A fuzzy optimization method in stereotactic treatment planning is presented. Instead of strict twovalued (1 or 0; YES or NO) assignments and decisions, the present algorithm affords a more flexible way to build treatment plans by using a continuous range of true values for constraints. Furthermore, multiple objectives can be considered simultaneously in the proposed fuzzy optimization. Compared with ordinary methods, the present method is desirable not only for its capability to reduce the risk in giving inconsistent constraints but also for its capability to obtain a better treatment plan.

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Emission computed tomography (ECT) has as its major emphasis the quantitative determination of the moment to moment changes in the chemistry and flow physiology of injected or inhaled compounds labeled with radioactive atoms in a human body. The major difference lies in the fact that ECT seeks to describe the location and intensity of sources of emitted photons in an attenuating medium whereas transmission X-ray computed tomography (TCT) seeks to determine the distribution of the attenuating medium. A second important difference between ECT and TCT is that of available statistics. ECT statistics are low because each photon without control in emitting direction must be detected and analyzed, not as in TCT. The following sections review the historical development of image reconstruction methods for imaging in nuclear medicine, relevant intrinsic concepts for image reconstruction on ECT, and current status of volume imaging as well as a unique approach on iterative techniques for ECT.

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The EM (Expectation-Maximization) algorithm is a general-purpose stable procedure for maximum likelihood estimation in a wide variety of situations described as incomplete-data problem. Incomplete- data problems where the EM algorithm has been succesfully applied include not only evidently incomplete- data situations, for example, there are missing data, grouped observations, but also a whole variety of situations where the incompleteness of the data is not natural or evident.
In this article, at first, I summarize maximum likelihood estimation and formulation of the EM algorithm. Subsequently, I briefly mention the properties of the EM algorithm, and two applications where the typical probablistic models are assumed. Lastly, I introduce some problems, which arise from applying the EM algorithm to the complex situations, and the examples of the solutions against them.

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Maximum likelihood criterion accounting for statistical behavior of photon is very useful for image reconstruction from projection at low count rate. EM (expectation maximization) algorithm is a powerful tool which provides such a maximum likelihood estimator successfully. In this article, the fundamental algorithms for emission and transmission image reconstruction originally proposed by Lange and Carson are reviewed. Here the likelihood incorporating the Poisson nature of photon counting is formulated and the procedure of the EM algorithm for maximizing the likelihood is shown.

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We investigated combined scatter and attenuation correction with maximum likelihood expectation maximization (ML-EM) algorithm for projections obtained with simultaneous transmission (^99nrTc) and emission (²⁰¹T1) data acquisition in ²⁰¹T1 SPECT. An energy window 15% in width centered at 140 keV was used for 999'C transmission photons. Measured linear attenuation coefficient was included in probability factor, cii, of ML-EM algorithm. Emission data acquisition was performed with three energy windows: a 30% photopeak energy window set symmetrically over 73-keV peak of ²⁰¹T1 and a 7.3% energy window set over 54.5-keV peak and a 4.4% energy window set over 91.0-keV peak. Latter two scatter windows were placed one full width half maximum (FWHM) below and above the photopeak centerline. The scatter fraction in the primary peak was estimated using trapezoid approximation and scatter was compensated intrinsically within ML-EM algorithm. Ordered subsets expectation maximization (OSEM) algorithm was used to accelerate ML-EM algorithm.²⁰¹T1 myocardial perfusion images and quantitative accuracy for activity concentration in uniform attenuating medium was reported.

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This paper describes the property of ordered subsets expectation maximization (OS-EM) method. This method divides projections into some subsets and acceralates the convergence of the reconstruction. In this paper, the OS-EM method was analized in terms of three parameters: (1) the number of projections which composes a subset, (2) the projection data which composes a subset, and (3) the subset access order among subsets.

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This paper intends to explain iterative image reconstruction methods for emission tomography based on least-squares objective functions. The methods estimate object distributions by minimizing the objective functions in iterative manners under constraints or no constraint. As for the constraints, non-negativity of pixel values is considered, as in most cases. For convenience the methods are divided into three groups. The first is the so called nonlinear optimization methods performing line searches, such as the conjugate gradient method for unconstrained optimization, and the gradient and augmented Lagrangian methods for constrained optimization. The second is the methods using the Gauss-Siedel and successive over-relaxation methods. The third is the method of applying Karush-Kuhn-Tucker conditions to image reconstruction. Algorithms for the three groups are described.

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