright at the source position directed toward the detector û points from right to left on the detector, ˆv points from bottom to top in the detector, and w points toward the detector from the focal point. In addition to this detector configuration we are also interested in the configuration in which the detector plane remains parallel to the x y axis so that the magnification is unaltered from view to view. Conveniently, the coordinate system located at the center of this detector is simply a translation of the global coordinate system: r = u û + v ˆv + w ŵ + O, (20) r = u î + v ĵ + w ˆk + O, (21) where the origin of this primed system is shared with the origin of the unprimed system. Now that we have defined our different coordinate systems we may use them to derive the quantities of interest. The unit vectors for the unprimed system are related to those of the global and primed systems as follows. sin t cos t 0 î û ˆv = cos t cos θ sin t cos θ sin θ ĵ (22) ŵ cos t sin θ sin t sin θ cos θ ˆk This relationship will be used directly to determine the parallel rebinning relationship from the primed system to the unprimed detector system. The parallel beam geometry here assumes rays which are parallel to w. If we have measurements of ray sums in the primed system (u,v,t,θ) and we would like to relate these to measurements in the unprimed system (u, v, t, θ) the following relation may be utilized: ( ( )( ) u sin t cos t u = v) cos t cos θ sin t cos θ v (23) Conversely, for parallel rays if we would like to determine a point in the primed detector from the unprimed detector we may use the following relationship: ( ) ( )( ) u sin t cos t v = cos θ u cos t sin t (24) cos θ v Note, that as θ goes to π/2 two of the matrix values in Eq. 24 go to infinity. However, when θ is π/2 the geometry corresponds to a cone-beam computed tomography acquisition, and thus in this limit we would use a cone-beam CT reconstruction formula rather than this planar tomosynthesis framework. 8, 9 Due to space constraints the cone-beam rebinning relationship between the two planes is not presented here but may be derived straightforwardly using this system geometry. 2.3. Reconstruction Steps In the case of planar tomosynthesis the acquisition will be of cone-beam projection data. Here we discuss applications where the X-ray source will remain in one given plane throughout the acquisition. Under these geometrical conditions we may design a unique virtual object which is an affine transformation of our original object. This conversion to a virtual object may be utilized with any of the reconstruction algorithms assuming parallel beams. 1 4 With the virtual object definition and the virtual projection scaling known we will outline the reconstruction steps below. The most basic implementation of the planar tomosynthesis reconstruction in the parallel beam framework is given in Figure 3 (a). This flow chart is provided for one to become familiar with this basic framework. This example is given for an acquisition where a flat panel detector lies parallel to the source trajectory. The virtual projection scaling may be applied directly on the acquired data in this case without the need for a re-binning operation (Eq. 14). One may then apply any filter derived assuming a parallel beam geometry. The data may then be backprojected via a parallel-beam backprojection in order to reconstruct the virtual object. After the virtual object is reconstructed one may re-grid from the virtual object space to the real object space using Eq. 8 and Eq. 11. 6510-79 V. 7 (p.6 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

Analytic Projections onto the unprimed Detector Cone-beam forward project onto a flat panel detector lying parallel to the source trajectory Cone-beam Re-binning from unprimed to primed detector Apply the virtual projection scaling Apply the virtual projection scaling Apply a filter if desired Parallel-beam backprojection to form a virtual object Re-grid from the virtual object to the real object Parallel-beam Re-binning from primed to unprimed detector Apply Parallel Beam Based Filter Parallel-beam backprojection from unprimed detector For each recon point Transform from real to virtual coordinates Find the backprojection location of the virtual point onto the detector Bi-linear interpolate among the filtered values Multiply the result by Q 2 end (a) (b) Figure 3. (a) Basic implementation steps. (b) Implementation steps for C-arm circular tomosynthesis. After describing the fundamental steps necessary for reconstruction in this framework we will present the required steps to perform reconstruction from the C-arm circular tomosynthesis geometry defined above. In this case the cone-beam projections are made directly onto the unprimed frame (real detector). A cone-beam rebinning maps the acquired cone-beam projections measured in the unprimed frame to the effective cone-beam projections which would have been measured in the primed frame (virtual detector). The virtual projection scaling may then be applied directly on the data in the virtual detector using Eq. 14. A parallel rebinning using Eq. 23 maps the scaled data to the real detector. On the real detector a parallel beam based filter is then applied. 1 3 Here we have implemented the kernel introduced by Pelc 1 : G θ (k u,k v )=( π sin θ ) k u W (k) (25) m θ where k u and k v are the conjugate variables of u and v respectively, k = k 2 u + k2 v, m θ is the number of projections and W (k) is a general window function added to control the tradeoff between resolution and the noise properties of the reconstruction. Finally, a parallel beam backprojection operation is utilized here. In this implementation we do not require an image space rebinning operation. The voxel driven backprojection algorithm utilized here directly reconstructs a point in the real object by: (1) finding the corresponding point in the virtual object (Eq. 8), (2) performing a bilinear interpolation along parallel rays, and (3) multiplying by Q 2 according to Eq. 11. Note this backprojection formalism via a virtual object has also been utilized in cone-beam CT reconstruction. 10 2.4. Parallel Beam Approximation The conventional approach which is employed in many tomosynthesis reconstruction algorithms is to assume parallel beams during the algorithm development and then approximately apply the derived algorithm to conebeam data. 2 4 The virtual object concept discussed here enables one to circumvent this approximation as parallel beams through the virtual object are directly generated in this framework. Thus, we will provide a comparison here using a filtered backprojection algorithm implemented both with the virtual object concept utilizing a parallel beam filter, and an approximation in which the parallel-beam filter is followed by a cone-beam backprojection. 6510-79 V. 7 (p.7 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

2.5. Reconstruction Parameters Numerical simulations have been conducted assuming cone-beam measurements are taken in the geometry described in Figure 2. Analytic projection data was generated through two phantoms: a small uniform sphere placed at the origin and a high contrast Shepp-Logan phantom. The radius of the small sphere was 0.05, and the parameters for the ellipsoids which make up the high contrast Shepp-Logan phantom are given in the Appendix. The scanning geometry was parameterized as: detector Size = 4 4, detector Sampling = 500 500, R=5, D=8, θ =15, and view Sampling = 40/(2π). The reconstruction of the small sphere was conducted with an in-plane image matrix of 512 512 and a total of 256 slices were reconstructed. The high contrast Shepp-Logan phantom was reconstructed with an in-plane image matrix of 256 256 and a total of 128 slices were reconstructed. No roll-off window was applied in the filtering step. 3. RESULTS 3.1. Virtual Object Concept For demonstration purposes the transformation of a simple geometrical object is included here to graphically visualize this operation. A simple cube of uniform linear attenuation is the sample object, and Figure 4 (a) demonstrates one given slice through this cube. The same slice is shown in Figure 4 (b) after the object has been transformed to a virtual object. Following this operation we apply the inverse transformation to arrive at the result given in Figure 4(c). Thus, as demonstrated in (d) after applying the forward and inverse transform the sample object is well recovered, with the exception of a slight overshoot at the boundary caused by slight interpolation error. Therefore, we have confidence that images reconstructed in the virtual object space may be transformed to the real object space in a straightforward manner. It is important to note that this example of image space rebinning is given to provide insight into the virtual object concept and in the actual implementation no interpolation is performed in image space (Figure 3 (b)). 1 0.8 0.6 0.4 0.2 0 20 40 60 80 100 120 140 160 180 200 (a) (b) (c) (d) Figure 4. The central x z slice of a centered uniform cube where each side has dimension 2 (a), the central x v z v slice of the virtual object (b), by applying the inverse transform to the image in (b) the central x z slice is recovered (c), a plot of the central vertical line through (a) and (c) is shown in (d) where the asterisk corresponds to the true value from (a) and the line corresponds to the recovered value from (c). After demonstrating the forward and inverse transform to the virtual object space we assess the effect of the magnification parameter on the virtual object. The phantom is demonstrated in Figure 5 (a). The phantom shown here is a cube with uniform attenuation value and a length of 0.5 on each side. The virtual object for three different magnification values will be investigated. By fixing D and scaling R the effective magnification of the system is varied here. It is clear that the ratio of R/D effects both the size and shape of the virtual object (Figure 5 (b-d)). As the ratio of R/D increases the shape of the virtual object more closely resembles the shape of the original object since the incident cone-beams are closer to the parallel beam limit. In the third set of numerical experiments we have demonstrated here the equivalence between the cone-beam projection of the original image object and the scaled parallel beam projection of the virtual image object Figure 6(a-c). The respective images of forward projections through the uniform cube using the standard Siddon ray tracing algorithm 11 are displayed in Figure 6(a) and Figure 6(b). A sample plot (Figure 6 (c)) demonstrates 6510-79 V. 7 (p.8 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

(a) (b) (c) (d) Figure 5. The x z cross section of the phantom (a) and three separate virtual objects for different locations of the source and detector plane (b-d): (b) R=1.1 and D=3, (c) R=1.5 and D=3, and (d) R=2 and D=3. the agreement between the projection values calculated for cone-beam projections of the original image object when compared with parallel beam projections of the virtual image object. Since we now have equivalent parallel projection measurements through the virtual object we can directly sample the Fourier space of the virtual object by utilizing the 3D projection slice theorem. 1.2 1 0.8 Ray Sum 0.6 0.4 0.2 0 (a) (b) 0.2 0 50 100 150 Projection Index (c) Figure 6. (a) The cone-beam projection of the original object where the focal point is (0,0,2.2). (b) Weighted parallel beam projection through the virtual object where the rays are parallel to the z axis. (c) A comparison plot of the projection onto the detector row v = 0 where the line corresponds to the cone-beam projection from (a) and the red circles to the parallel beam projection of the virtual object in (b). 3.2. Reconstruction Results Images have been reconstructed here using the virtual object concept in a filtered backprojection framework proposed by Pelc. 1 Since this algorithm utilizes a filter kernel derived in the parallel beam framework it is a good candidate to demonstrate the virtual object framework. However, we note that any algorithm derived with the parallel beam assumption may be utilized for planar tomosynthesis reconstruction. The two phantoms reconstructed were a uniform sphere and the high contrast Shepp-Logan phantom. A relatively small half tomographic angle of 15 was employed in this study and thus significant signal leakage between planes is expected. The filtered backprojection algorithm utilizing the virtual object concept is first compared with the standard backprojection reconstruction. The comparison for the case of a uniform sphere is given in Figure 7. Next the central slice of the high contrast Shepp-Logan phantom was reconstructed (Figure 8). As noted by Pelc 1 we expect that this FBP result will accentuate the high frequency content of the image. The density of sampling in frequency space has been normalized in the sampled regions. However, the circular tomosynthesis sampling in frequency space leaves large regions of frequency space unsampled. Thus, this tomosynthetic sampling effectively 6510-79 V. 7 (p.9 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

populates more of the high frequency components in frequency space. Therefore, the high spatial frequency components have been accentuate in Figure 7 and Figure 8. (a) (b) Figure 7. Reconstruction results for a uniform sphere placed at the origin: (a) the axial slices and (b) the coronal slices. The respective images: phantom (left), backprojection (center), and parallel beam filtered backprojection with the virtual object (right). Figure 8. Reconstruction results for the central slice of the high contrast Shepp-Logan phantom. The respective images: phantom (left), backprojection (center), and parallel beam filtered backprojection with the virtual object (right). The second set of reconstruction comparisons provided here demonstrate the difference between the parallel beam filtered backprojection algorithm utilizing the virtual object concept and the approximation to the algorithm of using the parallel beam filter and cone-beam backprojection. The approximation has the advantage that no rebinning is required on the detector plane, while the parallel beam reconstruction with virtual object has the advantage that the filter and backprojection operations are completely consistent. The reconstruction comparison for the small sphere are given in Figure 9. Additionally, the high contrast Shepp-Logan phantom has been reconstructed with both of these approaches (Figure 10). One may note an increase in streaking artifacts 6510-79 V. 7 (p.10 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

from the high contrast objects in Figure 9 (b) and Figure 10 when the parallel-beam filter is combined with the cone-beam backprojection operation. Based on the simulation results the approximation of treating the conebeam projections approximately as parallel beam projections in filter derivation is well justified. However, the streak artifact pattern is reduced using our new fully consistent algorithm. Quantitative comparisons between these two approaches are not provided here as the spatial resolution has not yet been matched. (a) (b) Figure 9. Reconstruction results for a uniform sphere placed at the origin: (a) the axial slices and (b) the coronal slices. The respective images: parallel beam filtered backprojection with the virtual object (left) and parallel beam filter with cone-beam backprojection. Figure 10. Reconstruction results for the central slice of the high contrast Shepp-Logan phantom. The respective images: parallel beam filtered backprojection with the virtual object (left) and parallel beam filter with cone-beam backprojection. 4. CONCLUSIONS A planar tomosynthesis framework has been proposed using a virtual object concept in the reconstruction algorithm. This framework may be utilized with any existing tomosynthesis algorithm derived assuming parallel beams. Additionally, new avenues for new algorithm development have been paved such as frequency space based iterative algorithms. 12 Reconstruction simulations for an FBP type algorithm demonstrate reduction in streaking artifacts from high contrast objects, while confirming the previous assertion that the cone-beams could be treated approximately as parallel beams. Future research includes quantitative resolution and noise measurements. Acknowledgments We would like to acknowledge our funding support: NIH R01 EB005712-01, #T32CA009206-24, and the Herman I. Shapiro Graduate Fellowship. Thanks to Professor Norbert Pelc for providing our group with a copy of his PhD thesis. Additionally, thanks to Charles Mistretta for helpful discussions, and Orhan Unal for computer network support. 6510-79 V. 7 (p.11 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM

Appendix The definition for the high contrast Shepp-Logan phantom used in the simulations presented here are given below. (x,y,z) β a b c ρ (0,0,.25) 0.69.92.9 1 (0,0,.25) 0.6624.874.88 -.8 (-.22,0,0) 108.41.16.21 -.2 (.22,0,0) 72.31.11.22 -.2 (0,0.35,0) 0.21.25.50.1 (0,.1,0) 0.046.046.046.1 (-.08,-.65,0) 0.046.023.02.1 (.06,-.650,0) 90.046.023.02.1 (.06,-.105,.875) 90.056.040.10.1 (0,.1,.875) 0.056.056.1.1 Table 1. The definition of the high contrast Shepp-Logan phantom where the conventions for the ellipsoids are taken from Kak and Slaney 13. REFERENCES 1. N. Pelc, A Generalized filtered backprojection algorithm for three dimensional reconstruction. PhD thesis, Harvard University, Boston, 1979. 2. G. Lauritsch and W. H. Harer, A theoretical framework for filtered backprojection in tomosynthesis, Proc. SPIE 3338, pp. 1127 1137, 1998. 3. G. M. Stevens, R. Fahrig, and N. J. Pelc, Filtered backprojection for modifying the impulse response of circular tomosynthesis, Med. Phys. 28, pp. 372 380, 2001. 4. H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, Three-dimensional image reconstruction by digital tomo-synthesis using inverse filtering, IEEE Tran. Med. Imaging 12, pp. 307 313, 1993. 5. G.-H. Chen, S. Leng, and C. Mistretta, A novel extension of the parallel-beam projection-slice theorem to the divergent fan-beam and cone-beam projections, Med. Phys. 32, pp. 654 665, 2005. 6. B. Nett, J. Zambelli, C. Riddel, B. Belanger, and G.-H. Chen, Circular tomosynthesis implemented with a clinical interventional flat-panel based C-Arm, SPIE Medical Imaging 6510-58, 2007. 7. P. R. Edholm and P.-E. Danielsson, Divergent X-ray projections may under certain conditions be treated as parallel projections, Computer Methods and Programs in Biomedicine 57, pp. 91 94, 1998. 8. L. A. Feldkamp, L. C. Davis, and J. W. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A 1, pp. 612 619, 1984. 9. B. E. Nett, T.-L. Zhuang, and G.-H. Chen, A cone-beam FBP reconstruction algorithm for short-scan and super-short scan source trajectories., Fully 3D Reconstruction in Radiology and Nuclear Medicine, 2005. 10. J. Zambelli, B. Nett, S. Leng, C. Riddell, B. Belanger, and G.-H. Chen, Novel C-arm based cone-beam CT using a source trajectory of two concentric arcs, SPIE Medical Imaging 6510-61, 2007. 11. R. L. Siddon, Fast calculation of the exact radiological path for a three-dimensional CT array, Med. Phys. 12, pp. 252 255, 1985. 12. J. Velikina, S. Leng, and G.-H. Chen, Limited-view-angle tomographic image reconstruction via total variation minimization, SPIE Medical Imaging 6510-71, 2007. 13. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE, Bellingham, WA, pp. 102, 1988. 6510-79 V. 7 (p.12 of 12) / Color: No / Format: Letter / Date: 3/3/2007 8:44:49 AM