This paper presents a new class of algorithms for penalized-likelihood reconstruction of attenuation maps from lowcount transmission scans. We derive the algorithms by applying to the transmission log-likelihood a version of the convexity technique developed by De Pierro for emission tomography. The new class includes the single-coordinate ascent (SCA) algorithmand Lange's convex algorithm for transmission tomography as special cases. The new grouped-coordinate ascent (GCA) algorithms in the class overcome several limitations associated with previous algorithms. (1) Fewer exponentiations are required than in the transmission ML-EM algorithm or in the SCA algorithm. (2) The algorithms intrinsically accommodate nonnegativity constraints, unlike many gradientbased methods. (3) The algorithms are easily parallelizable, unlike the SCA algorithm and perhaps line-search algorithms. We show that the GCA algorithms converge faster than the SCA algorithm, even on conventional workstations. An ex...

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.

Traditional space-invariant regularization methods in tomographic image reconstruction using penalized-likelihood estimators produce images with nonuniform spatial resolution properties. The local point spread functions that quantify the smoothing properties of such estimators are space-variant, asymmetric, and object-dependent even for space-invariant imaging systems. We propose a new quadratic regularization scheme for tomographic imaging systems that yields increased spatial uniformity and is motivated by the least-squares tting of a parameterized local impulse response to a desired global response. We have developed computationally e cient methods for PET systems with shift-invariant geometric responses. We demonstrate the increased spatial uniformity of this new method versus conventional quadratic regularization schemes in simulated PET thorax scans.

We introduce a plane, which we call the delta-sigma plane, that is indexed by the norm of the estimator bias gradient and the variance of the estimator. The norm of the bias gradient is related to the maximum variation in the estimator bias function over a neighborhood of parameter space. Using a uniform Cramer-Rao (CR) bound on estimator variance a delta-sigma tradeoff curve is specied which denes an "unachievable region" of the delta-sigma plane for a specified statistical model. In order to place an estimator on this plane for comparison to the delta-sigma tradeoff curve, the estimator variance, bias gradient, and bias gradient norm must be evaluated. We present a simple and accurate method for experimentally determining the bias gradient norm based on applying a bootstrap estimator to a sample mean constructed from the gradient of the log-likelihood. We demonstrate the methods developed in this paper for linear Gaussian and non-linear Poisson inverse problems.

ASPIRE 3.0 is a collection of ANSI C language programs for performing tomographic image reconstruction and image restoration using statistical methods. This user's guide describes how to compile and use the software. This work was supported in part by DOE grant DE-FG02-87ER60561 and NIH grants CA-60711 and CA-54362. ASPIRE 3.0 January 22, 1999 2 Notice ASPIRE 3.0 is copyright 1990-1998 Jeff Fessler and The University of Michigan ASPIRE 3.0 is available only to particular individuals for academic collaboration. Do not distribute this software to anyone else. ffl This code is provided as is, with absolutely no warranty. ffl Neither Jeff Fessler nor The University of Michigan assume any liability for the use or misuse of this software. There are no guarantees of its correctness, nor its efficacy for diagnostic imaging. ffl The copyright and disclaimer headers must remain in the source code. ffl We will be glad to answer a limited set of simple, precise questions. We welcome your fe...

This paper analyzes and compares image reconstruction methods based on practical approximations to the exact loglikelihood of randoms-precorrected PET measurements. The methods apply to both emission and transmission tomography; however in this paper we focus on transmission tomography. The results of experimental PET transmission scans and variance approximations demonstrate that the "shifted Poisson" (SP) method avoids the systematic bias of the conventional data-weighted least squares (WLS) method, and leads to significantly lower variance than conventional statistical methods based on the log-likelihood of the ordinary Poisson (OP) model. We develop covariance approximations to analyze the propagation of noise from attenuation maps into emission images via the attenuation correction factors (ACFs). Empirical pixel and region variances from real transmission data agree closely with the analytical predictions. Both the approximations and the empirical results show that the performanc...

Traditional space-invariant regularization schemes in tomographic image reconstruction using penalized-likelihood estimators produce images with nonuniform resolution properties. The local point spread functions that quantify the local smoothing properties of such estimators are not only space-variant and asymmetric, but are also object-dependent even for space-invariant systems. We propose a new regularization scheme for increased spatial uniformity and demonstrate the resolution properties of this new method versus conventional regularization schemes through an investigation of local point spread functions.

Positron Emission Tomography (PET) is a diagnostic imaging tool that provides images of radioactive substances injected into the body to trace biological functions. The radioactive substance emits a positron which annihilates with an electron to produce two 511 keV photons traveling in approximately opposite directions to be coincidentally detected by two detectors. Many photons are absorbed or scattered, reducing the number of detected emission events. Attenuation correction is crucial for quantitatively accurate PET reconstructions. PET transmission scans are performed to estimate attenuation parameters which are in turn used to correct the emission scans for attenuation effects. The noise in estimating the attenuation parameters propagates to the emission images affecting their quality and quantitative correctness. Thus, attenuation image reconstruction is extremely important in PET. Conventional methods of attenuation correction are suboptimal and ignore the Poisson nature of the data. We propose to use penalized likelihood image reconstruction techniques for transmission scans. Current algorithms for transmission tomography have two important problems: 1) they are not guaranteed to converge, 2) if they converge, the convergence is slow. We develop new fast and monotonic optimization algorithms for penalized likelihood image reconstruction based on a novel paraboloidal surrogates principle. We present results showing the speed of the new optimization algorithms as compared to previous ones. We apply the algorithms to PET data obtained from an anthropomorphic thorax phantom and real patient data. A transmission scan per...

2 distance. PCG - None PCG - Diagonal PCG - Fourier PCG - M9 0 10 20 30 40 50 60 70 80 90 100 10 -5 10 -4 10 -3 10 -2 10 -1 10 Iteration Normalized Max. Distance NPWLS Preconditioned CG Algorithms Initialized with FBP Image D.S. Thorax Coordinate Ascent 0.6 Coordinate Ascent 1.0 Coordinate Ascent 1.4 PCG - M9 0 10 20 30 40 50 60 70 80 90 100 10 -5 10 -4 10 -3 10 -2 10 -1 10 Iteration NPWLS Preconditioned CG Algorithms Initialized with FBP Image D.S. Thorax Fig. 47. As above for normalized l 1 distance. 21 PCG - None PCG - Diagonal PCG - Fourier PCG - Combined 0 5 10 15 20 25 30 35 40 10 -7 10 -6 10<F70.09

Previous methods for optimizing the scan times for PET transmission and emission scans under a total scan time constraint were based on linear non-statistical methods and used noise equivalent counts (NEC) criteria. The scan times determined by NEC analysis may be suboptimal when nonlinear statistical image reconstruction methods are used. For statistical image reconstruction, the predicted variance in selected regions of interest is an appropriate alternative to NEC analysis. We propose a new method for optimizing the relative scan times (fractions) based on analytical approximations to the covariance of images reconstructed by both conventional and penalized-likelihood methods. We perform simulations to compare predicted standard deviations with empirical ones. Results show that for statistical transmission image reconstruction, the optimal fraction of the scan time devoted to transmission scanning is shorter than for conventional transmission smoothing. I.