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

Nonuniform Fast Fourier Transforms Using Min-Max Interpolation

The fast Fourier transform (FFT) is used widely in signal processing for efficient computation of the FT of finite-length signals over a set of uniformly spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e., a nonuniform FT. Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85840/1/Fessler70.pd

Fast, Iterative, Field-Corrected Image Reconstruction for MRI

Magnetic field inhomogeneities cause distortions in the reconstructed images for non-cartesian k-space MRI (using spirals, for example). Several noniterative methods are currently used to compensate for the off-resonance during the reconstruction, but these methods rely on the assumption of a smoothly varying field map. Recently, iterative methods have been proposed that do not rely on this assumption and have the potential to estimate undistorted field maps, but suffer from prohibitively long computation times. In this abstract we present a min-max derived, time-segmented approximation to the signal equation for MRI that, when combined with the nonuniform fast Fourier transform, provides a fast, accurate field-corrected image reconstruction.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86011/1/Fessler175.pd

Maximum-Likelihood Transmission Image Reconstruction for Overlapping Transmission Beams

In many transmission imaging geometries, the transmitted "beams" of photons overlap on the detector, such that a detector element may record photons that originated in different sources or source locations and thus traversed different paths through the object. Examples include systems based on scanning line sources or on multiple parallel rod sources. The overlap of these beams has been disregarded by both conventional analytical reconstruction methods as well as by previous statistical reconstruction methods. The authors propose a new algorithm for statistical image reconstruction of attenuation maps that explicitly accounts for overlapping beams in transmission scans. The algorithm is guaranteed to monotonically increase the objective function at each iteration. The availability of this algorithm enables the possibility of deliberately increasing the beam overlap so as to increase count rates. Simulated single photon emission tomography transmission scans based on a multiple line source array demonstrate that the proposed method yields improved resolution/noise tradeoffs relative to "conventional" reconstruction algorithms, both statistical and nonstatistical.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85818/1/Fessler78.pd

Conjugate Phase MRI Reconstruction With Spatially Variant Sample Density Correction

A new image reconstruction method to correct for the effects of magnetic field inhomogeneity in non-Cartesian sampled magnetic resonance imaging (MRI) is proposed. The conjugate phase reconstruction method, which corrects for phase accumulation due to applied gradients and magnetic field inhomogeneity, has been commonly used for this case. This can lead to incomplete correction, in part, due to the presence of gradients in the field inhomogeneity function. Based on local distortions to the k-space trajectory from these gradients, a spatially variant sample density compensation function is introduced as part of the conjugate phase reconstruction. This method was applied to both simulated and experimental spiral imaging data and shown to produce more accurate image reconstructions. Two approaches for fast implementation that allow the use of fast Fourier transforms are also described. The proposed method is shown to produce fast and accurate image reconstructions for spiral sampled MRI.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85978/1/Fessler52.pd

Maximum Likelihood Transmission Image Reconstruction for Over lapping Transmission Beams

In many transmission imaging geometries, the transmitted “beams” of photons overlap on the detector, such that a detector element may record photons that originated in different sources or source locations and thus traversed different paths through the object, Examples include systems based on scanning line sources or on multiple parallel rod sources. The overlap of these beams has been disregarded by both conventional analytical reconstruction methods as well as by previous statistical reconstruction methods. We propose a new algorithm for statistical image reconstruction of attenuation maps that explicitly accounts for overlapping beams in transmission scans. The algorithm is guaranteed to monotonically increase the objective function at each iteration. The availability of this algorithm enables the possibility of deliberately increasing the beam overlap so as to increase count rates. Simulated SPECT transmission scans based on a multiple line source array demonstrate that the proposed method yields improved resolution/noise tradeoffs relative to “conventional” reconstruction algorithms, both statistical and nonstatistical.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85817/1/Fessler156.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

Fast Parallelizable Algorithms for Transmission Image Reconstruction

Presents a new class of algorithm for penalized-likelihood reconstruction of attenuation maps from low-count transmission scans. The authors derive the algorithms by applying to the transmission log-likelihood a variation of the convexity technique developed by De Pierro for the emission case. The new algorithms overcome several limitations associated with previous algorithms. (1) Fewer exponentiations are required than in the transmission EM algorithm or in coordinate-ascent algorithms. (2) The algorithms intrinsically accommodate nonnegativity constraints, unlike many gradient-based methods. (3) The algorithms are easily parallelizable, unlike coordinate-ascent algorithms and perhaps line-search algorithms. The authors show that the algorithms converge faster than several alternatives, even on conventional workstations. They give examples from low-count PET transmission scans and from truncated fan-beam SPECT transmission scans.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86006/1/Fessler136.pd

Spectral Analysis Using Regularized Non-Negative Least-Squares Estimation

The implementation of spectral analysis techniques involves solving a highly underdetermined linear system equation and is prone to the effect of measurement noise. The authors propose to use a regularized non-negative least-square estimator to stabilize the implementation of the technique. They introduce a penalty term in their formulation of the function to discourage disparities in tracer kinetics between neighboring pixels and use an iterative method to impose positivity constraints. The authors show results from analysis of FDG thorax images of patients suspected to have cancers and summarize their findings.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85892/1/Fessler137.pd