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...

We study linear smoothers and their use in building non-parametric regression models. In part Qfthis paper we examine certain aspects of linear smoothers for scatterplots; examples of these are the running mean and running line, kernel, and cubic spline smoothers. The eigenvalue and singular value decompositions of the corresponding smoother matrix are used to qualitatively describe a smoother, and several other topics such as the number of degrees of freedom of a smoother are discussed. In the second part of the paper we describe how Iinear-smoothers can be used to estimate the additive model, a powerful non-parametric regression model, using the "backfitting algorithm". We study the convergence of the backfitting algorithm and prove its convergence for a class of smoothers that includes cubic e:ttJlCl€~nt jJI:::Jll<l.li:6I;:U least squares. algorithm and ' dis.cuss ev'W()r(is: Nea-parametric, sean-parametric, regression, Gauss-Seidelalgorithm,

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e. for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shiftvariant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shiftvariant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.

Normal mixture models are being increasingly used to model the distributions of a wide variety of random phenomena and to cluster sets of continuous multivariate data. However, for a set of data containing a group or groups of observations with longer than normal tails or atypical observations, the use of normal components may unduly affect the fit of the mixture model. In this paper, we consider a more robust approach by modelling the data by a mixture of t distributions. The use of the ECM algorithm to fit this t mixture model is described and examples of its use are given in the context of clustering multivariate data in the presence of atypical observations in the form of background noise.

L1 regularized logistic regression is now a workhorse of machine learning: it is widely used for many classification problems, particularly ones with many features. L1 regularized logistic regression requires solving a convex optimization problem. However, standard algorithms for solving convex optimization problems do not scale well enough to handle the large datasets encountered in many practical settings. In this paper, we propose an efficient algorithm for L1 regularized logistic regression. Our algorithm iteratively approximates the objective function by a quadratic approximation at the current point, while maintaining the L1 constraint. In each iteration, it uses the efficient LARS (Least Angle Regression) algorithm to solve the resulting L1 constrained quadratic optimization problem. Our theoretical results show that our algorithm is guaranteed to converge to the global optimum. Our experiments show that our algorithm significantly outperforms standard algorithms for solving convex optimization problems. Moreover, our algorithm outperforms four previously published algorithms that were specifically designed to solve the L1 regularized logistic regression problem.

We present a new class of algorithms for edge-preserving restoration of piecewise-smooth images measured in nonGaussian noise under shift-variant blur. The algorithms are based on minimizing a regularized objective function, and are guaranteed to monotonically decrease the objective function. The algorithms are derived by using a combination of two previously unconnected concepts: A. De Pierro's convexity technique for optimization transfer, and P. Huber's iteration for M-estimation. Convergence to the unique global minimum is guaranteed for strictly convex objective functions. The convergence rate is very fast relative to conventional gradient-based iterations. The proposed algorithms are flexibly parallelizable, and easily accommodate nonnegativity constraints and arbitrary neighborhood structures. Implementation in Matlab is remarkably simple, requiring no cumbersome line searches or tolerance parameters. Keywords: Image restoration, non-Gaussian noise, deconvolution, Bayesian meth...

Partial Least Squares (PLS) is a highly efficient statistical regression technique that is well suited for the analysis of high-dimensional genomic data. In this paper we review the theory and applications of PLS both under methodological and biological points of view. Focusing on microarray expression data we provide a systematic comparison of the PLS approaches currently employed, and discuss problems as different as tumor classification, identification of relevant genes, survival analysis and modeling of gene networks. 2 1

Summary. In recent years, many methods have been developed for regression in highdimensional settings. We propose covariance-regularized regression, a family of methods that use a shrunken estimate of the inverse covariance matrix of the features in order to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing its log likelihood, under a multivariate normal model, subject to a constraint on its elements; this estimate is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso, and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyze gene expression data sets with multiple class and survival outcomes.

this paper, attention has been focused on methods that treat coe#cients at least as if they were independent. However, it is intuitively clear that if one coe#cient in the wavelet array is nonzero, then it is more likely #in some appropriate sense# that neighbouring coe#cients will be also. One way of incorporating this notion is by some form of block thresholding, where coe#cients are considered in neighbouring blocks; see for example Hall et al. #1998# and Cai & Silverman #1998#. An obvious question for future consideration is integrate the ideas of block thresholding and related methods within the range of models and methods considered in this paper.

Abstract not found