## 1 Introduction
Possible edge preserving penalties include total variation (TV) (e.g., [19]) Huber [20] and hyperbolic functions (e.g., [21]). Note that \(\rho(\cdot)\) is convex for all these options. The optimal choice of the penalty function \(J\) and the smoothing parameter \(h\) are unsolved problems in image processing and will not be further elaborated in this paper. We emphasize that smoothing by MPL indeed produce visually improved reconstructions over the tradition filtered-backprojection method particularly in dose-limited tomography such as low dose X-ray CT. The edge preserving penalties are extremely useful, such as TV and Huber penalties; see [22; 23; 24]. However, the MPL reconstructions can have unnatural noise textures very different from the familiar filtered-backprojection method. Its impact on diagnostic tasks is still unknown and this is an active research area; see [25] for examples and discussions. We adopt the following notations throughout this paper. Let \(\mathbf{x}^{(k)}\) be the estimate of \(\mathbf{x}\) obtained at iteration \(k\) of an algorithm. The notation \(\
abla b(\cdot)\) indicates the derivative of function \(b\) with respect to the variable in the brackets. For example, \(\
abla b(A_{i}\mathbf{x})\) represents the derivative of \(b\) with respect to \(A_{i}\mathbf{x}\) and \(\
abla b(\mathbf{x};\mathbf{x}^{(k)})\) the derivative of \(b\) with respect to \(\mathbf{x}\). We use \(\
abla_{j}b(\mathbf{x})\) to denote the derivative of \(b\) with respect to \(x_{j}\), the \(j\)-th element of vector \(\mathbf{x}\). We also let \(\
abla b(\mathbf{x}^{(k)})\) and \(\
abla_{j}b(\mathbf{x}^{(k)})\) represent, respectively, \(\
abla b(\mathbf{x})\) and \(\
abla_{j}b(\mathbf{x})\) evaluated at \(\mathbf{x}=\mathbf{x}^{(k)}\). Non-negatively constrained MPL image reconstruction algorithms can be classified into simultaneous and block-iterative (a.k.a. ordered subset (OS)) algorithms. For simultaneous algorithms, all elements in \(\mathbf{y}\) are used to update \(\mathbf{x}\) in each iteration, and for block-iterative algorithms, distinct portions of \(\mathbf{y}\) are used in turn to update \(\mathbf{x}\). We discuss in this paper some simultaneous algorithms for non-negatively constrained MPL reconstructions, and the block-iterative algorithms are not included in our discussions. The rest of this paper is arranged as follows. The expectation-maximization algorithm for emission tomography is discussed in Section 2. Section 3 explains the alternating minimization algorithm designed specifically for transmission tomography. Section 4 contains explanations on the optimization transfer algorithms and their applications to tomographic reconstructions. The multiplicative iterative (MI) algorithms for tomographic imaging are provided in Section 5 and the Fisher scoring based Jacobi or Gauss-Seidel over-relaxation algorithms are presented in Section 6. Section 7 explains another Gauss-Seidel method named the iterative coordinate ascent algorithm. Finally, Section 8 includes discussions and remarks about this paper. In this paper we focus on explaining and summarizing different non-negatively constrained tomographic imaging algorithms. Numerical comparisons of some of these algorithms are available in [26], and therefore will not be given in this paper.