R θ (t) 2. (θ) s. (t) θ 2. f(x,y) f(x,y) s 1. Fourier transform v. object

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1 Parallelisation of Tomographic Reconstruction Methods M.A. Westenberg and J.B.T.M. Roerdink Department of Mathematics and Computing Science, University of Groningen P.O. Bo 800, 9700 AV Groningen, The Netherlands Abstract In this paper we discuss the parallelisation of two different tomographic reconstruction methods: ltered backprojection and direct Fourier reconstruction. These algorithms are implemented on a Connection Machine CM-5. Filtered backprojection turns out to be more dicult to parallelise on a CM-5 than direct Fourier reconstruction. However, it is shown that for both methods a fast implementation is possible. Keywords: Computerised Tomography (CT), Connection Machine CM-5, Radon transform, ltered backprojection, direct Fourier reconstruction. Introduction Tomographic reconstruction is the art of image reconstruction from projections, i.e. line integrals of an unknown object function. The problem is to determine the internal structure of an object without having to damage it. Generally, various probes such as X-rays, gamma rays, visible light, electrons and other kinds of radiation are used to produce a set of parallel projections, called a prole. A set of such proles taken from dierent angles is called the Radon transform of the object. The mathematical framework of such reconstruction problems has been developed by Johann Radon in 97. The number of dierent elds in which reconstruction problems arise is quite large: astronomy, molecular biology, geophysics, optics and medicine, to name a few. The eld in which the most spectacular success has occurred is computerised tomography (CT) in medicine. This is re- ected in the awarding of the Nobel Prize in physiology or medicine to G. N. Hounseld and A. M. Cormack in 979. In diagnostic medicine, computerised tomography uses one or more X-ray sources and an array of detectors to measure the attenuation of X-rays along a number of parallel lines, resulting in a prole for some angle. These proles are measured for incremental values of and constitute a sampling of the Radon transform. By using an appropriate reconstruction algorithm, an image of a cross-sectional slice of some part of the human body is obtained. Tissues which have dierent attenuation coecients can be distinguished from each other and this property can be used to nd the presence of, for instance, tumours. This paper focuses on parallel ray CT and reconstruction methods for this technique. Section 2 gives the mathematical background of the Radon transform and its inversion. The net two sections describe two dierent reconstruction algorithms, viz. ltered backprojection and direct Fourier reconstruction. For both algorithms it is described how they can be parallelised on a Connection Machine CM-5. 2 The Radon transform and its inverse 2. The Radon transform in two dimensions Let f(; y) be a continuous function that is decreasing suciently fast at innity. Let be an angle between 0 and, t a real number, and? the line dened by? = f(; y) 2 IR 2 : cos + y sin = tg; () The integral of f over the line?, Rf(; t) := Z? f(; y) d dy; (2) is called the Radon transform of the function f. The integral Rf(; t), with and t ed, is called a projection and the function R : t 7! Rf(; t) a prole, cf. Fig. (a). To the Radon operator is associated a backprojection operator R # by R # g(; y) = Z g(; cos + y sin ) d: (3) Whereas R integrates over all points on a line, R # integrates over all lines through a point. There are two dierent modes used for sampling the line integrals. One is parallel beam scanning

2 t where parallel rays, i.e. line integrals, are determined for a ed direction. This process is then repeated for a number of dierent directions. The other is fan beam scanning where line integrals emanating from a source point are determined. This is repeated for a number of dierent source points, see Fig.. In this paper we restrict ourselves to parallel beam scanning. (t) R θ t f(,y) R θ (t) 2 (a) Parallel beam scanning y t θ θ 2 D (θ) s θ f(,y) s D s (θ) 2 y θ s2 (b) Fan beam scanning Figure : Scanning modes used in sampling the Radon transform. 2.2 Fourier slice theorem The relationship between the one-dimensional Fourier transform of a prole and a slice of the twodimensional Fourier transform of the original object function is stated in the Fourier slice theorem [2]: The Fourier transform of a prole obtained by parallel scanning of a function f(; y) taken at angle gives a slice of the two-dimensional Fourier transform, F (u; v), of f subtending an angle with the u-ais. In other words, the Fourier transform of R gives the values of F (u; v) along line BB in Fig. 2. Epressed in a formula, this theorem says br () = Z Z?? f(; y)e?i2( cos +y sin ) d dy; (4) where the right-hand side represents the twodimensional Fourier transform of the object function at a spatial frequency vector (u; v) = ( cos ; sin ). This result forms the basis of many dierent reconstruction techniques in parallel ray tomography. If innitely many angles and rays are taken, F (u; v) is known at all points in the uv-plane (and not only on a nite number of radial lines) and the object function f(; y) can be reconstructed using the inverse Fourier transform: f(; y) = Z Z?? F (u; v)e i2(u+vy) du dv: (5) 2.3 Sampling the Radon transform The standard parallel scanning procedure is to sample Rf(; t) on a polar grid, i.e. taking p directions (number of angles) uniformly distributed over the half-circle and 2q+ equally spaced values of t (number of rays), where p b and q b= [3]. Here b is the essential bandwidth of the function f, meaning that the Fourier transform b f() of f is negligible for jj > b. Insuciency of the data, either by undersampling a projection or by taking the number of projections too small, causes aliasing artifacts such as Gibbs phenomena, streaks (q too small) and Moire patterns (display resolution too small) [2]. The ideal relation between p and q is p = q [3]. In order to be able to assess the accuracy of the reconstruction algorithms which will be described in the following sections, one needs a reference image. We use the Shepp-Logan \head phantom" [8] which is common practice in the eld of CT. This phantom consists of 0 ellipses, each with dierent refractive inde, of which projection data can be computed analytically. s 3 Filtered backprojection R θ y θ Fourier transform v B θ u The ltered backprojection algorithm, see e.g. [2, 3], is currently the most used reconstruction method in the medical eld, and has proved to be very accurate. object Space domain B Frequency domain Figure 2: Relationship between the Fourier transform of a projection and the Fourier transform of the object along a radial line. 3. Algorithm By a change of coordinate system and rearranging the integral terms in (5), the algorithm can be derived. The rectangular uv-coordinate system in the frequency domain is transformed to a polar coordinate system. Using symmetry properties of the Fourier transform and the Fourier slice theorem, 2

3 equation (5) may be written as f(; y) = Z 0 Z? br ()jje i2t d d; (6) with t = cos + y sin. The multiplication with jj in the Fourier domain represents a ltering operation, called the ramp lter. The integral over corresponds to the backprojection operator. In practice one replaces this ideal ramp lter in the frequency domain by a function W b () which approimates jj for low frequencies but goes to zero for high frequencies, in order to suppress noise. This formula forms the basis of the ltered backprojection algorithm of inverting the Radon transform. In the discrete case the Radon transform Rf(; t) is assumed to be available for ( j ; s l ), j = ; : : : ; p, l =?q; : : : ; q, with j = (j? )=p; s l = hl; h = =q: (7) Here it is assumed that the support of f is the unit disk. The ltered backprojection algorithm in this case goes as follows. Step For j = ; : : : ; p carry out the convolutions v j;k = h qx l=?q w b (s k? s l )R j (s l ); k =?q; : : : ; q: (8) For the function w b, whose Fourier transform equals W b as mentioned above, see below. Step 2 For each reconstruction point (; y), compute the reconstruction f FBP by the discrete backprojection f FBP (; y) = 2 p px j= ((? u)v j;k + uv j;k+) ; (9) where, for each (; y) and j, k and u are determined by s = cos j +y sin j ; k s h k+; u = s h?k: Reconstruction may be performed on a (2q + ) (2q + ) grid, which corresponds to the sampling of the proles. One possible choice for w b is the Shepp- Logan lter. This is a lter with cut-o frequency b = =h, w b (s l ) = b2 4? 4l : (0) 2 Possible high frequency artefacts can be smoothed out by using a window function which goes smoothly to zero for high frequencies. Rowland [6] describes several windowing functions including the so-called Hanning window, which is used in this paper. 3.2 Compleity The number of operations needed for the convolutions is O(pq log q) if a FFT is used. The backprojection operation takes O(p) operations for each. If the reconstruction is performed on a (2q+)(2q+) grid, this leads to O(pq 2 ) operations. Using the optimal relation p = q, the total compleity is O(q 3 ). 3.3 Accuracy The relative error norm is dened by jo(; y)? R(; y)j2 P;y P;y jo(; ; () y)j2 where O(; y) represents the original phantom and R(; y) the reconstructed phantom. The relative error should always be considered together with a visual evaluation, because the L 2 -norm is not a very good error measure in CT [] Reconstruction of the Shepp-Logan phantom The results of the ltered backprojection algorithm applied to projections of the Shepp-Logan phantom are shown in Fig. 3. The projection data were generated using p = 256 and q = 27 and the reconstruction grid was of size The image on the left shows the reconstructed phantom and the image on the right the absolute value of the dierence between the original phantom and the reconstruction. The values from 0{0% of the range of the dierence image have been inverted and scaled to the full range of gray-values. As can be seen in the gure, there are only small edge eects inside the \head", and elsewhere the reconstruction is very accurate. Outside the largest ellipse Gibbs phenomenon artefacts appear which theoretically would disappear if an innite number of rays were used [2, 3]. Applying () to the part of the reconstruction shown in Fig. 4 results in a relative error of 0:002. This plot shows a part of the vertical line through the centre, i.e. the y = 0 line, of the Shepp-Logan phantom. The relative error computed over the whole image is 0:034. As can be seen in the gures, the ltered backprojection algorithm gives a very accurate reconstruction even though the ideal relation p = q is not eactly satised. 3.4 Implementation The FBP algorithm has been implemented on a Connection Machine CM-5 with 6 processors. There are two dierent implementations of the backprojection step of the algorithm. These are described in the sections below. In order to keep matters simple, only one input variable N is used for the number of rays, 3

4 N Filtering Backprojection Straightforward Lookup tables Table : Timing results in seconds on the CM-5 for dierent implementations of the backprojection step of the algorithm. The rst method is a straightforward implementation, the second one uses lookup tables. (a) (b) Figure 3: Reconstruction of the phantom (a) and the dierence with the original (b). The values from 0{0% are scaled to the full range of grey values to make the dierences visible. Projection data were generated for p = 256 and q = 27 and the reconstruction grid is of size Inside the \head" the reconstruction is very accurate and only small edge eects are visible. Outside the largest ellipse, the Gibbs phenomenon is visible although it is much reduced because of the use of a Hanning window. angles and the reconstruction grid. So, projections were generated using N rays over N angles, and the reconstruction grid was of size N N. Table shows the run-time of the algorithm for dierent values of N. The ltering step is easily implemented on the CM-5. The Connection Machine Scientic Software Library (CMSSL) has a specialised implementation of the FFT. This routine is highly optimised for the CM-5. So, ltering can be done eciently using FFT's Straightforward method Value Original Reconstruction Figure 4: A numerical comparison of a part of the y = 0 line of the reconstruction with the true values of the Shepp-Logan phantom, c.f. Fig. 4. In the rst method, formula (9) for the backprojection step is translated directly into CM-Fortran. In the implementation, three loops are needed: two loops over and y running over the reconstruction grid, and one loop running over the angular variable j. In contrast to what (9) suggests, the loop over j is chosen to be the outer loop. This leads to a more ecient implementation on the CM-5 for the following reason. If a number of loops is used on this machine, at least one of these loops must be serialised. It is faster to make one loop serial instead of two nested loops, because serial loops are eecuted by the front-end. So in this case, for each j the front-end instructs the processing nodes to eecute the loops on and y in parallel. Timing results are shown in Table. As can be seen, the backprojection step consumes a lot of time and grows roughly by a factor of 6 when N is doubled. Detailed investigation showed that most time was consumed by communications between dierent processors. The backprojection requires for each piel in the reconstruction dierent values of the ltered projections for ed angle. As these projections are distributed over dierent processors, this step requires a lot of communication, i.e. a maimum of O(N 2 ) communications for each angle. 4

5 3.4.2 Use of a lookup table The communicational demand can be decreased by making use of a so-called lookup table. This is a construct provided by the Fortran Library on the CM-5. When creating a lookup table for a given array of values, a copy of this table is assigned to each processor. Addressing can then be done locally on each processor and thus requires no communication. This reduces the number of communications for a given angle from O(N 2 ) to O(N). Table shows the dramatic reduction of run-time when using the lookup tables. A mean speedup by a factor of about 6 is achieved for values of N 28. Note the reduction in run-time for N = 024, from almost 7 minutes down to 57 seconds. Another method using lookup tables would be to create one large lookup table of all proles before the backprojection is computed. It seems quite plausible a priori that this would cause even more reduction of run-time. However, we have tried this and the results were a little disappointing. For N = 52 a reduction of almost a second is achieved which is only about 0%. A problem with this method is the memory usage, since a copy of all projection data is assigned to each processor. For N = 024 the memory requirements are too large and the program cannot reconstruct images of this size Conclusion The method using lookup tables can compute reconstructions fast. Due to the architecture of the CM-5, the ltered data are distributed over the processors and reconstruction cannot be computed locally. Partly, this problem is solved by using the lookup tables, but there remains a lot of communication in the creation of the table. Improvement could possibly be gained by using a shared memory computer instead of a distributed memory computer. object function. This theorem not only gives a basis for the ltered backprojection algorithm but also for the so-called direct Fourier methods. The problems with these methods are due to the discretisation of the equations involved. In the discrete case the Radon transform Rf(; t) is assumed to be sampled at ( j ; s l ), j = ; : : : ; p, l =?q; : : : ; q, with j and s l as given in (7). In the following, the largest value for l is chosen to be q? in order to have an even number of values for l. In the ideal case l has a number of values equal to a power of 2, so that standard FFT algorithms can be used to arrive at fast implementations. The polar coordinate grid G p;q is given by G p;q = fr j : r =?q; : : : ; q? ; j = ; : : : ; pg: The standard Fourier reconstruction algorithm goes as follows [3]: Step For j = ; : : : ; p, r =?q; : : : ; q?, compute approimations bg jr to d Rf(j ; r) by taking the one-dimensional Fourier transform of the pro- les. This rst step provides an approimation to the two-dimensional Fourier transform f b of the object function f on radial lines in G p;q, see Fig. 5. Step 2 Interpolate bg jr onto a Cartesian grid by using nearest neighbour interpolation. Step 3 Compute an approimation f ~ to f(h~), ~ 2 Z 2 by taking a discrete inverse two-dimensional Fourier transform. 4 Fourier reconstruction In this section the direct Fourier reconstruction method will be described. This has a lower compleity than ltered backprojection making it an interesting method to investigate. First the idea behind Fourier reconstruction will be given followed by the standard Fourier reconstruction algorithm. Then a new and better algorithm is proposed. The implementation of this new algorithm together with its accuracy will be considered at the end of this section. 4. Basic Fourier reconstruction The Fourier slice theorem from Section 2.2 links the one-dimensional Fourier transform of a prole to a slice of the two-dimensional Fourier transform of the Figure 5: The representation of bg jr on the polar grid for p = 8, q = 3. The origin of the coordinate aes lies in the middle of the gure. The circles on the polar lines indicate the points on which bg is known. The Cartesian grid is also shown for reference. The crucial step in this algorithm is the interpolation from the polar grid to the Cartesian grid. The nearest neighbour interpolation described above turns out to be too simple. It produces severe artefacts and is very inaccurate [3], so several other interpolation techniques have been proposed. Stark et al. [9] suggest a complicated sinc-interpolation step and they report to reach reconstructions of a quality 5

6 comparable to that of ltered backprojection. Natterer [3] gives two alternative methods. The rst one combines oversampling of bg with sinc-interpolation and the other is the so-called Pasciak algorithm [5]. This is the one also used by Schulte [7] and will be described in this report. 4.2 Pasciak algorithm The idea behind the method of Pasciak [5] is to alter the rst step in such a way that the values of bg( j ; ) lie on the vertical lines of the Cartesian grid for j cos j j j sin j j, j = (j? )=p, and on the horizontal lines otherwise. The values of bg thus obtained can then be assigned to the points of the Cartesian grid by either using nearest neighbour interpolation or linear interpolation. As the algorithm using linear interpolation gives better results this is the one described below. Step For j = ; : : : ; p compute approimations bg jr to bg( j ; rc(j)) by bg jr = p q? X h 2 l=?q e?ilc(j)r=q g( j ; s l ); (2) where r =?q; : : : ; q?, and c(j) = = ma(j sin j j; j cos j j). This corresponds to computing the discrete Fourier transform for arbitrary step-size in the frequency domain, which can be done by the chirp z-transform [4]. The result of this transform is shown in Fig. 6. As can be seen, the points of bg are much closer to the grid points of G p;q. Step 4 Compute the discrete inverse 2-D Fourier transform f ~, ~ 2 Z 2, of b F. 4.3 Compleity The compleity of the Pasciak algorithm can be estimated as follows. The rst step takes O(pq log q) operations. The second step, the interpolation, uses O(q 2 ) operations. The ltering with the cos-lter needs O(q 2 ) multiplications and nally the twodimensional inverse Fourier transform can be done in O(q 2 log q) steps. Using the relation p = q the total compleity is O(q 2 log q). 4.4 Accuracy The accuracy of the Pasciak algorithm was investigated by taking a simple Shepp-Logan phantom and the standard Shepp-Logan phantom. This simpler phantom has greater dierences between the refractive coecients of the ellipses, and therefore is easier to reconstruct The simple phantom The accuracy of the direct Fourier method was rst tested on the simple phantom. Projection data were generated for p = 400 and q = 27, so that p and q satisfy the relation p = q. The reconstructed image is shown in Fig. 7 on the left and the (scaled) dierence with the original on the right. The results are visually attractive, but in the dierence image there are high frequency artefacts visible. The error computed over the whole image is 0:078 and over the part of the y = 0 line 0:059, so the error is larger than for ltered backprojection. Figure 6: Result after applying the chirp z-transform to bg. The origin of the coordinate aes lies in the middle of the gure. In this case p = 8 and q = 3. (a) (b) Step 2 Interpolate bg jr onto the Cartesian grid by using linear interpolation. The values on the diagonals need not to be interpolated if the number of angles is divisible by 4. Step 3 To suppress the high frequencies, lter f b with the cos-lter. For ~ 2 Z 2, bf ~ = ( cos( jj~jj 2q ) b f~ jj~jj < q 0 jj~jj q (3) Figure 7: Reconstruction of the simpler phantom (a) and the dierence with the original (b). Projection data were generated for p = 400 and q = 27. Reconstruction is performed on a grid. The lower 0% of the gray values in the image were scaled to the full range in order to make artefacts visible. The reconstruction of the y = 0 line is compared with the original in Fig. 8 and from this plot it is clear that there are quite a few reconstruction artefacts. 6

7 Original Reconstruction Original Reconstruction Value Value Figure 8: Comparison of a part of the y = 0 line of the reconstruction with the original simpler Shepp- Logan phantom. In this case p = 400 and q = The standard Shepp-Logan phantom Net, the accuracy of the Pasciak method was tested on the standard Shepp-Logan phantom for the same projection data. The reconstructed image is shown in Fig. 9 on the left. There are a lot of high frequency artefacts visible in the form of lines over the whole image. Computing the error norm () over the whole image results in an error of (a) Figure 9: Reconstruction of the standard phantom (a) and the dierence with the original (b). Projection data were generated for p = 400 and q = 27. Reconstruction was performed on a grid. The lower 0% of the gray values in the image were scaled to the full range in order to make artefacts visible. The section of the line y = 0 shown in Fig. 0 has an error of From the plot, it is clear that the reconstruction artefacts for this phantom are even worse than for the simple phantom Conclusion It is clear from the results reported that the direct Fourier method cannot compete with ltered back- (b) Figure 0: A numerical comparison of a part of the y = 0 line of the reconstruction with the true values of the Shepp-Logan phantom. Projection data were generated for p = 400 and q = 27. Reconstruction was performed on a grid. projection on accuracy. A more dicult interpolation method, however, could be used to increase the accuracy. As far as time compleity is concerned, direct Fourier reconstruction is seen to be faster than ltered backprojection. 4.5 Implementation The Pasciak algorithm is very suitable for parallelisation. The chirp z-transform which can be implemented by FFT's can be transformed into CM- Fortran very easily. The interpolation step is also easy to parallelise using the available constructs of CM-Fortran like WHERE and FORALL. The WHERE construct corresponds more or less to a parallel IF statement, and FORALL to a parallel DO. So, this algorithm can be implemented on the CM-5 in a straightforward manner. Using lookup tables for the dierent values of and the sine and cosine of these angles, it is possible to arrive at fast implementations. The strategy from Section 3.4 is also used here to carry out timings on the CM-5 for dierent inputs. The parameter N has the same meaning. The results are shown in Table 2 for dierent values of N. In the interpolation step there is some communication between processors when at a certain point the actual interpolation is computed. The values needed in this step are distributed across dierent processors and for this method, just as for FBP, it is not possible to select the layout of the arrays in such a way that no communication is involved. The communication burden, however, is not as strong as for ltered backprojection. The latter algorithm iterates over the dierent angles (and involves communication in each iteration) and the Pasciak algorithm does not iterate, so the communication step is performed only once. 7

8 N Chirp z Interpolation IFFT Total Table 2: Timing results in seconds on the CM-5 for dierent implementations of the Pasciak algorithm. The three major steps in the algorithm are shown, were the ltering step is combined with that of the two-dimensional inverse Fourier transform. The last column shows the total time needed for reconstruction. When the timing results in Table 2 are compared to the timings for ltered backprojection (Table ), it can be concluded that the Pasciak method is indeed faster than ltered backprojection. 5 Conclusions Of the reconstruction algorithms considered, the ltered backprojection algorithm turns out to be the most dicult to parallelise. This is mainly due to the distributed memory of the CM-5, which is the source of a lot of communication between the processors. This communication problem is partly solved by using lookup tables for the ltered projections, but still most of the time is spent on communication. Using these lookup tables, the implementation on the CM-5 has led to a fast reconstruction algorithm: it is possible to reconstruct a image in about 8:7 seconds. The reconstruction quality of ltered backprojection is ecellent, which was already known from the literature. Direct Fourier reconstruction using the Pasciak algorithm is easier to parallelise on the CM-5. The problem of communication is still present, but the burden is not as strong. The time needed for reconstruction is much less than for ltered backprojection, not only epressed in time compleity, but also in actual run-time. The quality of the reconstruction, however, is not as good. Because there is quite a gap between the run-time of ltered backprojection and the Pasciak algorithm (a reconstruction takes about :3 seconds), some etra time can be spent for improvement of the interpolation method, see e.g. [7]. 5. Future research As eplained above, the distributed memory of the CM-5 poses some problems. Perhaps the use of a shared memory computer could improve the actual run-time of the reconstruction algorithms. It would be interesting to implement the algorithms which were described here on a shared memory architecture. This paper deals with two-dimensional reconstructions. If a three-dimensional reconstruction is wanted, the standard procedure is to use a stack of two-dimensional reconstructions. However, it is also possible to reconstruct a three-dimensional image directly. This method calls for dierent reconstruction algorithms. The combination with three-dimensional imaging methods, such as volume visualisation or even virtual environments opens a large eld of interesting problems to investigate. References [] G. T. Herman. Image Reconstruction from Projections: the Fundamentals of Computerized Tomography. Academic Press, 980. [2] A. C. Kak and M. Slaney. Principles of Computerized Tomographic Imaging. IEEE Press, New York, 988. [3] F. Natterer. The Mathematics of Computerized Tomography. B.G. Teubner and J. Wiley, 986. [4] H. J. Nussbaumer. Fast Fourier Transform and Convolution Algorithms. Springer-Verlag, 982. [5] J. E. Pasciak. A note on the Fourier algorithm for image reconstruction. preprint, Applied Mathematics Department, Brookhaven National Laboratory, Upton, New York, 973. [6] S. W. Rowland. Computer implementation of image reconstruction formulas. In G. T. Herman, editor, Image Reconstruction from Projections: Implementation and Applications, volume 32 of Topics in Applied Physics, pages 9{79. Springer- Verlag, 979. [7] J. Schulte. Fourierrekonstruktion in der computer-tomographie. Master's thesis, Westfalischen Wilhelms-Universitat Munster, 994. [8] L. A. Shepp and B. F. Logan. The Fourier reconstruction of a head section. IEEE Transactions on Nuclear Science, NS{2:2{43, 974. [9] H. H. Stark, J. W. Woods, I. Paul, and R. Hingorani. Direct Fourier reconstruction in computer tomography. IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP-29(2):237{ 245, April 98. 8