Penalized Weighted Least-Squares Image Reconstruction for Positron Emission Tomography

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, the author argues statistically that a least-squares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. The author proposes 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 eliminated by the PWLS+SOR method, and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over FBP.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85851/1/Fessler105.pd

Two-stage estimation in inverse problems using a combined wavelet thresholding and penalized maximum likelihood

Inverse problems occur in a wide range of practical scientific investigations where the variables of interest are only observed indirectly, such as magnetic and seismic imaging in geophysics, electrical tomography in industrial process monitoring, or PET scanning in medicine. Linear inverse problems can be thought of as highly multivariate regression problems with strong multicollinearity where the aim is to interpret regression parameters-prediction is not of interest. Estimation, to give a fitted model, is known as an inverse problem which can be ill-posed and ill-conditioned, making estimation using least-squares or maximum likelihood unstable or even impossible. Instead, one approach is to introduce additional constraints through a penalty term and a penalized least-squares or penalized maximum likelihood approach taken. The major cause of numerical problems in the estimation is noise in the data and hence using a pre-processing which reduces noise may be helpful. Wavelet thresholding has proven to be highly efficient at separating useful information from noise but there has been very little work considering the use of wavelet methods for inverse problems. Hence it is of great interest to investigate the usefulness of this as an additional step in estimation for inverse problems. In particular a two stage process is proposed combining inversion and wavelet thresholding. The thresholding will be considered as either a pre-inversion or post-inversion filter and the results compared. A simulation investigation is described and reported which compares these two alternative, and also which uses a minimum mean-squared error approach to choose the penalty parameter, in the inversion, and the threshold, in the wavelet thresholding, either sequentially or jointly. The results demonstrate that a combined approach is worthwhile and that for the piecewise constant test function considered, it is better to post-process after the inversion step than it is to use the more intuitive wavelet thresholding pre-processing step for noise reduction before inversion. This new approach hence has the potential to enhance the estimation results in a wide range of applied inverse problems

Direct estimation of kinetic parametric images for dynamic PET.

Dynamic positron emission tomography (PET) can monitor spatiotemporal distribution of radiotracer in vivo. The spatiotemporal information can be used to estimate parametric images of radiotracer kinetics that are of physiological and biochemical interests. Direct estimation of parametric images from raw projection data allows accurate noise modeling and has been shown to offer better image quality than conventional indirect methods, which reconstruct a sequence of PET images first and then perform tracer kinetic modeling pixel-by-pixel. Direct reconstruction of parametric images has gained increasing interests with the advances in computing hardware. Many direct reconstruction algorithms have been developed for different kinetic models. In this paper we review the recent progress in the development of direct reconstruction algorithms for parametric image estimation. Algorithms for linear and nonlinear kinetic models are described and their properties are discussed

Hybrid Poissoflolynomial Objective Functions for Tomographic Image Reconstruction from Transmission Scans

This paper describes rapidly converging algorithms for computing attenuation maps from Poisson transmission measurements using penalized-likelihood objective functions. We demonstrate that an under-relaxed cyclic coordinate-ascent algorithm converges faster than the convex algorithm of Lange (see ibid., vol.4, no.10, p.1430-1438, 1995), which in turn converges faster than the expectation-maximization (EM) algorithm for transmission tomography. To further reduce computation, one could replace the log-likelihood objective with a quadratic approximation. However, we show with simulations and analysis that the quadratic objective function leads to biased estimates for low-count measurements. Therefore we introduce hybrid Poisson/polynomial objective functions that use the exact Poisson log-likelihood for detector measurements with low counts, but use computationally efficient quadratic or cubic approximations for the high-count detector measurements. We demonstrate that the hybrid objective functions reduce computation time without increasing estimation bias.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86023/1/Fessler100.pd

Regularization for Uniform Spatial Resolution Properties in Penalized-Likelihood Image Reconstruction

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. The authors propose a new quadratic regularization scheme for tomographic imaging systems that yields increased spatial uniformity and is motivated by the least-squares fitting of a parameterized local impulse response to a desired global response. The authors have developed computationally efficient methods for PET systems with shift-invariant geometric responses. They demonstrate the increased spatial uniformity of this new method versus conventional quadratic regularization schemes in simulated PET thorax scans.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85867/1/Fessler79.pd

Mean and Variance of Implicitly Defined Biased Estimators (Such as Penalized Maximum Likelihood) : Applications to Tomography

Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear least squares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore, investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators. This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters. We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson statistics. We also describe a “plug-in” approximation that provides a remarkably accurate estimate of variability even from a single noisy Poisson sinogram measurement. The approximations should be useful in a wide range of estimation problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85819/1/Fessler99.pd

Grouped-Coordinate Ascent Algorithms for Penalized-Likelihood Transmission Image Reconstruction

Presents a new class of algorithms for penalized-likelihood reconstruction of attenuation maps from low-count 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) algorithm and 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 maximum likelihood-expectation maximization (ML-EM) algorithm or in the SCA algorithm. (2) The algorithms intrinsically accommodate nonnegativity constraints, unlike many gradient-based 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 example from a low-count positron emission tomography (PET) transmission scan illustrates the method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86021/1/Fessler93.pd

Conjugate-Gradient Preconditioning Methods for Shift-Variant PET Image Reconstruction

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 shift-variant, 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 shift-variant 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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85979/1/Fessler85.pd

Fully 3D PET Image Reconstruction Using A Fourier Preconditioned Conjugate-Gradient Algorithm

Since the data sixes in fully 3D PET imaging are very large, iterative image reconstruction algorithms must converge in very few iterations to be useful. One can improve the convergence rate of the conjugate-gradient (CG) algorithm by incorporating preconditioning operators that approximate the inverse of the Hessian of the objective function. If the 3D cylindrical PET geometry were not truncated at the ends, then the Hessian of the penalized least-squares objective function would be approximately shift-invariant, i.e. G'G would be nearly block-circulant, where G is the system matrix. The authors propose a Fourier preconditioner based on this shift-invariant approximation to the Hessian. Results show that this preconditioner significantly accelerates the convergence of the CG algorithm with only a small increase in computation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86015/1/Fessler139.pd

Quantitative Statistical Methods for Image Quality Assessment

Quantitative measures of image quality and reliability are critical for both qualitative interpretation and quantitative analysis of medical images. While, in theory, it is possible to analyze reconstructed images by means of Monte Carlo simulations using a large number of noise realizations, the associated computational burden makes this approach impractical. Additionally, this approach is less meaningful in clinical scenarios, where multiple noise realizations are generally unavailable. The practical alternative is to compute closed-form analytical expressions for image quality measures. The objective of this paper is to review statistical analysis techniques that enable us to compute two key metrics: resolution (determined from the local impulse response) and covariance. The underlying methods include fixed-point approaches, which compute these metrics at a fixed point (the unique and stable solution) independent of the iterative algorithm employed, and iteration-based approaches, which yield results that are dependent on the algorithm, initialization, and number of iterations. We also explore extensions of some of these methods to a range of special contexts, including dynamic and motion-compensated image reconstruction. While most of the discussed techniques were developed for emission tomography, the general methods are extensible to other imaging modalities as well. In addition to enabling image characterization, these analysis techniques allow us to control and enhance imaging system performance. We review practical applications where performance improvement is achieved by applying these ideas to the contexts of both hardware (optimizing scanner design) and image reconstruction (designing regularization functions that produce uniform resolution or maximize task-specific figures of merit)