NONLINEAR BACK PROJECTION FOR TOMOGRAPHIC IMAGE RECONSTRUCTION Ken Sauer and Charles A. Bouman Department of Eletrial Engineering, University of Notre Dame Notre Dame, IN 46556, (219) 631-6999 Shool of Eletrial Engineering, Purdue University West Lafayette, IN 4797-51, (317) 494-34 ABSTRACT The objetive of this paper is to investigate a new non-iterative paradigm for image reonstrution based on the use of nonlinear bak projetion filters. This method, whih we all nonlinear bak projetion (NBP), attempts to diretly model the optimal inverse operator through off-line training. Potential advantages of the NBP method inlude the ability to better aount for effets of limited quantities and quality of measurements, image ross-setion properties, and forward model non-linearities. We present some preliminary numerial results to illustrate the potential advantages of this approah and to illustrate diretions for future investigation. 1. INTRODUCTION In reent years, onsiderable effort has been put into the development Bayesian model-based approahes to tomographi image reonstrution[1, 2, 3]. While these methods an substantially improve reonstrution quality, these iterative methods an be omputationally demanding, requiring at least 1 to 2 times the omputation of filtered bak projetion. Even the best reonstrution algorithms may produe artifats that an be visually identified in limited data problems. In fat, several tehniques have been proposed to take advantage of restritive a priori knowledge of the objet to ompensate for these limitations in nonlinear geometri reonstrution [4, 5, 6]. The fat remains that onventional filtered bak projetion (FBP) an produe surprisingly good quality reonstrutions despite its limitation to simple averaging of projetion data. In this researh, we propose a more diret and nonlinear approah to the Bayesian tomographi inverse problem. Rather than trying to develop an aurate THIS WORK WAS SUPPORTED BY A GRANT FROM THE NATIONAL SCIENCE FOUNDATION. forward model that an be inverted, our approah is to diretly model the inverse operator. The goal is to develop an non-iterative Bayesian reonstrution method whih requires omputation omparable to onventional CBP methods, but ahieves quality omparible to or better than that of urrent Bayesian methods. The method we propose forms a bak projeted image ross-setion by applying non-linear filters to the projeted data. This method, whih we all non-linear bak projetion, attempts to diretly model the optimal inverse operator through off-line training of these non-linear filters using example training data. This diret approah to modeling of the inverse operator has a number of potential advantages whih make it interesting: Better modeling of image ross-setion behavior - Current Bayesian models, suh as MRF s are limited in their omplexity by the diffiulty of estimating model parameters. A diret model an be more effetively trained for the attributes of typial image rosssetions. Better modeling of non-linear forward models - Non-linear forward models are diffiult to inorporate in urrent Bayesian methods. This problem is avoided by diret modeling of the inverse operator. Less omputation - Sine diret non-linear inversion is not iterative, it has potentially muh lower omputational requirements. 2. NON-LINEAR BACK PROJECTION METHOD The nonlinear bak projetion tomography (NBP) method is illustrated in Fig. 1. Conventional CBP works, as shown in Fig. 1(a), by filtering the projetions along a speifi angle, θ, and then bak projeting the result. Theoretially, this method yields perfet reonstrution with a ontinuum of data, but in pratie it is well known to produe artifats, and either overly smooth
θ Projetion or exessively noisy reonstrutions. The NBP method works by applying a nonlinear filter to a window in the sinogram spae as illustrated in Fig. 1(b). The behavior of this filter depends on the loal harateristis of the sinogram. In addition, the filter s harateristis may depend on the speifi pixel being bak projeted; however, we will not onsider this dependene in the present investigation. The non-linear filter is then formed by ombining M distint linear filters, eah of whih is designed to minimize mean squared error for some lass of input values. A related model has been proposed by Popat and Piard [7] for appliation in image restoration, and ompression. 3. DERIVATION OF NBP ALGORITHM { t Bakprojet Bakprojet Filter (a) Sinogram nonlinear filter window θ t 2π Let Y be a vetor ontaining all the sinogram data, and let X be a pixel in the unknown image ross-setion. Furthermore, let Y j be the vetor of sinogram samples taken from the j th window. In other words, j indexes the projetion angle and Y j ontains the samples from the filter window of Fig. 1(b). Notie that the position of the window along both the t and θ oordinates will depend on the partiular pixel being reonstruted. For simpliity, we will assume that the enter pixel of the window falls preisely on the on the path integral required for bak projetion. In pratie, the bak projetion is interpolated by a weighted ombination of neighboring projetion windows, but this does not substantively effet the resulting analysis. The onventional CBP reonstrution may be expressed as X = FY j j (b) Figure 1: This figure illustrates that basi onept of nonlinear bak projetion. (a) Conventional CBP works by filtering the projetions at eah angle and then bak projeting the result. (b) Nonlinear bak projetion applies a nonlinear filter to a window of data in the sinogram and then bak projets this result. where F is the matrix whih implements a filter along t, and the sum over j is the bak projetion operation. For onventional CBP, the matrix F does not depend on j, the projetion angle. Moreover, most of the elements in F are zero sine only the samples at a single angle are filtered. For NBP, we will make the assumption that eah set of samples, Y j, has assoiated with it a disrete lass, C j, whih takes on values between and M 1. Instead of a single filter, we will have M filters denoted by F where < M. We will make two assumptions expressed in the following two equations. E[X Y,C] = j F Cj Y j P{C j = Y} = f( Y j ). The first equation states that given the lass information, the minimum mean squared error (MMSE) estimate may be obtained by applying an appropriate filter
to eah window. The seond equation states that the distribution of eah lass is only dependent on the pixels in the assoiated window. Using these two assumptions, we ompute the MMSE estimator of X given Y. E[X Y ] = E[E[X Y,C] Y ] (1) = E F Cj Y j j Y = j = j E [ F Cj Y ] Y j ( ) F f( Y j ) Intuitively, this optimal estimator is formed by applying a spatially varying filter, F f( Y j ) to the sinogram. Sine the filter depends on the data in the window through f( Y j ), this is atually a nonlinear filter. The image is then reonstruted by bak projeting the nonlinearly filtered sinogram. To apply this strategy, the filters F and the distribution f( Y j ) must be estimated. To do this, we first rewrite (1) in the form E[X Y ]= F f( Y j )Y j = F Ȳ j where Ȳ = j f( Y j)y j. By defining, F =[F,F 1,,F M 1 ] Y Y 1 Ȳ =. Y M 1 the optimal filters may be easily omputed as the least squares solution to [ X min E F Ȳ 2] F In order to ompute the probabilities f( Y j ), we use a Gaussian mixture model for Y j so that Y j 4. EXPERIMENTAL RESULTS In this setion we present preliminary experimental results to illustrate the method of NBP. Figures 2 (a) and (b) show the syntheti phantom together with the filter bak projetion reonstrution. The ross setion was reonstruted at a resolution of 128 128 from 16 uniformly spaed projetion angles. Figure 2 () and (d) show the result of training and applying NBP with 18 and 6 lusters respetively. Notie that the NBP reonstrutions have redued artifats beause eah of the filters is designed to smooth these errors. However, some sharpness is also lost along the edges of features. We believe that this sharpness an be reovered by allowing filters to vary depending on the spatial loation of pixels in the image. Figure 3 illustrates how the filters for various lusters vary. Eah slie through the 3-D plot shows the filter values for a speifi luster. Notie that some filters are impulsive while others are smoothing funtions. Figure 4 shows how the mean squared error (MSE) varies with the number of lusters used in the NBP reonstrution. The MSE is normalized with respet to filtered bak projetion reonstrution; so an MSE of 1 is equal to that of filtered bak projetion. Notie that as the number of lusters is inreased the MSE dereases beause the projetions with different behaviors an be treated differently. This effet is more notieable when the MSE is omputed on the entire image beause of the large disontinuity generated at the support boundary of the objet. 5. CONCLUSION We presented a novel nonlinear image reonstrution algorithm whih is oneptually similar to filtered bak projetion, but is not limited by a restrition to linear filters. Preliminary results indiate that the method an redue reonstrution artifats. We expet that the method an be improved by making the filters a funtion of both the ross-setion pixel and the projetion data. p(y j )= p(y j )π 6. REFERENCES where p(y j ) is a multivariate Gaussian distribution, and M 1 π = 1. The parameters of p(y j ) andthe probabilities π may be estimated using the EM algorithm [8, 9, 1]. Given the mixture model, we have f( Y j )= p(y j )π p(y j )π. [1] T. Hebert and R. Leahy. A generalized EM algorithm for 3-d Bayesian reonstrution from Poisson data using Gibbs priors. IEEE Trans. on Medial Imaging, 8(2):194 22, June 1989. [2] Gabor T. Herman and D. Odhner. Performane evaluation of an iterative image reonstrution algorithm for positron emission tomography.
IEEE Trans. on Medial Imaging, 1(3):336 346, September 1991. 2 1.5 18 Filters Used for Nonlinear Bak Projetion [3] Peter J. Green. Bayesian reonstrution from emission tomography data using a modified EM algorithm. IEEE Trans. on Medial Imaging, 9(1):84 93, Marh 199. 1.5.5 1 1.5 [4] Y. Bresler, J. A. Fessler, and A. Maovski. A Bayesian approah to reonstrution from inomplete projetions of a multiple objet 3D domain. IEEE Trans. on Pattern Analysis and Mahine Intelligene, 11:84 858, August 1989. 2 1 8 6 4 2 5 1 Filter Number 15 2 [5] J. Prine and A.S. Willsky. A hierarhial algorithm for limited-angle reonstrution. In Pro. of IEEE Int l Conf. on Aoust., Speeh and Sig. Pro., pages 1468 1471, Glasgow, Sotland, May 23-26 1989. Figure 3: Plot of 18 different filters used in bak projetion operation. Eah filter has 9 taps. Notie that some filters are impulsive while others are smoothing funtions. [6] P. Milanfar, W.C. Karl, and A.S. Willsky. A moment-based variational approah to tomographi reonstrution. IEEE Trans. on Image Proessing, 5:459 47, Marh 1996. [7] K. Popat and R. W. Piard. Cluster-based probability model and its appliation to image and texture proessing. Tehnial Report 351, MIT Media Laboratory, Cambridge, MA. 1. [8] C. A. Bouman. Cluster: an unsupervised algorithm for modeling Gaussian mixtures. Available from http://www.ee.purdue.edu/ bouman, April 1997. Normalized MSE.5 Entire Image Center Only [9] L. E. Baum and T. Petrie. Statistial inferene for probabilisti funtions of finite state Markov hains. Ann. Math. Statistis, 37:1554 1563, 1966. [1] Murray Aitkin and Donald B. Rubin. Estimation and hypothesis testing in finite mixture models. Journal of the Royal Statistial Soiety B, 47(1):67 75, 1985.. 1 2 3 4 5 6 Number of Clusters Figure 4: Plot of mean squared reonstrution error versus number of lusters. A value of represents the mean squared error ahieved by filtered bak projetion Notie that the error dereases more rapidly for the entire image sine this inludes the large disontinuity along the support boundary of the objet.
(a) (b) () (d) Figure 2: (a) Original phantom used for simulations; (b) Filtered bak projetion reonstrution; () Nonlinear bak projetion result using 18 lusters; (d) Nonlinear bak projetion result using 6 lusters.