Fast Projectors for Iterative 3D PET Reconstruction

Transcription

1 2005 IEEE Nuclear Science Symposium Conference Record M05-3 Fast Projectors for Iterative 3D PET Reconstruction Sanghee Cho, Quanzheng Li, Sangtae Ahn and Richard M. Leahy Signal and Image Processing Institute, University of Southern California, Los Angeles, CA Abstract We describe a fast forward and back projector pair based on inverse Fourier rebinning for use in iterative fully 3D image reconstruction. The fast projector pair is used as part of a factored system matrix that models detector-pair response with a shift variant sinogram blur kernel. In this way we are able to combine the computational advantages of Fourier rebinning with iterative reconstruction using accurate system models. The forward projector uses the exact inverse rebinning equation to map stacked transaxial projections into a full 3D sinogram. The back projector is simply the transpose of the forward projector and differs from the true exact rebinning operator in the sense that it does not require reprojection to compute missing lines of response (LOR). The inverse rebinning projector differs in performance by less than 1% in RMS error from the fully 3D geometric projector computed using solid angle calculations of voxel-wise sensitivity along each LOR. The cost of forward and back projection for the micropet Focus220 small animal scanner using the inverse rebinning projector is 5 to 15 times less than that for the fully 3D geometric projector, depending on the reconstructed field of view and voxel size. We compare performance between the inverse rebinning and 3D geometric projectors using simulated point and phantom data and in vivo FDG mouse data. I. INTRODUCTION ITERATIVE methods based on a Poisson noise model have been widely demonstrated to provide superior image quality compared to analytic reconstruction. The dominant computation cost in iterative fully 3D PET image reconstruction is the calculation of forward and back projections between the image and sinogram spaces. Consequently, the availability of fast projection methods for use with iterative reconstruction methods would facilitate more routine use of fully 3D iterative reconstruction. A very successful approach to combining iterative methods with 3D data is to first rebin the data into a set of sinograms using Fourier rebinning and then apply a iterative reconstruction method such as OSEM [1]. While this solution results in relatively short reconstruction times, the one time data dimensionality reduction that occurs during the initial rebinning step has the potential to induce a loss in resolution or noise performance compared to a method that iterates between the image and original data. Mapping between the data and image space can be achieved in a number of different ways. The standard approach is to use geometric projection operations entirely in the space domain. These projectors can be based on a line integral model, a This work was supported by NIBIB under Grant no R01 EB detection tube intersection model [2] or a potentially more accurate model that accounts for detector pair sensitivity through consideration of the solid angle subtended at the detectors from each voxel [3]. Costs can be reduced using Fourier-based projectors in which the FFT algorithm is used in combination with the Fourier slice theorem to map the calculations from the spatial to the frequency domain [4]. The major limitation to this approach is that it is not well suited to use with algorithms that iterate over subsets of the data. An alternative, which achieves a similar speed up to Fourier methods, is based on a hierarchical decomposition that exploits the property that the angular bandwidth of the sinogram is proportional to the image size. Bresler et al. used this decomposition to develop fast forward [5] and back projection [6] operators that can be applied to fully 3D PET, but to date published fully 3D applications are limited to conebeam tomography. A third approach to development of fast projectors is to make use of one of the Fourier rebinning methods developed over the past several years by Defrise and colleagues [7] [9]. In our approach, the exact inverse rebinning equation [7] is applied to develop the fast projectors. We evaluate this fast projector pair in comparison to our factored system model which has been widely used in small animal PET systems [3]. II. METHODS A. MAP Reconstruction and the Factored System Model The projectors described below can be used with many of the different iterative fully 3D reconstruction methods, but here we focus on MAP reconstruction based on the shifted-poisson model [10], [11]. The log-likelihood function L(y x) can be written as L(y x) = [ (ŷ i + 2r i + s i ) + (y i + 2r i ) log(ŷ i + 2r i + s i )] (1) i where ŷ = P x is the forward projection of image x using the system matrix P, y i are the measured sinogram data, and r i and s i are the estimated means of the randoms and scatter, respectively. We find a maximum a posteriori (MAP) solution as the maximizer of the objective function: ˆx(y) = arg max L(y x) βφ(x) (2) x 0 where φ(x) is a quadratic penalty or Gibbs energy function that penalizes the squared differences between each pair of neighboring voxels and β is the smoothing or hyper parameter /05/$ IEEE 1933

2 The projection or system matrix P models the sensitivity of each detector pair to positron emissions from each voxel. In fully 3D systems this matrix is huge (> elements for the Focus 220 small animal scanner described in Section III- A) but can be stored efficiently through the use of geometrical symmetries and sparse matrix structures in combination with the factorization [3]: P = P det.sens P det.blur P attn P geom P positron (3) where P geom is the geometric sensitivity matrix representing the probability that an emission from each voxel will result, in the absence of attenuation and noncolinearity effects, in a photon pair entering the surfaces of the detector pair associated with each LOR and the detail description for other operators is in [3]. Here, projections (i.e. multiplication by P geom and P T geom) are still time consuming, requiring a mapping between each sinogram element and all those voxels that intersect the corresponding LOR. B. Exact Inverse Rebinning The earliest Fourier rebinning method, FORE, is approximate. Exact rebinning methods, FOREX and FOREJ, were later derived, respectively from the Fourier slice theorem and John s equation [8], [9]. While the cost for exact rebinning using FOREX can be high because of the need for reprojection to compute missing data, exact inverse rebinning is of lower cost since this does not require reprojection. Here we use inverse rebinning based on the FOREX result [8], summarized as follows. We define f(x, y, z) as the 3D source distribution. The sinogram p(s, φ, z, δ) can be written as the line integral: p(s, φ, z, δ) = dtf(s cos φ t sin φ, s sin φ + t cos φ, z + tδ) where s and φ are the radial and angular direction variables, respectively, for a single sinogram, t is the line integral variable and δ is the tangent of the oblique angle θ as shown in Fig. 1. y φ s (a) Transaxial plane LOR x θ δ = tan θ (b) axial oblique angle Fig. 1. (a) Top and (b) side view of the 3D PET geometry indicating variables used in the definitions of the sinograms. For δ = 0, p(s, φ, z, 0) represents the stacked sinograms of the transaxial slices. The relationship between the 3D z (4) sinograms p(s, φ, z, δ) and the sinograms p(s, φ, z, 0) is given by the following exact inverse rebinning equation: (, k, z, δ) = { ( )} δz exp ik arctan 1 + ( δz ) 2, k, z, 0 (5) where ( ) is the 3D Fourier transform (FT) of p(s, φ, z, δ) at a fixed angle δ. Variables and z are the radial and axial frequency variables, respectively, and k is the Fourier series index in the angular direction (the Fourier transform is discrete with respect to angle φ because the function is 2π periodic). C. Forward and Back Projection using Inverse Rebinning The projection operator P geom in (3) can be approximately factored into the product of two operators: the projector that maps the 3D image data into the stack of sinograms p(s, φ, z, 0) and the inverse rebinning operator that maps these sinograms into the full 3D data p(s, φ, z, δ). For inverse rebinning, we use a modified version of (5). Using the shift property of the FT and taking the inverse FT with respect to the angular frequency index k, the exponential term on the right hand side drops out leaving the FT relationship: P (, φ, z, δ) = ( ) 2 ( ) δz δz P 1 +, φ arctan, z, 0 (6) where P (, φ, z, δ) is the FT of p(s, φ, z, δ) with respect to the variables s and z. This alternative form has several advantages for our application. Generally, the angular direction is the largest dimension in the sinogram space and the adjustment of the number of sample points to some favorable number for efficient computation of the FFT (e.g. powers of 2) requires resampling rather than zeropadding because of the angular periodicity of the data. Since we work directly with the angular variable φ we avoid the need for resampling. More importantly, as we describe below, this form facilitates the use of rebinning in combination with ordered-subsets methods. The additional cost associated with working with respect to φ rather than k is that we need to perform a interpolation in (φ, ) rather than just as would be the case when using (5). Combining inverse rebinning with a projector, we can write the forward geometric projection operation in factored matrix form as: y = δ A δ T δ N δ B φ φ I φ,δ F P x (7) where x is the 3D image vector and y is the full 3D sinogram data. This operation replaces only the P geom component of the full system matrix (3) and is combined with the other terms to 1934

3 model the entire data acquisition process. We now define each of the terms in (7). P is the geometric projection operator that computes the sinogram data p(s, φ, z, 0) from each slice of the 3D image data. F is a block matrix that computes the FFT in (s, z) for all angles φ to yield P (, φ, z, 0). The remainder of the calculations are performed separately for each pair (φ, δ). The 3D oblique sinogram P (, φ, z, δ) can be obtained from P (, φ, z, 0) using (6) where = 1 + ( δ z ) 2 and φ = φ arctan ( δ z ). The matrices Iφ,δ perform the interpolation of P (, φ, z, 0) with respect to variables (, φ ). The term in parenthesis computes those components of the 3D sinogram corresponding to a fixed value of (φ, δ). is a inverse FFT with respect to (, z ), yielding p(s, φ, z, δ). B φ is a concatenation operator such that the sum over φ combines the components of the 3D sinogram for all values of φ at a fixed value of oblique angle δ. N δ is a diagonal matrix that compensates for differences in sensitivity between the line integral model and our solid angle model as a function of oblique angle δ, as described in Section II-D. T δ is a truncation matrix which removes the unobserved or missing data in the oblique sinograms and, finally A δ is a second concatenation operator which when summed over δ assembles all oblique sinograms into a single vector. x P F IF B i IF B 1 N 1 T 1 IF B 2 N 2 T 2... IF B δm N δm T δm (a) New forward projector I 1,i I 2,i I φm,i... A 1 A 2 A δm B 1 B 2 B φm y 2 (b) Detail for IF B i operator Fig. 2. Block diagram of the forward projector: (a) each dotted line boxes represents inverse rebinning for a single oblique angle; the summation operator indicates concatenation of all oblique sinograms. (b) Detail for IF B i in (a); each dotted line box indicates calculation for a single angle φ; the summation operator indicates concatenation over all angles φ.; δ m and φ m are the largest indices for δ and φ, respectively A block diagram for the forward projector is shown in Fig. 2. We have included this detailed description and accompanying figure since it clearly shows the degree to which the calculations can be parallelized: each dotted box in Fig. 2a and b can y δm y 1 y be computed independently. Similarly, the decomposition in Fig. 2b with respect to φ indicates that the method can be readily adapted to use with ordered subsets algorithms, although we note that a full set of stacked sinograms will be required for the computation for each subset. The back projection operator is obtained by transposing the whole operator in (7): ˆx = P T F δ φ I T φ,δ BT φ N δ Tδ T AT δ y. (8) It is interesting to note that the backprojector in (8), which is the adjoint of the inverse rebinning procedure in (5), is not the equivalent exact Fourier rebinning operator. In exact rebinning we would need to perform reprojection to compute the missing oblique data, while in (8) the missing data remain zero through the transpose of the truncation operator T δ which simply adds zeroes to produce arrays of the correct dimension before applying the FFTs. D. Normalization Factors The inverse rebinning operator is based on the ideal line integral model and does not account for reduced sensitivity as the oblique angle increases. This drop in sensitivity is caused by a reduced solid angle and is modeled explicitly in the geometric calculation of P geom [3]. To match P geom we must therefore scale each sinogram computed through inverse rebinning. To compute these factors we forward project a uniform cylinder of radius 25.6mm and calculate the ratio between the results based on P geom and those based on inverse rebinning. For each oblique angle, the scale factor is then computed as the average of these ratios over the nonzero portion of the sinogram. To evaluate the performance of the new projector, we define the error metric: E RMS = y r y p y r where is the Euclidean norm, y r and y p are the sinograms computed using P geom and inverse rebinning, respectively. Fig. 3 shows the error as a function of oblique angle with and without normalization for solid angle. The error was calculated for the Focus 220 scanner [12] with 48 rings for maximum ring difference (MRD) 47 and span of 3 resulting in 16 different oblique angles in the fully 3D sinograms. When normalization is applied, the error is within 1% for all oblique angles. III. SIMULATION RESULTS A. Computation Cost and Accuracy We compared the new inverse rebinning projector with our fully 3D geometric projector, P geom, in terms of accuracy and run time. All calculations were based on the micropet Focus 220 small animal scanner [12]. A voxel Hoffman brain phantom with voxel size 0.4mm was used as ground truth. We used the 3D geometric projector to compute the fully 3D (9) 1935

4 Normalized Error w/o Normalization w Normalization Oblique Angle Fig. 3. Forward projection error using inverse rebinning with and without normalization to account for variations in solid angle with ring difference. B. Application to the MAP Reconstruction We conclude by illustrating performance when using the inverse rebinning projector in MAP reconstruction of phantom data and of in vivo mouse data. We use the preconditioned conjugate gradient method described in [11] and compare performance when we replace the fully 3D geometric projector, P geom with the inverse rebinning projector. In these studies we do not include positron range modeling in the factorization (3). The only user specified parameters in the MAP algorithm are the smoothing parameter β in (2) and the number of iterations. All results below use the image from 2 iterations of 3D OSEM with 6 subsets as an initial estimate. For a matched comparison between the two projectors we use identical values of β. We reconstructed images from sinogram data for the digital 3D Hoffman brain phantom. Pseudo-Poisson data were generated to simulate total counts with a uniform 5% randoms background. We ran 30 iterations with a hyperparameter β = 0.1. TABLE I COMPARISON OF COMPUTATION TIME FOR PROJECTION(UNIT:SECS) Image Size 128x128x65 256x256x97 3D Geometric(A) Forward IRB + Geometric(B) A/B D Geometric(A) Backward IRB + Geometric(B) A/B (a) 3D Geometric (b) IRB + Geometric sinogram for the Focus 220 with a maximum ring difference of 47 and a span of 3 for a total of 1,567 sinograms, each of dimension For the inverse rebinning operator we first used the stacked form of the geometric projector to compute the 95 transaxial sinograms. The inverse rebinning operator was then used to compute the fully 3D sinograms. As shown in Fig. 3, the error is less than 1% and increases only slightly with oblique angle. We also compared the accuracy of backprojections using the same error metric and achieved an error of less than 1% for this data. Run times for forward and backprojection are shown in Table I. The code was run on an AMD Opteron 250 workstation with dual 2.4GHz CPU and 8Gbytes RAM with only one CPU used in these benchmark tests. Forward and back projectors were computed for images of size and voxels, in both cases with a transaxial voxel size of 0.4mm. We list the forward and backprojection times for the inverse rebinning projector in two parts: the cost for inverse rebinning (IRB) and the cost for computing the stacked sinograms. The table shows a speed up factor relative to the fully 3D calculation of approximately five for the smaller image size and 15 for the larger image size. (c) True (d) Difference Fig. 4. Reconstructed transaxial sections through the brain phantom (c) using the 3D geometric (a) and inverse rebinning (b) projectors; (d) is the difference between (a) and (b); [min max] = (a) [ ] (b) [ ] (c) [ ] (d) [ ] Figs. 4 shows a transaxial section through the reconstructed images. Visually the images appear very similar and the difference image reveals little structure in the errors. Finally we compare reconstructions for an in vivo mouse study using 18 FDG. The data were collected from the Focus 220 scanner. The sinogram data were collected for a total of one hour starting 30mins post injection for a total of counts. In Fig. 5 we show sagittal, coronal and transaxial sections through reconstructions obtained using MAP with both projectors. For comparison we also show a reconstruction using Fourier Rebinning (FORE)+ OSEM(4 iterations) and 1936

5 the analytic 3DRP method [13] with a ramp filter. The OSEM algorithm uses line integrals and linear interpolation for computing projections and does not include a detector response model, as is the case in [1]. These results indicate much larger differences between MAP and both FORE+OSEM and 3DRP than there is between the two projectors used in MAP reconstruction. IV. CONCLUSIONS We have developed an approach for combining Fourier rebinning with iterative 3D PET reconstruction as a means of reducing reconstruction time while retaining the advantages of fully 3D iterative reconstruction. The overall speed up in the current implementation of the code ranges between 5 and 15 depending on the image and voxel size. By simple oblique angle dependent normalization, the new projectors matched the fully 3D geometric projector within a 1% rms error. When implemented as part of a MAP reconstruction algorithm, the results indicate small differences between the geometric and Fourier based 3D projectors. The method can also be applied to fully 3D ordered subsets algorithms although in the case the speed up is reduced because of the requirement for computation of the fully stacked transaxial projection or backprojection for each subset calculation. ACKNOWLEDGMENT The authors would like to thank David Stout of UCLA, Bing Bai of Siemens Preclinical Solutions for providing the mouse data and Michel Defrise for his helpful insights on the low computational cost of exact inverse rebinning. REFERENCES [1] K. Lee, P. E. Kinahan, J. A. Fessler, R. S. Miyaoka, M. Janes and T. K. Lewellen, Pragmatic fully 3D image reconstruction for the MiCES mouse imaging PET scanner, Phys. Med. Biol., vol. 49, pp , [2] A. Terstegge, S. Weber, H. Herzog, H. W. Muller-Gartner and H. Halling, High Resolution and Better Quantification by Tube of Response Modeling in 3D PET Reconstruction, IEEE Nuclear Science Symp. and Medical Imaging Conf., pp , [3] J. Qi, R.M. Leahy, S. R. Cherry, A. Chatziioannou and T. H. Farquhar, High-resolution 3D Bayesian image reconstruction using the micropet small-animal scanner, Phys. Med. Biol., vol. 43, pp , [4] S. Matej, J. A. Fessler and I. G. Kazantsev, Iterative Tomographic Image Reconstruction Using Fourier-Based Forward and Back-Projectors, IEEE Trans. Med. Imag., vol. 23, pp , [5] A. Boag and Y. Bresler, A Multilevel Domain Decomposition Algorithm for Fast O(N 2 log N) Reprojection Tomographic Images, IEEE Trans. Image Processing, vol. 9, pp , Sept [6] S. Basu and Y. Bresler, O(N 3 log N) Backprojection Algorithm for the 3-D Radon Transform, IEEE Trans. Med. Imag., vol. 21, pp , Feb [7] M. Defrise, P. E. Kinahan, D. W. Townsend, C. Michel, M. Sibomana and D. F. Newport, Exact and Approximate Rebinning Algorithms for 3-D PET Data, IEEE Trans. Med. Imag., vol. 16, pp , April [8] Xuan Liu, M. Defrise, C. Michel, M. Sibomana, C. Comtat, P. E. Kinahan and D. W. Townsend, Exact Rebinning Methods for Three-Dimensional PET, IEEE Trans. Med. Imag., vol. 18, pp , Aug [9] M. Defrise and Xuan Liu, A fast rebinning algorithm for 3D positron emission tomography using John s equation, Inverse Problems, vol. 15, pp , Fig. 5. Transaxial, sagittal and coronal views through an 18 FDG mouse study on the Focus. The images were reconstruced by using (a) MAP with fully 3D geometric projector; (b) MAP with inverse rebinning projector; (c) FORE + OSEM; (d) 3DRP with a ramp filter. [10] M. Yavuz and J. A. Fessler, Statistical image reconstruction methods for random-precorrected PET scans, Med. Image Anal., vol. 2, no. 4, pp , [11] J. Qi, R.M. Leahy,C. Hsu, T. H. Farquhar and S. R. Cherry, Fully 3D Bayesian Image Reconstruction for the ECAT EXACT HR+, IEEE Trans. Nucl. Sci., vol. 45, pp , June [12] Y. Tai, A. Ruangma, D. Rowland, S. Siegel, D. F. Newport and P. L. Chow, Performance Evaluation of the micropet Focus: A Third-Generation micropet Scanner Dedicated to Animal Imaging, Journal of Nucl. Med., vol. 46, pp , [13] P. Kinahan and J. Rogers, Analytic 3D image reconstruction using all detected events, IEEE Trans. Nucl. Sci., vol. 36, pp ,