D. W. Wilson, B. M. W. Tsui, and H. H. Barrett, "Noise properties of the EM algorithm: II. Monte Carlo simulations, " Phys. Med. Biol., vol. 39, pp. 847--872, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....plays a significant role. The nonnegativity constraint is a severe nonlinearity, whereas the approximations are based on a local linearization of the estimator [1] Further work on this difficult problem is needed. Our empirical histograms of j show that the log normal model proposed in [11,12] for ML EM does not appear applicable to the penalizedlikelihood case. Further work is also needed to develop analytical autocorrelation function approximations that reflect the asymmetric properties of the empirical autocorrelation functions. We have also observed such asymmetries in the ....

D. W. Wilson, B. M. W. Tsui, and H. H. Barrett, "Noise properties of the EM algorithm: II. Monte Carlo simulations, " Phys. Med. Biol., vol. 39, pp. 847--872, 1994.

....such as higher resolution in high count regions etc. in a manner similar to methods for space varying regularization [8, 9] but in this paper we focus on the goal of providing uniform resolution. This paper is somewhat in the spirit of previous studies that used the local impulse response [10 14] or an effective local Gaussian resolution [15] to quantify the resolution properties of the unregularized maximum likelihood expectation maximization (ML EM) algorithm for emission tomography. However, there is an important difference in our approach: since the ML EM algorithm is rarely iterated ....

....response if we could avoid performing extensive numerical simulations. The remainder of this paper is devoted to approximations suitable for likelihood based estimators in tomography. C. Linearized Local Impulse Response In the context of emission tomography, several investigators have observed [13,14,36,37] that the ensemble mean of a likelihood based estimator is approximately equal to the value that one obtains by applying the estimator to noiseless data: E [ Y ) Y ( 4 = 7) Here Y ( E [Y ] Z yf(y; dy (8) denotes the mean of the measurement vector, ....

D. W. Wilson, B. M. W. Tsui, and H. H. Barrett, "Noise properties of the EM algorithm: II. Monte Carlo simulations," Phys. Med. Biol., vol. 39, pp. 847--872, 1994.

.... filtered backprojection [10] the ordered subsets expectation maximization algorithm [11] or weighted least squares conjugate gradient [12] Except in simple linear cases [13] it is generally difficult to analyze the performance of methods based on stopping rules, although Barrett et al. 14] [15] have analyzed the periteration behavior of the maximum likelihood expectation maximization algorithm for emission tomography. The approximations we derive are somewhat easier to use since they are independent of number of iterations (provided sufficient iterations are used to maximize the ....

.... Y ) 13) This approximation is simply the value produced by applying the estimator (1) to noise free data. This approach requires minimal computation, and works surprisingly well for penalized likelihood objectives. It has been used extensively by investigators in emission tomography [14] [15], 20] Apparently, the principal source of bias in penalizedlikelihood estimators is the regularizing penalty that one includes in Phi, so (13) allows one to examine the effects of the penalty separately from the effects of noise. However, the approximation (13) is certainly not always adequate, ....

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D. W. Wilson, B. M. W. Tsui, and H. H. Barrett, "Noise properties of the EM algorithm: II. Monte Carlo simulations," Phys. Med. Biol., vol. 39, pp. 847--872, 1994.

....such as higher resolution in high count regions etc. in a manner similar to methods for space varying regularization [8, 9] but in this paper we focus on the goal of providing uniform resolution. This paper is somewhat in the spirit of previous studies that used the local impulse response [10 14] or an effective local Gaussian resolution [15] to quantify the resolution properties of the unregularized maximum likelihood expectation maximization (ML EM) algorithm for emission tomography. However, there is an important difference in our approach: since the ML EM algorithm is rarely iterated ....

....response if we could avoid performing extensive numerical simulations. The remainder of this paper is devoted to approximations suitable for likelihood based estimators in tomography. C. Linearized Local Impulse Response In the context of emission tomography, several investigators have observed [13, 14, 37, 38] that the ensemble mean IEEE TRANSACTIONS ON IMAGE PROCESSING, EDICS 2.3, TO APPEAR. VERSION January 23, 1996 4 of a likelihood based estimator is approximately equal to the value that one obtains by applying the estimator to noiseless data: E [ Y ) Y ( 4 = 7) ....

D. W. Wilson, B. M. W. Tsui, and H. H. Barrett, "Noise properties of the EM algorithm: II. Monte Carlo simulations," Phys. Med. Biol., vol. 39, pp. 847--872, 1994.

....variances of MAP reconstruction algorithms. Since the EM algorithm for ML reconstruction [3] is rarely iterated to convergence, Barrett et al. [2] derived approximate formulae for the mean and covariance of the reconstructed image as a function of the iteration number. Monte Carlo validations [4] showed that these theoretical predictions matched Monte Carlo estimates for the earlier iterations at 1 This work was supported by the National Cancer Institute under Grant No. R01 CA59794. which the algorithm is usually terminated. In high count situations, these results were also accurate ....

....the CRC approximation. Three points of interest were selected as shown in Fig. 4(b) Since several investigators have observed that the ensemble mean of ML and MAP estimators are approximately equal to the reconstruction that is obtained by applying the estimator to the expectation of the data [2, 4, 5, 9, 37], we measured the CRC from reconstructions of two noiseless data sets: i) the original phantom sinogram, and (ii) the sinogram of the phantom after perturbation of a single voxel. Fig. 5 shows a comparison of the measured and theoretically predicted CRCs for each point of interest. The ....

D. W. Wilson, B. M. W. Tsui, and H. H. Barrett, "Noise properties of the EM algorithm: II. Monte Carlo simulations," Physics in Medicine and Biology, vol. 39, pp. 847--872, 1994.

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D. W. Wilson, B. M. W. Tsui, and H. H. Barrett, "Noise properties of the EM algorithm: II. Monte Carlo simulations," Physics in Medicine and Biology, vol. 39, pp. 847--872, 1994.

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