D. L. Snyder and M. I. Miller, "The use of sieves to stabilize images produced with EM algorithm for emission tomography," IEEE Trans. Nucl. Sci., vol. NS-32, pp. 3864--3872, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....restoration seek the solution which best matches the probabilistic behavior of the data. Maximum likelihood (ML) estimation selects the reconstruction which most closely matches the data available, but may yield solutions which do not have many of the properties expected in the original function[7]. Maximum a posteriori (MAP) estimation allows the introduction of a prior distribution which reflects knowledge or beliefs concerning the types of images acceptable as estimates of the original cross section. This work was supported by an NEC Faculty Fellowship. The Markov random field (MRF) ....

D.L. Snyder and M.T. Miller, "The Use of Sieves to Stabilize Images Produced with the EM Algorithm for Emission Tomography," IEEE Trans. Nucl. Sci., vol. NS-32, pp. 3864-3872, 1985.

....behavior raises serious questions as to the applicability of the solution procedure, especially in the presence of attenuation, scatter, and accidentals that present significant difficulties in the physical model. Many smoothing methods have been introduced to compensate for this deficiency. Sieve [11], maximum a posteriori (MAP) 6] and other smoothing [5] methods are able to reduce speckling and compensate for data inaccuracies due to attenuation, scatter, and accidentals, to some degree. However, none of these methods address the problem of divergence in the basic mathematical model, in ....

D. L. Snyder, and M. I. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Trans. on Nucl. Science, Vol. NS-32, No. 5, pp. 3864--3872, Oct. 1985.

....(FBP) However, methods based purely on the maximum likelihood estimate produce overly noisy images. This noise may be reduced by stopping the iterative procedure used to find the maximum likelihood estimate before conver gence [1] by iterating until convergence followed by postsmoothing [2], or by including a roughness penalty term in the objective function [3] It is difficult to control resolution properties with stopping criteria. Post smoothing methods allow for better resolution control but require iteration until convergence. Since unregularized algorithms converge slowly, ....

....that the r j coefficients of our penalty design will also be smoothly varying. This is also implied by the above locally shift invariant approximation. For this reason we use the approximation RSyme J B Jr j. To illustrate this approximation, consider a simple 1 dimensional example with a single [ 1 2 1] basis. For a single basis function there is a single coeffi cient r j for each position j. In terms of (2) this means wj,j l = r j and Wj l,j = r j l. If r j is smoothly varying (i.e. r j rJ l) then wj,j l Wj l,j and R is nearly symmetric. Substituting Syme J B Jr j into (9) yields lJ(tr) ....

D.L. Snyder and M. I. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE . Nuc. Sci., vol. 32, pp. 3864-3871, Oct. 1985.

....will show the noise and edge artifacts. Therefore, it is important to regularize MLE for a 2 better reconstructed image. A variety of regularization methods have been studied in literature, like the early stopping rule in Veklerov and Llacer [12] the method of sieves in Snyder and Miller [13], the Bayesian approaches with different kinds of priors in Hebert and Leahy [14] Green [15] Herman et al. 16] and so on. Recently, Ouyang et al. 17] proposed to use the correlated structure information as the prior information and obtain the Bayesian reconstruction of PET via the weighted ....

....to reduce the edge and noise artifacts is to use the regularization methods. Several regularization methods have been proposed previously. Silverman et al. 19] and Green [20] added in a smoothness penalty in the regularization method and modify the EM algorithm accordingly. Snyder and Miller [13] used the method of sieves together with the EM algorithm. Green [15] also considered the modified EM algorithm for Bayesian approach with prior information about the patterns of target images. In addition, the minimax estimator based on tapered orthogonal series in Johnstone and Silverman [21] ....

D. L. Snyder and M. I. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Transactions on Nuclear Science, vol. NS-32, no. 5, pp. 3864--3872, 1985.

.... human bodies, the estimator without regularization has edge and noise artifacts as shown in Snyder, Miller, and Politte [5] A variety of regularization methods have been studied in literature, like the early stopping rule in Veklerov and Llacer [6] the method of sieves in Snyder and Miller [7], Chang and Hsiung [8] the Bayesian approach with different kinds of priors in Hebert and Leahy [9] Green [10] Herman, Pierro and Gai [11] and so forth. In order to have better reconstruction, the regularization methods shall not based solely on mathematical reasons. Geman and Geman [12] ....

D. L. Snyder and M. I. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Transactions on Nuclear Science, vol. NS32, no. 5, pp. 3864--3872, 1985.

....ill behaved. This phenomena, called dimensional instability by Tapia and Thompson [TT78] would manifest itself in any algorithm that maximizes the loglikelihood. These issues have been extensively studied in the context of Poisson intensity estimation in applications such as medical imaging (see [SM85], SMLTP87] and Chapter 3 of [SM91] Several solutions have been proposed for Poisson imaging which we can adapt to our radio astronomy problem. 4.1 The Method of Sieves One approach is to restrict the solution to lie in a restricted subset called a sieve [Gre81] One possibility is to require ....

....is not at all like (1. 4) Good s Roughness Good s roughness penalty [GG71] was originally formulated for smoothing estimates in nonparametric probability density estimation; a thorough analysis in this context is given by Tapia and Thompson [TT78] Following the suggestion of Snyder and Miller ([SM85], Section II.1) Good s roughness was later applied to closely related problems of Poisson intensity estimation in PET [MR91] SPECT [MM91, BM93, MB93, BMMW94] and optical sectioning microscopy [JM93] In the next few subsections, in order to conveniently express the penalties, we will index # ....

D.L. Snyder and M.I. Miller. The use of sieves to stabilize images produced with the EM algorithm for emission tomography. IEEE Trans. on Nuclear Science, 32:3864--3872, 1985.

....after a number of iterations, and (ii) the algorithm is computationally expensive in that it requires many time consuming iterations as well as significant amounts of memory. The checkerboarding has been addressed in numerous ways, e.g. regularization by the method of kernel of sieves [9, 10, 11], smoothing [12] and MAP estimation which finds the that maximizes the posterior probability P (jn ) P (n j) P ( using Poisson and Gaussian priors [13, 14, 15, 16, 17] penalized likelihoods [18] Good s measure of roughness [19] and Gibbs priors that induce Markov random fields [20, 21, ....

D. L. Snyder and M. I. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Trans. Nucl. Sci., vol. 32, pp. 3864--3870, 1985.

....not guarantee good quality images. In fact, after approximately 100 EM iterations to an image of 128 128 pixels, image quality begins to deteriorate and continuously gets worse as the likelihood function approaches its maximum, see [22] Various methods of regularization have been used before [32], 34] 16] and [12] We will show in this paper how to adapt the TV method to restore PET images by combining it with the EM algorithm. We will demonstrate that the use of TV can dramatically improve the quality of reconstructed images. However, as simulations will show, the di erence between ....

D. L. Snyder and M. I. Miller, The use of sieves to stabilize images produced with the EM algorithm for emission tomography, IEEE Trans. Nucl. Sci. NS-32:3864-3872, 1985.

....transmission case. For the emission problem, maximum likelihood (ML) estimation of x from y yields the optimization problem xML = arg min x M X i=1 (P i x Gamma y i log(P i x) 1. 6) For low signal to noise ratio medical imaging problems, the ML estimate has well documented shortcomings [36, 37, 38]. Noise and sampling limitations can produce high frequency noise in the ML reconstruction that is not present in the original cross section. It is therefore desirable to regularize tomographic inversion by some means. Maximum a posteriori probability (MAP) estimation addresses this problem by ....

D. Snyder and M. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Trans. on Nuclear Science NS-32, 3864--3871 (1985).

....the ML solution is better approximated. One approach to reducing noise is to start with a smooth image and terminate ML EM reconstruction before noise becomes too severe [3, 4, 5, 6] Alternatively, constraints may be placed on the estimated image. Such approaches include 1 the method of sieves [7], penalized maximum likelihood [8, 9] and Bayesian reconstruction [10, 11, 12, 13, 14, 15, 16, 17] Within the Bayesian framework, the prior density for the image is generally specified via Gibbs distributions [18, 19] Several proposed Gibbs priors assign higher potentials (lower probabilities) ....

D. L. Snyder and M. I. Miller. The use of sieves to stabilize images produced with the EM algorithm for emission tomography. IEEE Transactions on Nuclear Science, 32:3864--3872, 1985.

....(FBP) However, methods based purely on the maximum likelihood estimate produce overly noisy images. This noise may be reduced by stopping the iterative procedure used to find the maximum likelihood estimate before convergence [1] iterating until convergence followed by post smoothing [2], or by including a roughness penalty term in the objective function [3] It is difficult to control resolution properties with stopping criteria. Post smoothing methods allow for better resolution control but require iteration until convergence. Since unregularized algorithms converge slowly, ....

D. L. Snyder and M. I. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Tr. Nuc. Sci., vol. 32, pp. 3864--3871, Oct. 1985.

....parameters in the discretized reconstruction which best match the data. Due to the typical limits in fidelity of data, however, ML estimates are unstable, and have been improved upon by regularized estimation, such as maximum a posteriori probability (MAP) estimation [1] or the method of sieves [2]. Both the ML and MAP reconstructions may be formulated as the solution to an optimization problem. However, this optimization problem is a formidable numerical task due to both the number of parameters in the estimate (pixels or voxels) and the number of observations (photon counting ....

....associated with each image pixel, and K is the average number of iterations required for the half interval search. requires the computation of two iterations of filtered back projection. For low signal to noise ratio medical imaging problems, the shortcomings of ML estimation are well documented[2, 26, 27]. Therefore, many researchers have resorted to some form of regularized estimation for tomographic inversion. Maximum a posteriori probability (MAP) estimation addresses this problem by adding regularization in the form an a priori density for #. The MAP estimate has been shown to substantially ....

D. Snyder and M. Miller, "The use of Sieves to Stabilize Images Produced with the EM Algorithm for Emission Tomography," IEEE Trans. on Nuclear Science, vol. NS-32, no. 5, pp. 3864-3871, Oct. 1985.

....septa between z axis slices, and due to the angle of incidence with the detector. Miller and Wallis state that resolution in PET is almost completely independent of depth [9] Snyder and Miller also typically use a spatially invariant Gaussian point response to approximate their PET system [10]. 2.5.3 Clinical Implications Unless objects are large enough so that the point response induces a negligible effect, object shape and size can significantly effect image intensity. Researchers in PET [11] and in planar imaging [12] single photon imaging from only one view) have shown that ....

....iterating. Another constraint arises from the use of interpolation in the projection and backprojection steps. The interpolation smooths the data, effectively regularizing the solution. This idea is investigated more fully in Section 6.3. The other two constraints are proposed by Snyder et al. [10, 29] to reduce noise and alleviate edge artifacts. The idea is to constrain both the estimate and the quantity to be estimated. Although the notions are abstract, the implementation is to use different Gaussian filters in the projection and backprojection steps. With appropriate selection of a kernel ....

[Article contains additional citation context not shown here]

D. L. Snyder and M. I. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Trans. Nucl. Sci., vol. 32, no. 5, pp. 3864--3872, 1985.

....weights implicit in FBP D. Objective Function Objective functions based solely on the measurement statistics, be they Poisson or Gaussian, perform poorly due to the illconditioned nature of tomographic reconstruction. Unregularized methods produce increasingly noisy images with iteration [32]. To remedy this problem, several regularization methods have been investigated that impose smoothness constraints on the image estimate. One approach is the method of sieves [33,34] When AC effects are included in the Poisson case, the ML IB method of Politte and Snyder apparently requires that ....

D. L. Snyder and M. I. Miller. The use of sieves to stabilize images produced with the EM algorithm for emission tomography. IEEE Transactions on Nuclear Science, 32(5):3864--3871, October 1985.

....does not guarantee good quality images. In fact, after approximately the initial 100 EM iterations, 128 Theta 128 pixels, image quality begins to deteriorate and continuously gets worse as the likelihood function approaches its maximum [19] Various methods of regularization have been used before [28], 30] 15] and [11] We will show in this paper how to adapt the TV method to restore PET images, combining it with the EM algorithm. We will demonstrate that the use of TV can dramatically improve the quality of reconstructed images. The outline of the paper is as follows. In Section 2, we ....

Snyder DL and Miller MI, The use of sieves to stabilize images produced with the EM algorithm for emission tomography, IEEE Trans. Nucl. Sci. NS-32:3864-3872, 1985.

.... the iteration before convergence [18] using quadratic approximations to the likelihood with a penalty [19, 20] using a separable (non smoothness) prior [21 23] and introducing a smoothing step into the ML EM iteration [24 26] Perhaps the most popular alternative is the method of sieves [27, 28]. Sieves are usually implemented by post smoothing, even though the commutability requirement [28, eqn. 12) is rarely met in practice. However, recent studies, e.g. 29] have found that MAP (or equivalently PML) methods outperform the method of sieves. Therefore, in this report, we focus on ....

D L Snyder and M I Miller. The use of sieves to stabilize images produced with the EM algorithm for emission tomography. IEEE Transactions on Nuclear Science, 32(5):3864--3871, October 1985.

....case convergence of the Kullback discrimination to a positive value rather than 0 is expected. Initial image 0 is a uniform image in which each pixel has the same positive value. The EM ML iterations are carried out without employing regularization, filtering, or prior probabilities (cf. [33, 24, 12, 7, 14, 29, 39] for some of these methods in PET or SPECT) 4.1.3 Network and Algorithm Parameters The empirical results concern convergence, global and local (region of interest) error evaluation, and time of computation for distributed EM ML iteration under the following conditions. All runs on the CM 5 and ....

Snyder, D. L. and Miller, M. J. The use of sieves to stabilize images produced with the EM algorithm for emission tomography. IEEE Trans. Nucl. Sci., 32:3864--3870, 1985.

....but seek the solution which best matches the probabilistic behavior of the data. Maximum likelihood (ML) estimation selects the reconstruction which most closely matches the data available, but may yield solutions which do not have many of the properties expected in the original function[8]. Bayesian estimation allows the introduction of a prior distribution which reflects knowledge or beliefs concerning the types of images acceptable as estimates of the original cross section. This prior distribution weights the likelihood function and may even impose hard constraints on the ....

D.L. Snyder and M.T. Miller, "The Use of Sieves to Stabilize Images Produced with the EM Algorithm for Emission Tomography," IEEE Trans. Nucl. Sci., vol. NS-32, pp. 3864-3872, 1985.

....[13] Thomason 6 Materials and Methods Initial Image, Sinograms, and P Matrix In this study, initial image 0 is a uniform image in which each pixel has the same positive value. 512 iterations are carried out for EM ML without employing regularization, filtering, or prior probabilities (cf. [14, 15, 16, 17, 18, 19, 20] for some of these methods in PET or SPECT) No explicit terms for scatter or random counts are included in these computations. Scan data was obtained on an ECAT 921 PET scanner (CTI Siemens, Knoxville, TN) at the University of Tennessee Medical Center, Knoxville (UTMCK) 2 D reconstructions are ....

D L Snyder and M J Miller. The use of sieves to stabilize images produced with the EM algorithm for emission tomography. IEEE Trans. Nucl. Sci., 32:3864--3870, 1985.

....(FBP) However, methods based purely on the maximumlikelihood estimate produce overly noisy images. This noise may be reduced by stopping the iterative procedure used to find the maximum likelihood estimate before convergence [1] iterating until convergence followed by postsmoothing [2], or including a penalty term in the likelihood objective function [3] Penalized likelihood methods have the advantage of allowing arbitrary regularizations including edge preserving penalties and penalties incorporating anatomical side or boundary information. Regularization can also improve the ....

D. L. Snyder and M. I. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Tr. Nuc. Sci., vol. 32, pp. 3864--3871, Oct. 1985.

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D. L. Snyder and M. I. Miller, "The use of sieves to stabilize images produced with EM algorithm for emission tomography," IEEE Trans. Nucl. Sci., vol. NS-32, pp. 3864--3872, 1985.

No context found.

D. L. Snyder and M. I. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Tr. Nuc. Sci., vol. 32, pp. 3864--3871, Oct. 1985.

No context found.

D. Snyder and M. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Trans. on Nuclear Science 32, pp. 3864--3872, 1985.

No context found.

D. Snyder and M. Miller, "The use of sieves to stabilize images produced with the EM algorithm for emission tomography," IEEE Trans. Nuc. Scie. NS-32, pp. 3864--3872, 1985.

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