REAL-TIME and high-quality reconstruction of cone-beam

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1 Real-Time 3D Cone Beam Reconstruction Dzmitry Stsepankou, Klaus Kornmesser, Jürgen Hesser, Reinhard Männer Abstract The paper presents a comparison of filtered backprojection and iterative approaches (modified SIRT and OSEM) where backprojection is performed on FPGA hardware and forward projection is realized by a graphics card. The FPGAarchitecture allows scaling parallel processing of the backprojection with a constant memory bandwidth and it performs its task about a factor of 6 faster than conventional graphics card architectures (using a Xilinx Virtex II 6000 compared to a floating-point implementation on a graphics hardware). For the forward projection a graphics card (with NVIDIA GeForce FX 5900) is used. The system reconstructs a volume of 56 3 voxels and 60 projections (51 ) within 6 seconds for each backprojection step and 1.5 s for the forward projection. As assumed filtered backprojection is the fastest method but shows most artifacts. SIRT and OSEM generate results with suppressed artifacts and require between 4 (SIRT) and 3 (OSEM) minutes for reconstruction. The limited precision of on the FPGA does not lead to any degradation in the reconstructed image quality. Index Terms Cone-beam reconstruction, FBP, iterative reconstruction, real-time reconstruction, FPGA, hardware-acceleration. I. INTRODUCTION REAL-TIME and high-quality reconstruction of cone-beam data is highly relevant for operation control and online registration in radiation therapy; and it becomes more and more a limiting factor for further progress. Additionally, the reconstruction algorithms should be able to cope with the limited data problem in order to be able to reduce the dose for the patient. In this paper we discuss an evaluation of some modern reconstruction techniques implemented on our hardware system that fulfills the aforementioned requirements. One standard approach to reconstruct volumes from x-ray images is the filtered backprojection algorithm (FBP) [1]. It essentially performs first a filtering of the projection data and then the backprojection of them towards the source. This algorithm is often used due to its good compromise between reconstruction quality and speed []. If more reconstruction time is available, there are more sensitive approaches like ART (Algebraic Reconstruction Technique) that both can reduce reconstruction artifacts and the necessary data that is required [3]. The reconstruction time for a 56 3 volume typically requires about seconds for single processor machines [4]. There are several approaches to speed up processing. For example [4] uses a PCI-board with 8 RISC like processors. Others implemented the backprojection algorithm on a special-purpose processor [5] or on FPGAs [6] [7]. Texture mapping on graphics hardware was used by Cabral [8] for FBP and Mueller D. Stsepankou (dstepank@rumms.uni-mannheim.de) is with Institute for Computational Medicine, Universities Mannheim and Heidelberg, B6, 6, D Mannheim, Germany. [9] for several iterative techniques (ART and SART), and most recently [10] [11] OSEM (Ordered Subsets Estimation Maximization) [1] using an NVidia GeForce FX graphics hardware. The main problem of backprojection is the memory bandwidth-limitation, i.e. increasing the internal processor speed does not increase the performance of the whole system. While the performance of the processor doubles every 18 months, the memory bandwidth does not cope with that performance increase and therefore there is a widening gap. All approaches so far have to cope with this limitation. In our approach we have overcome this limitation by a dedicated new reconstruction architecture that is implemented on an FPGA system. The performance increase of FPGAs over years is by a factor of 4 every 18 months - even larger than for PCs and therefore it is assumed that in the near future they will dominate high-performance applications. II. ALGORITHMS In our work we consider the transmission X-ray cone-beam CT case with the flat-panel detector. Y P oisson(b 0 e a iµ i i + r) (1) where b 0 is a blank count, µ i is the absorption coefficient, a i are the interpolation coefficients and r stands for both background and detector noise. After acquisition the negative logarithm usually applied on data. p = log (Y/b 0 ) () To evaluate proposed architecture, we have applied several different reconstruction approaches. The Filtered backprojection is the analytical reconstruction method. The FDK [1] algorithm accumulates the weighted projection data on the voxels of the 3D grid where the weighting of the projection data accounts for the divergent beam for flat-panel: D p uv = p uv (3) D + u + v Here D is a source to detector distance, u and v are the offsets from the detector center. Then the projection data are filtered one dimensionally with the ramp filter [13] along lines perpendicular to the rotation axis. And finally the accumulation is performed: µ xyz = 1 N φ 1 k=0 R p(φ, u(φ, r),v(φ, r)) (4) L(r) The functions u(φ, r) and v(φ, r) map the object coordinates r(x, y, z) into detector coordinates u, v for particular view

2 angle φ. R - source to isocenter distance, L(r) is the distance from the source to the point r(x, y, z) to be reconstructed, projected onto the central ray. All these geometric parameters are conveniently obtained from the CT projection matrices. The iterative approaches make use of projection operation: p i = N 3 1 k=0 µ k w ki, (5) where w k are the weighting coefficients, and the backprojection operation: µ k = N φ 1 i=0 p i l ik (6) Here l i is the combination of factors from equations (3) and (4). The simulated projections are used then to calculate the correction factors for the reconstructed volume, which is incrementally updated and backprojected on each iteration. The algebraic reconstruction approaches assume the following equation system: ˆPµ = p (7) Here ˆP is the projection operator, µ is the volume data vector, p is the projection data vector. The gradient descent method solves this equation by the iterative formula: µ n+1 = µ n λ Jµ n, (8) where Jµ n is the gradient of the following functional: J(µ) = 1 ˆPµ p = 1 (µ), (9) and λ is the iteration step, which could be fixed at the simplest case. Adapting λ for each iteration however leads to better convergence. In this work we have used the method of minimum errors, firstly described by Fridman [14]. Here the iteration step is chosen by the condition of minimum the following functional µ n+1 µ, where µ is the solution of equation (7). Minimizing that functional gives: n λ n = Jµ n (10) Since we have the ill-posed problem, we can incorporate some penalty term into equation. N 3 1 R(µ) = w kh V (µ h µ k ), (11) k=0 h N k where N k is the neighborhood of voxel k. Here we use the MRF Gibbs penalty [15] with the edge-preserving Huber potential function: V (µ) = { µ δ, for µ δ, µ δ/ δ, for µ δ. (1) SIRT Simultaneous Iterative Reconstruction Technique) [16] performs the update for whole volume each step. Let label the forward projection operation (5) by FP(µ) and the backprojection (6) by BP(p). Combining the above equations with SIRT we get the following iterative formula: µ n+1 = µ n λ n (BP(FP(µ n ) p 0 )+βr(µ n )), (13) where p 0 is initial projections, β - is the regularization parameter. As an example of statistical iterative approaches we have implemented the OS-EM [1] algorithm. µ n+1 = µ n BP( )/BP (1) (14) FP(µ n ) This algorithm was originally derived for emission tomography, however some researchers had successfully applied it for transmission as well [17]. Since the poisson distribution of the raw scanner data is damaged by log in the transmission case (see ()), we had implemented the NEC scaling, proposed in [18], to restore the poisson distribution of the sinogram. III. IMPLEMENTATION Most of the reconstruction approaches could be subdivided into following steps: preprocessing (i.e. filtering), backprojection, forward projection (for iterative algorithms) and postprocessing (i.e. volume update calculation in iterative approaches). The most time consuming parts are backprojection and (forward) projection. Therefore, these two steps require hardware acceleration. Since graphics hardware is best suited to the projection step we choose the backprojection for FPGA acceleration. A. Backprojection In backprojection the contribution of each projection pixel to each affected voxel has to be computed leading to an overall complexity of the operation of O(N 4 ), assuming a volume of O(N 3 ) and O(N) projections. Similarly, we require the same amount of memory accesses defining the memory bandwidth. Since the internal clock-frequency of FPGAs is much smaller than that of normal CPUs one has to introduce parallelism to cope for that disadvantage. One typical approach is pipelining, i.e. the innermost loop of the reconstruction algorithm, i.e., how to compute the contribution of a projection to one voxel, is done on a hardware pipeline, that produces one result per clock cycle. In order to further increase the reconstruction speed, several pipelines are used where each pipeline processes the contribution of one projection plane to the considered voxel. However, since each pipeline needs access to projection and voxel data a higher number of pipelines require a proportional higher memory bandwidth, i.e. more memory devices need to be accessed in parallel. The main innovation that decouples memory bandwidth from the degree of parallelization is a suited caching mechanism. We essentially assume that the source-detector-unit rotates about one fixed axis about the imaged object. Next, we assume that we reconstruct slice-wise (see Fig 1). The rotation axis is p 0

3 is reduced by a factor of M. Fig 3 represents the block-diagram of the architecture. The backprojection architecture is voxel-driven and uses bilinear interpolation in the projection data. For this system we have developed a schema that assumes the rotation of the sourcedetector-unit about a given axis. Slabs perpendicular to this axis are backprojected using several projections in parallel. In the current system 1 pipelines are implemented. This means that we process the projection data into bundles of 1, i.e. we first apply backprojection on the first bundle obtaining a volume that is improved by the next bundle etc. Fig. 1. Slice-wise backprojection parallel to the slice normal and therefore the projection of one slice onto the acquired projections covers two or more lines depending on the cone beam angle. The larger the angle, the larger the number of such lines. For an efficient caching strategy we should be able to store all lines locally on the FPGA in order to minimize the cache reloading (see Fig ). We assume a pipelined system with identical pipelines and where each pipeline computes the contribution of a projection to the same voxel of the considered volume slice. Thus, for M pipelines we update M contributions for one voxel in parallel before considering the next voxel in the slice. Fig. 3. Block diagram of FPGA design B. Projection With the appearance of the new generation of consumer graphic cards (such as Nvidia GeForce FX), it becomes possible to accelerate the high quality projection of 16bit images using 3bit floating point calculations. The projection is the common operation for such a hardware, therefore the implementation of it is straightforward. For the line integral 3D texture mapping was selected: the volume is loaded into texture memory as a 3D texture with 16bit greyscale precision. Next, the geometry transformation (i.e. CT projection matrix) is applied where all the slices are interpolated tri-linearly. Finally, the accumulation into a high precision 3bit floating point off-screen buffer generates the projection (see Fig 4). Since we have 8 pipelines in current graphics card (GeForce FX 5900), here we have 8 pixels processed at one time. Fig.. For cache efficiency, we have to be able to store all the projection lines necessary for single volume slice Since for that configuration we can process the contribution of M pipelines to N voxels in the slice within N /M steps we have to read O(N +k N M) data. For the whole volume this amounts to O(N 3 +k N M) data accesses. Since we have only processed M instead of O(N) projections, this process has to be repeated N/M times leading to O( N M MN + N M N 3 )= O(N 3 + N 4 M )=O( N 4 M ). In other words, the memory bandwidth IV. RESULTS In our approach we programmed backprojection into a Xilinx Virtex II 6000 FPGA with 1 pipelines operating in parallel [19] [0]; it is approx. a factor of 6 faster than graphics card implementations: for a 56 3 volume and 60 projections (51 ) it requires 3s for backprojection and 3s for configuring the FPGA and data transfer. The latter is due to the fact that the FPGA is not large enough to implement the logic for parallel I/O during reconstruction.

4 Fig. 6. The results for FBP, OSEM and SIRT for sparce projection data (60 projections 51, volume size 56 3 ). Fig. 4. Accumulating of the 3D texture slices into pixel buffer Fig. 7. The results of the iterative algorithms (1, - OSEM, 3 - SIRT) for real CT datasets (10 projections 51, volume size 56 3 ). For evaluating the performance and quality we used several CT volumes of size 56 3 as basis and generate virtual projections of size 51. We have used the 3D extended Shepp- Logan phantom to demonstrate the reconstruction quality and several clinical CT datasets to show how the proposed system could reconstruct the real data. Table I shows the timings for the various algorithms. FBP shows fastest result, but has the disadvantages when the small number of projections is used. Iterative approaches consumes a longer time, however they could be used with the less projection number. TABLE I TIMINGS FOR VARIOUS ALGORITHMS Iterations Overall time Backprojection Projection FBP - 6s 6s - SIRT 10 4min 6 s/iteration 1.5 s/iteration OSEM 6 3min 1 s/iteration 15 s/iteration Firstly, we have shown that the performance of the FPGAdesign scales linearly with the amount of resources that is quadrupling approximately every two years. Currently, the fastest FPGA has four times the resources of the one we used; thus, using the same design, we could reduce backprojection including reload and transfer to 0.75 s. Secondly, the forward projection is ideally suited to graphics hardware, thus with the most recent graphics card we can cut the forward projection time by a factor of. However we have observed that reading the projections from GPU memory is currently a slow operation and it could be a bottleneck for further speedup. In addition, the quality of the results is not affected by the limited precision of computing on FPGAs since we apply a problem specific scaling before performing backprojection. ACKNOWLEDGMENT This project was supported by BMBF grant 01EZ004. REFERENCES Fig. 5. The original of 3D extended Shepp-Logan phantom and the FBP reconstructions using software and hardware implementations (40 projections 51, volume size 56 3 ). V. CONCLUSIONS AND FUTURE WORK In the paper we have shown how our proposed hardware architecture could be used to accelerate different reconstruction techniques. The main advantage of this system is twofold. [1] L. A. Feldkamp, L. C. Davis, and J. W. Kress, Practical cone-beam algorithm, J. Opt. Sco. Amer, vol. 1, no. A6, pp , [] T. H. Lin, G. Wang, and P. C. Weng, A multiple cone beam reconstruction algorithm for X-Ray microtomography, Springer Series in Optical Science, vol. 67, X-Ray Microscopy III, 199. [3] R. Gordon, R. Bender, and G. T. Herman, Algebraic reconstruction techniques for three-dimensional electron microscopy and X-Ray photography, J. Theor. Biol, vol. 9, pp , [4] [5] T. Schmitt, D. Fimmel, M. Kortke, and R. Merker, Parallel processor arrays for filtered backprojection, Computer Aided Systems Theory- EUROCAST 99, pp , Springer, 000. [6] I. Goddard and M. Trepanier, High-speed cone-beam reconstruction an embedded systems approach, Proc. SPIE Medical Imaging, vol. 4681, pp , 00. [7] S. Coric, M. Leeser, E. Miller, and M. Trepanier, Parallel-beam backprojection: an FPGA implementation optimized for medical imaging, Proceedings of the 00 ACM/SIGDA tenth international symposium on Field-programmable gate arrays, pp. 17 6, 00.

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