H. J. Trussell. Convergence criteria for iterative restoration methods. IEEE Trans. Acoust., Speech, Signal Processing, 31:129-136, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....image of a single point source. In this case, the structure of A depends on the boundary condition: Periodic boundary conditions imply that A is a block circulant matrix with circulant blocks (BCCB) 2] Zero boundary conditions imply that A is a block Toeplitz matrix with Toeplitz blocks (BTTB) [14, 20, 30]. Reflexive boundary conditions imply that A is a sum of a BTTB matrix and a block Hankel matrix with Hankel blocks (BHHB) 27] In the first case, matrix vector multiplication is done using the two dimensional discrete Fourier transform. All BCCB matrices can be written as: C = where b r is ....

H. J. Trussell. Convergence criteria for iterative restoration methods. IEEE Trans. Acoust., Speech, Signal Processing, 31:129-136, 1983.

....obtained with our approach. In the next set of experiments we illustrate that preconditioning can be effective in reducing the computational cost of the approach used in this paper. The test problem we use is a simulation of an application in which a degraded gamma ray spectrum is to be restored [20, 21]. The true solution fexact (i.e. the original spectrum) is shown on Figure 5.l a, and the noise free right hand side g = Kfexact (i.e. the degraded spectrum) is shown on Figure 5.1 b. The matrix K is a symmetric Toeplitz matrix with entries in the first column given 12 Newton method Secant ....

H. Joel Trussell. Convergence criteria for iterative restoration methods. IEEE Transactions on Acoustic, Speech, and Signal Processing, Vol. ASSP31, NO. 1, 1983.

....Many methods exist in the literature [44] 47] for constrained optimization. In this section, few of the methods used in the applications of iterative image restoration and blind equalization are discussed. Projection Method The first method, commonly referred to as the projection method [39] [55] involves projecting the parameters onto the convex constraint set after 47 each parameter update of the descent routine. This is one of the simplest and computationally efficient methods of incorporating deterministic constraints. The application of constraints in this manner is not consistent ....

....the signal restoration process. Deterministic regularization techniques are usually implementedby iterative procedures to alleviate the 53 computational problems. Regularization can also take the form of terminating an iterative restoration procedure before it converges to the inverse solution [55], 60] 61] 62] Nearly all concepts used in regularization are based on incorporating knowledge about either the true solution or the noise into the algorithm. In this sense the procedures used for truncating the number of iterations are called regularization as well [40] 4.2.4 Regularizing ....

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H. J. Trussell, "Convergence criteria for iterative restoration methods," IEEE Trans. Acoust. , Speech, Signal Process. ,vol. 31(1), pp. 201-212, Feb. 1983.

....4 4 4 5 455 3919 10 11 29 48 4 4 6 446 3693 7 11 29 48 4 4 general the circulant preconditioner can be cheaper to construct, cheaper to apply, and requires less storage than the Cholesky or LDL T factorizations. Example 3. The Toeplitz matrix here arises in an application in signal restoration [99]. Specifically, an n Theta (n Gamma 2 ) convolution matrix K is constructed with entries in the first column given by k i;1 = 8 : 1 p 2ff 2 exp h Gamma( Gammai 1) 2 2ff 2 i i = 1; 2; 2 1 0 otherwise ; and entries in the first row given by k 1;j = 8 : ....

H. J. Trussell. Convergence criteria for iterative restoration methods. IEEE Trans. Acoust., Speech, Signal Processing, 31:129--136, 1983.

....to using no preconditioner. Moreover, in general the circulant preconditioner can be cheaper to construct, cheaper to apply, and require less storage than the Cholesky or LDL T factorizations. Example 3. The Toeplitz matrix used in this example comes from an application in signal restoration [30]. Specifically, an n Theta (n Gamma 2 ) convolution matrix T is constructed to have entries in the first column given by t i;1 = 1 p 2ff 2 exp h Gamma( Gammai 1) 2 2ff 2 i i = 1; 2; 2 1 0 otherwise ; and entries in the first row given by t 1;j = ae t 1;1 if j = 1 0 ....

H. J. Trussell. Convergence criteria for iterative restoration methods. IEEE Trans. Acoust., Speech, Signal Processing, 31:129--136, 1983.

....image estimate to follow a previously defined probability distribution. We discuss four constraints here that are closely tied with the implementation of ML EM. First, stopping the iterations prior to convergence can be viewed as a type of regularization. Veklerov and Llacer [30, 31] and Trussel [32] formalize this idea by proposing stopping the iterations before the estimate follows the noise too closely, resulting in a solution not having the variance required to be a true Poisson process. They term the set of solutions that could generate the observed Poisson distributions the set of ....

H. J. Trussell, "Convergence criteria for iterative restoration methods," IEEE Trans. ASSP, vol. 31, no. 1, pp. 129--136, 1983.

....the error due to blurring decreases as the error due to noise amplification increases. At some point in the algorithm, this total error reaches a minimum and the procedure should be stopped before convergence. This phenomenon is explained and experimentally investigated in [32] Existing research [33], 34] 35] 36] has demonstrated the effectiveness of premature algorithm termination in combating noise amplification. A drawback of the proposed NAS RIF algorithm is that the convergence point is not necessarily the best estimate of the original image in the presence of noise. The iterative ....

H. J. Trussell, "Convergence criteria for iterative restoration methods," IEEE Trans. Acoust., Speech, Signal Process., vol. 31(1), pp. 201--212, Feb. 1983.

....to using no preconditioner. Moreover, in general the circulant preconditioner can be cheaper to construct, cheaper to apply, and require less storage than the Cholesky or LDL T factorizations. Example 3. The Toeplitz matrix used in this example comes from an application in signal restoration [26]. Specifically, an n Theta (n Gamma 2 ) convolution matrix T is constructed to have entries in the first column given by t i;1 = 1 p 2 ff 2 exp h Gamma( Gammai 1) 2 2ff 2 i i = 1; 2; 2 1 0 otherwise ; and entries in the first row given by t 1;j = t 1;1 if j = 1 0 ....

H. J. Trussell. Convergence criteria for iterative restoration methods. IEEE Trans. Acoust., Speech, Signal Processing, 31:129--136, 1983.

....Unfortunately, the point at which this minimum occurs is not generally known a priori. Therefore, we would like to have some method of determining the iteration where this minimum total error occurs. Previous work in this area has relied on having an estimate of the noise variance available [43, 44]. Since the number of iterations plays a role equivalent to a regularization parameter [31, 41] cross validation can be used to evaluate the restoration process as a function of number of iterations without requiring an estimate of the noise variance. In light of this, researchers have ....

....was computed after each iteration, and the procedure was stopped when the estimate began to increase. For comparative purposes, the average residual error and the MSE were also computed at each iteration. The average residual error was compared to noise variance, as recommended by Trussell [43]. When the residual error fell below the noise variance, the iteration was stopped. Furthermore, the iteration at which the MSE reached a minimum was noted. Table 3 compares the performance of the stopping rules. Restored images for the 30 dB case are shown in Figure 6. This experiment confirms ....

H. J. Trussell, "Convergence criteria for iterative restoration methods," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-31, pp. 129-- 136, February 1983.

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