One-step Backprojection Algorithm for Computed Tomography

Transcription

1 2006 IEEE Nuclear Science Symposium Conference Record M4-438 One-step Bacprojection Algorithm for Computed Tomography Dosi Hwang and Gengsheng L. Zeng, Senior Member, IEEE Abstract The iteratie algorithm can incorporate arious physical aspects into its reconstruction process, leading to a more accurate image than the filtered bacprojection algorithm. One drawbac of the iteratie algorithm is that it requires many iterations to produce the final images. In this paper, we show that we can reconstruct the same images as the Landweber iteratie method in one step. We reformulate the iteratie procedure into a matrix multiplication, build the matrix and store it on dis. We use this matrix to reconstruct images in one step. The reconstructed images are exactly the same as the ones that are reconstructed by the Landweber iteratie method. I I. INTRODUCTION N computed tomography, iteratie algorithms are receiing more attention as the speed of computers increases. This is because they can incorporate into their reconstruction process arious physical aspects, especially the measurement statistics, the geometric response model, and scatter or attenuation correction, leading to a more accurate image than the filtered bacprojection algorithm (FBP. Iteratie algorithms gie better reconstructions than the FBP methods een when no physical compensations are modeled into them []-[6]. One drawbac of the iteratie algorithm is that it taes many iterations, and thus long reconstruction times, to produce the final image. As more physical compensations are modeled in the reconstruction process, the iteratie algorithms may not be considered adequate for clinical applications een with contemporary high-speed computers. Een one iteration may tae seeral hours if arious complicated compensation schemes are included [7]. There are seeral acceleration methods to reduce reconstruction time, such as an orderedsubset expectation-maximization algorithm [8], a rescaled bloc iteratie algorithm [9], and a space-alternating generalized EM [0]. Howeer, they still require many iterations to reconstruct images. Furthermore, the accelerated algorithms usually do not produce the same reconstruction as the unaccelerated algorithms. In this paper, we show that we can reconstruct images from measurements in one step, just lie the filtered bacprojection methods, but produce the same images as the Landweber iteratie algorithm. We reformulate the iteratie procedure into a matrix multiplication, build the matrix, and store it on Manuscript receied Noember 5, This wor was supported in part by NIH Grants EB00489, EB003298, and EB 002. Dosi Hwang is with the Department of Bioengineering, Uniersity of Utah, Salt Lae City, UT 8402 USA, and with the Utah Center for Adanced Imaging Research, 729 Arapeen Dr., Salt Lae City, UT 8408 USA ( dosi.hwang@gmail.com. Gengsheng L. Zeng is with the Department of Radiology and Bioengineering, Uniersity of Utah, Salt Lae City, UT 8408 USA, and with the Utah Center for Adanced Imaging Research, 729 Arapeen Dr., Salt Lae City, UT 8408 USA ( larry@ucair.med.utah.edu. dis. We then use this matrix to reconstruct images in one step. In Section II of this paper we show how this one-step bacprojection can be implemented and describe the differences from the filtered bacprojection method. Computer simulation studies and real experiments in Sections III and IV show that our one-step bacprojection images are exactly the same as the ones that are reconstructed by the iteratie method. In Section V we discuss more about the one-step bacprojection method including implementation issues and future studies. II. THEORY We will consider a generalized Landweber iteratie algorithm []-[3] to sole FX=P, where F is a projector, X is an array of the unnowns to be reconstructed, and P is an array of projection data. The iteratie algorithm is expressed as ( ( X = X + T( P FX, ( where T is a bacprojector. For a simple Landweber algorithm, T=αF T, where α is a step size to mae the algorithm conerge. The iteratie algorithm ( can be reformulated as follows: ( ( X = X + T( PFX (2 ( = ( I TF X + TP. Let B=I-TF, then (2 can be expressed as in (3. ( X = BX + TP ( = BBX ( + TP + TP 2 ( = B X + ( B+ I TP 2 ( 2 = B ( BX + TP + ( B+ I TP (3 3 ( 2 2 = BX + ( B + B+ ITP + (0 = B X + ( B + B + + B+ I TP. Therefore, if we use X (0 =0, then the reconstructed image at (+ th iteration becomes X = CTP, (4 where h C = B. (5 h= 0 If we can precompute the matrix C, then the reconstruction process is just a one-step bacprojection as shown in (4 with post-filtering implemented as a matrix multiplication using C. A closer loo at (5 reeals that C does not depend on the projection data P or any other intermediate estimates. Therefore, we can actually precompute the matrix C and store /06/$ IEEE. 3453

2 it before scanning, and use it later to reconstruct images in one step when projection data are obtained. When the one-step bacprojection method is applied to a high-resolution imaging system, it may be difficult to use the whole matrix C at once due to memory limitations of the computer. In this case, we can build a matrix D =C T in a iew-by-iew order and process indiidual portions of D separately for an indiidual or a group of iews as in (6. T = = =, θ, θ 2, θ θ 2 X C TP D P D D D P P P = D P + D P + + D P, θ θ, θ2 θ2, For the implementation of a large C matrix, we may need to compute indiidual components or a group of components of C one at a time by loading into memory the minimum amount of data necessary to compute it. This can be done as follows: Build the projection matrix F and sae it to dis (each element f ij or a group of elements can be calculated indiidually and saed to dis. f f2 fm f2 f 2M F =, (7 fn fn2 fnm where M is the number of image pixels and N is the number of data points. 2 Build the bacprojection matrix T or use T=F T (we will use T=F T hereafter. 3 Calculate B=I-TF=I-F T F element-by-element: N fti ftj if i = j t = [ B] = b ij ij = (8 N fti ftj if i j t = 4 Update C (=: + bij if i = j [ C] = c ij ij = (9 bij if i j 5 Calculate B 2 element-by-element 6 Update C (=2: M 2 (2 ij ij it tj t = (6 B = b = b b (0 c = c + b ( (2 ij ij ij 7 In the same way, calculate B 3, B 4, B 5,, and update C=C+B. Further analysis of the one-step bacprojection method (4 can be made if we consider the case of letting go to infinity. If the bacprojector T, thus the matrix B in (5 is designed to mae the algorithm conerge, C becomes (2 as goes to infinity: C = B = I TF = TF assuming (TF - becomes h h ( ( (2 h= 0 h= 0 exists. Then, the reconstructed image ( ( X = C TP = TF TP (true = ( TF T( FX + NM (true = ( TF ( TF X + ( TF TN = + (true X ( TF TNM M (3 where N M is the measurement noise in P. If there is no noise, N M = 0, then we can reconstruct the true image, X (true. Howeer, if there is noise, N M 0, the noise will be amplified by (TF - T. Therefore, if the matrix TF is ill-conditioned, it will amplify the noise significantly. On the other hand, if we design TF as well-conditioned, the noise amplification will be reduced. Or if we regularize (TF - such that the noise can be suppressed while the true image can be presered, we can reconstruct better images. Computing C (5 can be understood as finding a regularized (TF -. Different regularized (TF - can be obtained for different alues of. The optimum alue for remains in question as in many other iteratie algorithms. So far, is usually selected based on experience. There are other ways to obtain a regularized (TF - such as using the singular alue decomposition (SVD; small singular alues are discarded when they are inerted. Howeer, SVD requires a lot of memory and is impractical for the large matrices of computed tomography. This analysis can be applied to a bacprojection-filtering (BPF method. In BPF, projection data are first bacprojected into the image space, and the transfer function of the projection-bacprojection operator is inersed. The theoretical relationship between the original image and the bacprojected image is gien by (4 in the frequency domain [4]-[7]: H = H = H BP 2 2 X X + 2 (4 where H X is the Fourier transform of the original image, H BP is the Fourier transform of the bacprojected image, and is the spatial frequency. Therefore, the bacprojected image is filtered by a ramp filter to reconstruct the original image as in (5 HBP = HX = H. (5 X That is, the projection data are bacprojected then filtered by a 2D ramp filter. The transfer function / corresponds to the Fourier transform of the point response function of TF in (3, and the ramp filter corresponds to the Fourier transform of the point response function of (TF -. If there is noise, it will be amplified by the ramp filter, as explained before. As a regularization method to suppress noise, the magnitudes of the ramp filter at high frequencies can be reduced using a window function such as a cosine, Hamming, or Hann window etc. Howeer this regularization sacrifices the resolution of images. It is unclear whether C is equialent to a certain window function for ramp filtering. Another important issue should be addressed. Both FBP and BPF assume that the transfer function of the projection- 3454

3 bacprojection operator is / and use the ramp filter to inert it during reconstruction. Howeer, / is a theoretically deried function based on continuous sampling, which is impossible in computed tomography. Therefore the actual transfer function of the projection-bacprojection operator is not /. Thus if the ramp filter is used to inert the actual transfer function, there is a model mismatch, which produces errors in reconstruction. We can express this model mismatch problem in matrix notation as (6. ( X = TFest TP (true = ( TFest ( TFtrue X + TNM ( true = ( TFest (( TFest + TFdiff X + TNM = TF TF X + TF X + TN = X + ( TF ( TF X + TN (true (true ( est ( est diff M ( true (true est diff M (6 where TF est is an estimated point response function of the projection-bacprojection operator which is used in reconstruction, TF true is the true point response function and TF diff is the difference between TF true and TF est. Equation (6 implies that the reconstructed images hae two inds of noise: one resulting from measurement noise and the other from model mismatch. These two types of noise are amplified by (TF est -. In the conentional FBP and BPF, the noise resulting from the model mismatch is difficult to reduce unless more accurately matched inerse filters are introduced instead of a simple ramp filter with window functions. This model mismatch problem in FBP and BPF becomes more significant when it comes to SPECT due to the geometric response of the collimator. In contrast, the one-step bacprojection method can incorporate the optimum projection-bacprojection operator into C, so that the model mismatch problem can be minimized. Therefore, along with the iteratiely regularized (TF -, one-step bacprojection can reconstruct better images than the conentional FBP and BPF methods. In Sections III and IV, we show that one-step bacprojection with the precomputed C produces exactly the same images as those reconstructed using the iteratie algorithm (. We also compare our proposed method with the conentional filtered bacprojection method and show that the one-step bacprojection method produces better images. III. COMPUTER SIMULATION STUDIES We applied the one-step bacprojection method to computer generated projection data. The phantom consisted of one large bacground disc (actiity= and one small hot lesion (actiity=2. The projection data were generated using the analytical formula and Poisson noise was added. The dimensions of the image were 38x38 and the number of projection iews was 56 oer 360. Three different count leels were simulated: high count (total counts = 623,392, medium count (24,98 and low count (62,6. Fig.. Comparison of reconstructed images. Three different count leels were simulated. Images were reconstructed by the iteratie method at the 00 th iteration, one-step bacprojection with the precomputed C 00, FBP, and FBP with Hann window. For all count leels, the iteratie algorithm (a and onestep bacprojection method (b produced exactly the same images, and they are less noisy than the image reconstructed by FBP (c. FBP with Hann window (d produced less noisy images, but could not presere the sharp edges. Fig. 2 shows line profiles of the reconstructed images for the high count experiments Iteratie Onestep FBP Fig. 2. Line profiles of the reconstructed images for high count experiments. IV. APPLICATION TO REAL DATA We applied the one-step bacprojection method to real SPECT data. A cardiac phantom (Data Spectrum Corporation, Hillsborough, NC, USA was used in this experiment. The emission data were acquired in a 64x64 array (the dimensions of each detector pixel were 2.32x2.32 mm 2. The number of projection iews was 60 oer 360. The total photon count for the selected slice was 3,238. We incorporated the geometric response model in the projector as well as in the bacprojector of ( and (4. We used the Zeng and Gullberg s method [8] for the geometric response modeling. Fig. 3 compares the reconstructed images using the iteratie method at the 00 th iteration, one-step bacprojection with C 00, FBP, and FBP with Hann window for noise suppression. Fig. 4 shows the line profiles for the reconstructed images. The reconstruction 3455

4 times for the iteratie method, one-step bacprojection, and FBP were 35, 0.33, and 0.08 sec, respectiely (.2 GHz Intel Celeron with 768 MB memory. (a image pixels (b D 50, θ (a iteratie (b one-step (c FBP (d FBP (Hann Fig. 3. Reconstructed images of the cardiac phantom. The total count was 3,238. Images were reconstructed by the iteratie method at the 00 th iteration, one-step bacprojection with precomputed C 00, FBP, and FBP with Hann window. 6 5 Iteratie Onestep FBP (c D 50, θ20 (d D 50, θ shifted Fig. 5. Symmetric design of image pixels. (a radially-symmetric pixels. D 50, θ20 (c can be obtained from D 50, θ (b by shifting its elements (d. Fig. 6 (a shows the reconstructed image by one-step bacprojection method with the radially-symmetric image pixels and the precomputed D 50. The noise has been reduced without sacrificing its spatial resolution, compared to the images reconstructed by FBP (Fig. 6 (b and (c Fig. 4. Line profiles of the reconstructed images of the cardiac phantom. The one-step bacprojection method produced exactly the same image as the iteratie method. Both images are much less noisy than the image reconstructed by FBP in Fig. 3 (c. The regularized FBP with Hann window in Fig. 3 (d reduced the noise but sacrificed resolution of the image, and the image became blurred. V. REDUCTION OF THE MATRIX SIZE The size of the matrix C or D can be reduced using image symmetry with special image representation such as a blob or natural pixel methods. In this study, we designed radiallysymmetric image pixels as shown in Fig. 5 (a, such that one small projection matrix for a single iew can represent all other iews by a simple rotation or shift of matrix elements. For example, D 50,θ20 (Fig. 5 (c can be obtained by shifting D 50,θ (Fig. 5 (d. Therefore the size of N pixels x (N bins x N iews of D can be reduced to N pixels x N bins (N bins is the number of bins in each iew. (a one-step (b FBP (c FBP (Hann Fig. 6. Simulation studies with symmetric image pixels. (a one-step bacprojection with precomputed D 50. The reconstructed image was transformed into regular square pixel format for comparison with FBP images (b and (c. VI. DISCUSSIONS The one-step bacprojection method produces exactly the same images as the Landweber iteratie algorithm, but in onestep, saing much computation time. Computing the matrix C (or D before scanning obiates the iteratie procedure that usually taes place after patients get scanned. The one-step bacprojection method produced better images than the conentional FBP algorithm. There are two reasons for this result: One-step bacprojection proides better regularization with C than the windowing method in FBP. This method also reduces the model mismatch by accurately modeling discrete angular sampling and the imaging system. For example, it is difficult to incorporate the geometric response of the collimator into the reconstruction process of FBP. In our experiments FBP resulted in significant noise compared to the one-step bacprojection with the geometric response model. An artifact in cone-beam geometry using a 3456

5 circular trajectory can also be improed by the one-step bacprojection method, as iteratie algorithms outperform FBP in reducing this artifact [9]-[22]. The size of the matrix C (or D itself can be reduced using image symmetry; the one-step bacprojection method can be considered a bacprojection filtering algorithm with postfiltering implemented as a matrix multiplication with C. Een though it is not certain whether the matrix C can be expressed as a function of the distance between two points, as in the conentional BPF, it is a function of radial distances and the angle difference of two points in spherical coordinates. Therefore, the matrix C can be compressed using this symmetry. Howeer, a square pixel or cubic oxel representation of an image maes it difficult to find a general expression for C. Using a blob representation of an image [22], [23] or a natural pixel method [24] may be useful lines of inestigation. If a general expression for the matrix can be obtained, it can replace a simple ramp filter in BPF and FBP to produce better images with arious compensation techniques included. [7] Bronnio AV. A filtering approach to image reconstruction in 3D SPECT. Phys Med Biol 2000;45: [8] Zeng GL, Gullberg GT. Three-dimensional iteratie reconstruction algorithms with attenuation and geometric point response correction. IEEE Trans Nucl Sci 99;38: [9] Manglos SH, Jaszcza RJ, Floyd CE. Maximum lielihood reconstruction for cone beam SPECT: deelopment and initial tests. Phys Med Biol 989;34: [20] Zeng GL, Gullberg GT. A study of reconstruction artifacts in cone beam tomography using filtered bacprojection and iteratie EM algorithms. IEEE Trans Nucl Sci 990;37: [2] Jaszcza RJ, Li J, Wang H, Coleman RE. Three-dimensional SPECT reconstruction of combined cone beam and parallel beam data. Phys Med Biol 992;37: [22] Lewitt RM. Alternaties to oxels for image representation in iteratie reconstruction algorithms. Phys Med Biol 992;37: [23] Matej S, Lewitt RM. Practical considerations for 3-D image reconstruction using spherically symmetric olume elements. IEEE Trans Med Imag 996;5: [24] Hsieh Y, Zeng GL, Gullberg GT. Projection space image reconstruction using strip functions to calculate pixels more natural for modeling the geometric response of the SPECT collimator. IEEE Trans Med Imag 998;7: REFERENCES [] Chornoboy ES, Chen CJ, Miller MI, Miller TR, Snyder DL. An ealuation of maximum lielihood reconstruction for SPECT. IEEE Trans Med Imag 990;9:99-0. [2] Liow J, Strother SC. Practical tradeoffs between noise, quantitation, and number of iterations for maximum lielihood-based reconstructions. IEEE Trans Med Imag 99;0: [3] Tourassi GD, Floyd Jr CE, Munley MT, Bowsher JE, Coleman RE. Improed lesion detection in SPECT using MLEM reconstruction. IEEE Trans Nucl Sci 99;38: [4] Rosenthal MS, Cullom J, Hawins W, Moore SC, Tsui BMW, Yester M. Quantitatie SPECT imaging: a reiew and recommendations by the focus committee of the society of nuclear medicine computer and instrumentation council. J Nucl Med 995;36: [5] Groch MW, Erwin WD. SPECT in the year 2000: basic principles. J Nucl Med Tech 2000;28: [6] Lewitt RM, Matej S. Oeriew of methods for image reconstruction from projections in emission computed tomography. Proc IEEE 2003;9: [7] Laurette I, Zeng GL, Welch A, Christian PE, Gullberg GT. A threedimensional ray-drien attenuation, scatter and geometric response correction technique for SPECT in inhomogeneous media. Phys Med Biol 2000;45: [8] Hudson HM, Larin RS. Accelerated image reconstruction using ordered subsets of projection data. IEEE Trans Med Imag 994;3:60-9. [9] Byrne CL 996 Bloc-iteratie methods for image reconstruction from projections. IEEE Trans Imag Proc 996;5: [0] Fessler JA, Hero AO. Penalized maximum-lielihood image reconstruction using space-alternating generalized EM algorithms. IEEE Trans Imag Proc 995;4: [] Landweber L. An iteration formula for Fredholm integral equations of the first ind. Amer J Math 95;73: [2] Jiang M, Wang G. Conergence studies on iteratie algorithms for image reconstruction. IEEE Trans Med Imag 2003;22: [3] Zeng GL, Gullberg GT. Unmatched projector/bacprojector pairs in an iteratie reconstruction algorithm. IEEE Trans Med Imag 2000;9: [4] Wernic MN, Aarsold JN. Emission tomography: the fundamentals of PET and SPECT Elseier Academic Press 2004; [5] Suzui S, Yamaguchi S. Comparison between an image reconstruction method of filtering bacprojection and the filtered bacprojection method. Appl Opt 988; 27: [6] Zeng GL, Gullberg GT. A bacprojection filtering algorithm for a spatially arying focal length collimator. IEEE Trans Med Imag 994;3: