Xi = where r is the residual error. In order to find the optimal basis vectors {vj}, the residual norms must be minimized, that is

Transcription

1 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 45, NO. 4, AUGUST Reconstructions cif Truncated Projections Using an Optimal Basis Expansion Derived from the Crops-Correlation of a Knowledge Set of a priori Cross-sections V. Y. Panin, StudentIMember ZEEE, G. L. Zeng, Member ZEEE and G. T. Gullberg, Senior Member IEEE University of Utah, Department of Radiology 729 Arapeen Dr., Salt Lake City, UT 84108, USA. Abstract An algorithm was developed to obtain reconstructions from truncated projections by utilizing cross-correlation of a knowledge set of a pn;ori nontruncated cross-sections with a similar structure. A cross-correlation matrix was constructed for the known set of crciss-sectional images. The eigenvectors of this matrix form a sift of orthogonal basis vectors for the reconstructed image. Th e basis set is optimal in the sense that the average of the differbnces between members of a given set of a priori images, and their truncated linear expansion for any basis set, is minimal fo1 this particular basis set. A procedure for finding optimal basis vectors is fundamental for deriving the Karhunen-Loeve (K-L) transform. Therefore, one can represent an image not in the knowledge set, but of similar structure by a linear combination o 1 basis vectors corresponding to the larger eigenvalues; thus, the number of basis vectors is reduced to a number less than the total number of pixels. The projection of an image represented by this linear combination of basis vectors is a linear combiination of projected basis vectors which are not necessarily ortlhogonal. A constrained least-squares method was used to evaluate the coefficients of this expansion by minimizing the sunk of squares difference between the expansion and the projection measurements taking into account the distribution of Coefficients over basis vectors. The constrained least-squarep estimates of the coefficients were used in an expansion olf the orthogonal basis to obtain the reconstructed image. The constrained solution has a reduced noise level in this invlkrse problem. It is shown that the reconstruction of truncated projections can be significantly improved over that of cctmmonly used iterative reconstruction algorithms. I. IN I RODUCTION The truncation prothem due to a relatively small detector size is well known in single photon emission computed tomography (SPECT). The image projections are greatly truncated during transmibsion imaging of the chest by a threedetector SPECT system, with fan-beam collimators [ 11. The truncation problem resull s in solving a rank deficient system of linear equations, which lieads to reconstruction artifacts when common reconstruction qlgorithms are applied [2]-[7]. In some reconstruction algorithms for truncated The research work presented in this manuscript was partially supported by NIH Grant RO1 HL and Picker International projections, the truncated projections were reconstructed by data extrapolation methods to complete the projections [5] Another approach is to determine the solution with the information available by solving only the system of linear equations given by available truncated projections at a different view [lo]. Considering the pixel values as unknowns, a fully determined problem can be ill-conditioned. The singular value decomposition (SVD) method has been applied to obtain a solution to this ill-conditioned problem [11,12]. It has been shown that accurate image reconstruction can be obtained with truncated projections, provided the projection model is accurate, that is, the coefficients in the linear equations model are accurate. However, this generalized inverse method by using SVD is very sensitive to model-mismatch and systematic errors in projections, due to the ill-conditioness of the problem. This paper describes a method to reconstruct the image from truncated projections by solving a system of linear equations, which is neither ill-conditioned or rank deficient (underdetermined). The system is better-conditioned by reducing the number of unknowns in the system of linear equations based on the fact that images of defined anatomical objects have similar forms. The unknowns are coefficients of a truncated basis vector expansion of the projected image, derived from the Karhunen-Loeve (K-L) transform. The leastsquare method is applied to find the unknowns, Le., the coefficients of the K-L transform. Since a truncated basis expansion does not represent an image precisely, the leastsquares method can be unstable, especially in cases of existing model-mismatching and systematic error. To regularize this inverse problem, a constraint based on the distribution of the magnitude of the expansion coefficient is explored. 11. METHODS A. Karhunen-Loeve Transform Suppose one has a set {Xi} of n members of a knowledge set of images of similar structure, where each Xi has a dimension N. One wants to find an optimal orthonormal basis (vj, j=l,..., N}, for {Xi}. In our development we will consider ian approximation by using m basis vectors Xi = m j= 1 b..v.+ri,,i V I = 1,2,..., n (1) where r is the residual error. In order to find the optimal basis vectors {vj}, the residual norms must be minimized, that is /98$ IEEE

2 2120 must be minimized. Here R is the correlation matrix for the given set. Next, we minimize the quadratic form (2) with T constraint v v = 1. It is known [I41 that extremums of this J J quadratic form correspond to eigenvectors of the matrix R and that eigenvectors should be chosen corresponding to the smaller eigenvalues of R that minimize (2). This procedure finds the optimal basis set in the sense that the average of the differences between members of a given set of a priori images, and their truncated linear expansion for any basis set, is minimal for this particular basis set. This procedure is the foundation for deriving the Karhunen-Loeve (IC-L) transform [ 131, Its basis vectors are often referred to as principal components. The Karhunen-Loeve transform has some important properties. One of them is that the eigenvalue Ij, normalized by the number of images from a knowledge set, represents the average value of b; for the whole knowledge set ensemble, that is, which implies that. n 1 - C b2ij = I for allj. jj= 1 Equations (3) and (4) are valid only for images from the knowledge set, and the distribution property (4) for the magnitudes of the expansion coefficients is expected for each (4) image of similar structure. B. Application to Reconstruction The K-L transform has found many applications in inverse problems [14]. Let Z be a test image approximately represented as a truncated linear expansion of basis vectors using only those basis vectors corresponding to the largest eigenvalues: N I = c, pjvl= c, pjvj. (5) j= 1 j=1 This linear relationship is retained in the projection space. Let P be the linear projection operator. Then m m P(Z)= Pj.P(vj) (6) j=1 is the expansion in the projection space. In practice, one has acquired projections P(Z) and can generate projections of basis vectors {P(v,)}. Because the generated projections of the basis vectors are no longer orthogonal, a linear least-squares method is used to find the coefficients pj of the linear expansion. Because relation (6) is not accurate and residual errors exist, the solution { Pi } is unstable. Note that the solution from (6) may not be the same solution from (5) due to nonorthogonality of basis projections. To stabilize the solution, a constrained linear inversion method is applied. Equation (3) indicates that if our image is similar to those 2 images in the knowledge set, p, /IJ would be approximately a constant for all j. This can be seen by interpreting the summation in (4) as the statistical average over the knowledge set ensemble. One can define the normalized energy of an 2 T expansion aszp./aj which can be rewritten as p Ap, where A is a diagonaf matrix of inverse eigenvalues. Thus, this Figure 1. Nine members of the knowledge set represent approximately the same cross section of nine different patients. They have the same position and differ in size and structure. Figure - 2. Eigenvalue - spectrum of the correlation matrix of the knowledge set in decreasing order up to 500 values. The eigenvalues are scaled logarithmically. Note the fast decay rate to smaller values at the beginning of the spectrum. This reflects the fact that the members of the knowledge set are similar.

3 2121 normalized energy can be used to regularize the least-squares problem. It is clear that this minimal energy constraint 2 2 encourages large pi for a large eigenvalue hj, and small pj for a small eigenvalue hi. One can look at expression (3) from a different point of view. The first eigenvector, corresponding to the largest eigenvalue of the correlation matrix of the knowledge set, can be considered as the average image over all images in the knowledge set. The average of the expansion coefficients, excluding the first coefficient, for the knowledge set is zero. The corresponding eigepvalues are the standard deviations of distribution of expansion coefficients. For a the solution to inverse problem, the expansion coefficient magnitude should not be much larger than the corresponding standard deviation and the regularized solution should be biased towards the a priori estimate of the average image. The coefficients of the expansion in (5) can be obtained by solving the following mihimization problem min{llp(z,i -AP1I2 + Y. PT 7 where columns of the matrix A are P(v$ and y is the constraint parameter. The constrained linear inversion solution is p = (A *A + YA)- A~P(z). We use a QR decomposition method to find the solution in (8). Once the coefficimts p are estimated, the image is reconstructed by using (ti). IID. RESULTS A. Knowledge Set We obtained approximately 1200 PET transmission images of actual patient data filom the database of Emory Crawford Long Hospital. From this a knowledge set or image-family consisting of 64x64 crass-sectional images of 933 male and female patients of diffxent ages were selected. Those not selected (approximately 150 patients) were excluded from the family or knowledge set in order to use them as test images. All cross-sectional irnages were obtained from PET reconstructions, which initially were 96x96 and had already been scaled to (0,255). Therefore the true values of the attenuation coefficients ire lost. However, our approach only (7) (8) Figure 3. The eigenvector of the correlation matrix corresponding to the largest eigenvalue requires that all cross-sections have approximately the same tissue attenuation coefficients, regardless their values. All images were refined by removing truncated edges and reconstruction artifacts. Initially the PET images contained emission data. After removing emission data (of the heart), almost all cross-sections displayed a black hole on the right side, which corresponds to the left ventricle of the heart. Also, all images had a noisy inner circle. Figure 1 shows several images from the knowledge set to illustrate the similarities and differences between them. Figure 2 represents the eigenvalue spectrum of the largest 500 values, scaled logarithmically. The largest eigenvalue was approximately 20 times larger than the second one. This reflects the fact that all cross-sections had a similar structure. The successive eigenvalues decayed rapidly in the spectrum. This again reflects the fact that images of the knowledge set are similar. From the magnitudes of the eigenvalues, one can estimate relative residual errors of the truncation of the expansion by using property (4). Figure 3 displays the first eigenvector of the correlation matrix of the knowledge set corresponding to the largest eigenvalue. This first eigenimage (Le. first eigenvector) can be considered as the average image. It does not have negative values; the other eigenvectors necessarily have negative values to insure orthogonality. The image of the first eigenvector is useful for evaluating the typical size and position of members in the knowledge set. B. Computer Simulations Computer simulations were used to demonstrate how the reconstructions were affected by severely truncated fan-beam projections. Test images of cross-sections had sizes that were larger than, the same as, and smaller than what is the typical, cross-sectional image defined by the first eigenvector of the correlation matrix. Parameters of fan-beam projections such as the noncircular (4 (b) (c) Figure 4. Each pair of images represent test image itself (left) and 50 iterations of the MLEM algorithm reconstruction (right) from truncated projections with the fan-beam geometry. (a) Image smaller than, (b) larger than, (c) same size as the typical image in the knowledge set.

4 2122 Figure 5. Reconstructions used 100 basis vectors. (a) Truncated basis expansions of test images, (b) Reconstructions by nonconstrained least-squares method, (c) Reconstructions by constrained least-squares method with y =1000. Figure 6. Reconstructions used 500 basis vectors. (a) Truncated basis expansions of test images. (b) Reconstructions by nonconstrained least-squares method. (c) Reconstructions by constrained least-squares method with y =1000. detector scanning orbit radii, the number of bins in the detector, and the number of views were taken from typical clinical patient data acquisitions. The number of detector bins was 56, the number of views was 120 (uniformly distributed from 0 to 2n) and the detector bin size was equal to the pixel size. The focal length was 65 cm, the typical value of orbit radius was 30 cm and the pixel size was assumed to be 0.7 cm. Therefore, fan-beam projections were truncated by 40%. The projection data were simulated by projecting a test image using a line-length weighted projector [15]. We refer to these data as pseudoprojection daia. The condition number and the spectrum of the singular values of matrix A in (7) showed that the problem in (7) was not ill-conditioned. The generalize inverse of A and the solution to the normal equation for the least-squares method gave the same: results. Figures 4, 5, and 6 represent the reconstructions of test images with different sizes. Figure 4 shows the test images and

5 2123 Figure 8. The reconstructed emission image of cardiac phantom and the line along, which profiles are drawn, see Fig. 9. Figure 9. Profiles of reconstructed emission image and magnified region of peaks. Reconstructions were obtained by 50 iterations of ML-EM algorithm, using a) the true attenuation map of Fig 4(b), left image, b) the ML-EM reconstructed map from the truncated projections, see Fig. 4(b), right image, and c) reconstructed by proposed method using 500 eigenimages, see Fig. 6(b), middle image. (a) reconstructions by M L-EM algorithm (50 iterations) from truncated projections. The ML-EM algorithm solved a system of a linear equations of 4096 unknowns. The system was illconditioned. Figure 5 displays reconstructions by nonconstrained (i.e., y = 0 in (7)) least-squares method and constrained method (with y = 1000). The constrained leastsquares method gave better results. Only 100 basis vectors were used in the reconstructims in Fig. 5. In other words, there were 100 unknowns. The same studies were repeated with 500 basis vectors and the results are shown in Fig. 6. It is observed that Fig. 6 represents better reconstructions than those in Fig. 5, especially for the smallest image in the first row. Using a larger number of basis vectors reduced residual errors in the truncated expansion. In both Figs. 5 and 6, column (a) shows truncated expansions as in (5) with m = 100 and m = 500, respectively. No projection data were used. The coefficients p were calculated by evaluating dot-products of the test image and the eigenvectors. The images in column (a) serve as the best possible reconstructions for the truncated expansion. Figure 7 gives mare details for the small image of Fig 6. The truncated basis explansion had ripples at the sharp edges of the reconstructions (see 7b), which are known as Gibbs phenomenon. Negative values in this expansion are observed. Constraints helped to r1:gularize the solution and reduce these artifacts. The ultimate goal of obtaining attenuation maps is to correct for photon attenuation in the emission image. Fig. 8 represents an emission reconstruction of cardiac phantom from attenuated projections, which were generated using a test (i.e., true) attenuation map of a big patient, see Fig. 4(b). Three attenuation maps were used to reconstruct the emission image, using 50 I iterations of the ML-EM algorithm. The three attenuation maps were 1) the true map, 2) the ML-EM reconstruction, and 3) the map reconstructed with proposed method (see Fig. 6(b)). Fig.9 indicates that the proposed method outperforms the ML-EM method. However, the quality of the reconstructed images is approximately the same. All three attenuation maps satisfied the same transmission projections. The proposed method uses the a priory information about transmission map, while the ML-EM may converge to a solution that satisfies the transmission projections but depends on initial condition. In addition, our proposed method has a clinical advantage of having an accurate registered transmission image for anatomical location. C. Physical Jaszczak Phantom Study The Jaszczak torso phantom was first scanned using a parallel collimator and a scanning line-source. Picker s PRISM 2000XP SPECT scanner was used. The reconstruction of one slice of the Jaszczak phantom from the nontruncated parallel collimator acquired transmission projections is shown in Fig. 10. This image can be considered a reference image. The phantom was scanned again, using a fan-beam collimator and a fixed line-source on a Picker PRISM 3000XP SPECT system. The fan-beam projections were severely truncated. The phantom image was very different from the images in the knowledge set. Figure 11 represents the ML-EM reconstruction from severely truncated fan-beam projections. Figure 12 represents the solution via the proposed method and the same patient knowledge set as in the previous computer simulations. Figure 13 displays a reconstruction by using 10 iterations of the ML-EM algorithm with the initial image taken from Fig. 12(c). The 10 iterations of ML-EM further refined the interior region

6 2124 (a> (b) (c> (4 Figure 12. Reconstruction using 100 basis vectors with coefficients determined by (a) non constrained least-squares method, (b), (c), (d) constramed least-squares method with y = 100, 1000, 10000, respectively. Figure 13. Reconstruction by ML-EM algorithm (10 iterations) with Fig. 12(c) as initial image. of the image. IV. DISCUSSION Using a priori knowledge of imaging objects can help solve ill-conditioned inverse problems, as in the case of reconstruction from truncated projections. Computer simulations, where model mismatch and systematic errors do not exist, display good quality of reconstructed images. It is shown that the solution is not greatly sensitive to residual errors in the truncated expansion. The situation is changed when one tries to obtain reconstruction from real projections where model mismatch and systematic errors exist. The Jaszczak phantom study shows that one can obtain a much better reconstruction than what is commonly done using iterative ML-EM algorithm. The reconstruction is not perfect. The reasons can be found in the answer to: 1) how well a particular phantom approximates the features of real patient cross-sections and 2) how realistic the projector P is in simulating the projection physics when projecting the eigenvectors (see Eq. (6)). One can improve the reconstruction by using a knowledge set of images of the same position and size. This can be achieved by three methods: 1) Some patient features (e.g., weigh, age, sex, etc ) can be used to choose a subset from the knowledge set according to the expected image. 2) All images can be transformed by rotating, translating, and scaling to one particular pattern of the interior region. Since the interior region can be used to identify the right slice, a matched knowledge set is used to reconstruct that particular slice. 3) The projection operator can be improved so that it models the projection process accurately to avoid model mismatch artifacts. Also, we suggest to construct separate knowledge sets for each slice. Some knowledge sets may contain two lungs, and other knowledge sets may contain one lung or no lungs. The first eigenimage from each knowledge set will provide a typical interior region. The proposed use of knowledge sets to improve the reconstruction of truncated projections has important clinical applications. Specifically, the method has potential with simultaneous transmission and emission studies where transmission data are truncated when converging collimators are used to increase the geometric sensitivity for emission imaging of small organs such as the heart. The principles presented may provide for SPECT in the future an attenuation correction method that does not require the acquisition of a transmission study. ACKNOWLEDGMENTS The authors are grateful to Andy B. Barclay and Dr. Robert L. Eisner at Emory Crawford Long Hospital for providing the patient database in the present work and to Dr. Edward Di Bella for his help in processing the patient data. Also the authors would like to thank the Biodynamics Research Unit of the Mayo Foundation for use of the Analyze software package. REFERENCES [l] G. T. Gullberg, G. L. Zeng, F.L.Datz, P. E. Christian, C.-H. Tung and H.T. Morgan, Review of convergent beam tomography in single photon emission computed tomography, Phys. Med. Biol., vol. 37, no. 3, pp , [2] J. C. Gore and S. Leeman, The reconstruction of objects from incomplete projections, Phys Med. Biol., vol. 25, no. 1, pp , [3] G. L. Zeng and 6. T. Gullberg, A study of reconstruction artifacts in cone beam tomography using filtered backprojection and iterative EM algorithms, IEEE Trans. Nucl. Sei., vol. 37, no. 2, pp , [4] S. H. Manglos, Truncation artifact suppression in cone-

7 2125 beam radionuclide transmission CT using maximum likelihood techniques: Evaluation with human subjects, Phys. Med. Biol., vol. 37, no. 3, pp , [5] G. L. Zeng, C.-13. Tung, and G.T. Gullberg, New approaches to reconstructing truncated projections in cardiac fan beam and cone beam tomography, J. Nucl. Med., vol. 31, no. 5, p. 867, 1990 (abstract). [6] G. T. Gullberg, C.-:H. Tung, B. M. W. Tsui, and J. R. Perry, Correction for truncated fan beam projections in cardiac SPECT imaging, Europ. J. Nucl. Med., vol. 15, no. 8, p. 559, 1989 (abstraci). [7] B. M. W. Tsui, X. D. Zhao, P. Vemon, D. Nowak, J. R. Perry and W. H. McCartney, Cardiac SPECT reconstructions WI th truncated projections in different SPECT system designs, J. Nucl. Med., vol. 33, no. 5, p. 831, 1992 (abstraci). [8] D.-S. Luo, T.-S. Pan, M. A. King, and W. Xia, Investigation of the use of conjugate views to correct for truncation in SPECT, J. Nucl. Med., vol. 36, no. 5, p. 40P, 1995 (abstract). [9] D. J. Kadrmas, R. J. Jaszczak, J. W. McCormic, R. E. Coleman and C. B Lim, Truncation artifact reduction in transmission CT for improved SPECT attenuation compensation, Phys. Med. Biol., vol. 40, no. 6, pp ,1995. [lo] W. Chang, G. Huang, S. Loncaric and E. Holindger, Rationales for sequential application of AsF transmission scan, J. Nucl. Med., vol. 36, no. 5, p. 169P, 1995 (abstract). [ll] G. T. Gullberg and G. L. Zeng, A reconstruction algorithm using singular value decomposition of discrete representation of the exponential Radon transform using natural pixels, IEEE Trans. Nucl, Sci., vol. 41, no. 6, pp , [12] G. L. Zeng and G. T. Gullberg, An SVD study of truncated transmission data in SPECT, IEEE Trans. Nucl. Sci., vol. 44, no. 1, pp , [ 131 W.K. Pratt, Digital Image Processing, second edition, John Wiley & Sons, Inc., New York, 1991 [ 141 S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensin): and Indirect Measurements, Dover Publications, Inc., IMineola, [ ls]r.h.huesman, Ci.T.Gullberg, W.L.Greenberg, and T.F.Budinger, Users Manual - Donner Algorithms for Reconstruction Tomography, Lawrence Berkley Laboratory Publication PUB-214, 1977