Abstract-A new method of reconstruction from SPECT data is proposed, which builds on the EM approach to maximum likelihood reconstruction from emission tomography data, but aims instead at maximum posterior probability estimation, that takes account of prior belief about “smoothness ” in the isotope concentration. A novel modification to the EM algorithm yields a practical method. The method is illustrated by an application to data from brain scans. I.

The Support Vector (SV) method was recently proposed for estimating regressions, constructing multidimensional splines, and solving linear operator equations [Vapnik, 1995]. In this presentation we report results of applying the SV method to these problems. 1 Introduction The Support Vector method is a universal tool for solving multidimensional function estimation problems. Initially it was designed to solve pattern recognition problems, where in order to find a decision rule with good generalization ability one selects some (small) subset of the training data, called the Support Vectors (SVs). Optimal separation of the SVs is equivalent to optimal separation the entire data. This led to a new method of representing decision functions where the decision functions are a linear expansion on a basis whose elements are nonlinear functions parameterized by the SVs (we need one SV for each element of the basis). This type of function representation is especially useful for high dimensional...

We define ordered subset processing for standard algorithms (such as Expectation Maximization, EM) for image restoration from projections. Ordered subsets methods group projection data into an ordered sequence of subsets (or blocks). An iteration of ordered subsets EM is defined as a single pass through all the subsets, in each subset using the current estimate to initialise application of EM with that data subset. This approach is similar in concept to block-Kaczmarz methods introduced by Eggermont et al [1] for iterative reconstruction. Simultaneous iterative reconstruction (SIRT) and multiplicative algebraic reconstruction (MART) techniques are well known special cases. Ordered subsets EM (OS-EM) provides a restoration imposing a natural positivity condition and with close links to the EM algorithm. OS-EM is applicable in both single photon (SPECT) and positron emission tomography (PET). In simulation studies in SPECT the OS-EM algorithm provides an order-ofmagnitude acceleration ...

This paper presents an image reconstruction method for positron-emission tomography (PET) based on a penalized, weighted least-squares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a least-squares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. We propose a simple data-based method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive over-relaxation (+SOR) algorithm converges rapidly to the global minimum of the PWLS objective. Quantitative simulation results demonstrate that the bias/variance tradeoff of the PWLS+SOR method is comparable to the maximum-likelihood expectation-maximization (ML-EM) method (but with fewer iterations), and is improved relative to the conventional filtered backprojection (FBP) method. Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminat...

The nonparametric multiscale platelet algorithms presented in this paper, unlike traditional wavelet-based methods, are both well suited to photon-limited medical imaging applications involving Poisson data and capable of better approximating edge contours. This paper introduces platelets, localized functions at various scales, locations, and orientations that produce piecewise linear image approximations, and a new multiscale image decomposition based on these functions. Platelets are well suited for approximating images consisting of smooth regions separated by smooth boundaries. For smoothness measured in certain H older classes, it is shown that the error of m-term platelet approximations can decay significantly faster than that of m-term approximations in terms of sinusoids, wavelets, or wedgelets. This suggests that platelets may outperform existing techniques for image denoising and reconstruction. Fast, platelet-based, maximum penalized likelihood methods for photon-limited image denoising, deblurring and tomographic reconstruction problems are developed. Because platelet decompositions of Poisson distributed images are tractable and computationally efficient, existing image reconstruction methods based on expectation-maximization type algorithms can be easily enhanced with platelet techniques. Experimental results suggest that platelet-based methods can outperform standard reconstruction methods currently in use in confocal microscopy, image restoration, and emission tomography.

The EM algorithm and its extensions are popular tools for modal estimation but are often criticised for their slow convergence. We propose a new method that can often make EM much faster. The intuitive idea is to use a 'covariance adjustment ' to correct the analysis of the M step, capitalising on extra information captured in the imputed complete data. The way we accomplish this is by parameter expansion; we expand the complete-data model while preserving the observed-data model and use the expanded complete-data model to generate EM. This parameter-expanded EM, PX-EM, algorithm shares the simplicity and stability of ordinary EM, but has a faster rate of convergence since its M step performs a more efficient analysis. The PX-EM algorithm is illustrated for the multivariate t distribution, a random effects model, factor analysis, probit regression and a Poisson imaging model.

This paper describes a statistical modeling and analysis method for linear inverse problems involving Poisson data based on a novel multiscale framework. The framework itself is founded upon a multiscale analysis associated with recursive partitioning of the underlying intensity, a corresponding multiscale factorization of the likelihood (induced by this analysis), and a choice of prior probability distribution made to match this factorization by modeling the \splits" in the underlying partition. The class of priors used here has the interesting feature that the \non-informative" member yields the traditional maximum likelihood solution; other choices are made to reect prior belief as to the smoothness of the unknown intensity. Adopting the expectation-maximization (EM) algorithm for use in computing the MAP estimate corresponding to our model, we nd that our model permits remarkably simple, closed-form expressions for the EM update equations. The behavior of our EM algorit...

In this dissertation we explore the use of wavelets in certain linear inverse problems with discrete, noisy data. We observe discrete samples of a process y(u) = (Kf)(u)+ z(u), where K is a linear operator, z is a noise process, and f is a function we wish to recover from the data. In the problems that we consider, the inverse of K, K \Gamma1 , either does not exist or is poorly behaved. Such problems are termed ill-posed i.e., ones in which small changes in the data may lead to large changes in the recovered version of f . Our methods are most effective for problems where the operator K is homogeneous with respect to dilations, such as integration, fractional integration, convolution, and the Radon transform. The theoretical framework in which we work is that of Donoho's (1992) WaveletVaguelette Decomposition (WVD). The WVD uses wavelets and vaguelettes (almost wavelets) to decompose the operator K. Although this formally resembles the Singular Value Decomposition (SVD), the use o...

A Bayesian method is presented for simultaneously segmenting and reconstructing emission computed tomography (ECT) images and for incorporating high-resolution, anatomical information into those reconstructions. The anatomical information is often available from other imaging modalities such as computed tomography (CT) or magnetic resonance imaging (MRI). The Bayesian procedure models the ECT radiopharmaceutical distribution as consisting of regions, such that radiopharmaceutical activity is similar throughout each region, and it estimates the number of regions, the mean activity of each region, and the region classification and mean activity of each voxel. Anatomical information is incorporated by assigning higher prior probabilities to ECT segmentations that more nearly resemble an anatomical segmentation. This approach is effective because anatomical tissue type often strongly influences radiopharmaceutical uptake. The Bayesian procedure is evaluated using physically acquired single...

A variety of sophisticated radiotracer models are available for the quantitative interpretation of dynamic positron emission tomography (PET) data. Parameters in these models are used to define quantities, such as metabolic rate, blood volume and flow, etc., characterizing the functional physiological and/or biochemical status of tissue, in vivo. We consider two methodologies for fitting radiotracer models on a pixel-wise basis to PET data. The first method does parameter optimization for each pixel considered as a separate region of interest. The second method also does pixel-wise analysis but incorporates an additive mixture representation to account for heterogeneity effects induced by instrumental and biological blurring. Several numerical and statistical techniques including cluster analysis, constrained non-linear optimization, sub-sampling, and spatial filtering are used to implement the methods. A computer simulation experiment, modeling a standard [F-18] deoxyglucose imaging protocol using the UW-PET scanner, is conducted to evaluate the statistical performance of the parametric images obtained by the two methods. The results obtained by mixture analysis are found to have substantially improved mean square error performance characteristics. The total compute time for mixture analysis is on the order of 0.7 seconds per pixel on a 16 MIPS workstation. This results in a total compute time of about 1 hour for a typical FDG brain study.