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

Monotonic Algorithms for Transmission Tomography

Presents a framework for designing fast and monotonic algorithms for transmission tomography penalized-likelihood image reconstruction. The new algorithms are based on paraboloidal surrogate functions for the log likelihood, Due to the form of the log-likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity. Unlike previous methods, the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events, such as scatter and random coincidences. The gradient and the curvature of the likelihood terms are evaluated only once per iteration. Since the problem is simplified at each iteration, the CPU time is less than that of current algorithms which directly minimize the objective, yet the convergence rate is comparable. The simplicity, monotonicity, and speed of the new algorithms are quite attractive. The convergence rates of the algorithms are demonstrated using real and simulated PET transmission scans.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85831/1/Fessler83.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

Preconditioning Methods for Shift-Variant Image Reconstruction

Preconditioning methods can accelerate the convergence of gradient-based iterative methods for tomographic image reconstruction and image restoration. Circulant preconditioners have been used extensively for shift-invariant problems. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. For inverse problems that are approximately shift-invariant (i.e. approximately block-Toeplitz or block-circulant Hessians), circulant or Fourier-based preconditioners can provide remarkable acceleration. 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 (penalized) weighted least-squares objective function is quite shift-variant, and the Fourier preconditioner performs poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that more accurately approximate the Hessian matrices of shift-variant imaging problems. Compared to diagonal or Fourier preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. Applications to position emission tomography (PET) illustrate the method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85856/1/Fessler147.pd

Joint Estimation of Attenuation and Emission Images from PET Scans

In modern PET scanners, image reconstruction is performed sequentially in two steps regardless of the reconstruction method: 1. Attenuation correction factor computation (ACF) from transmission scans, 2. Emission image reconstruction using the computed ACFs. This reconstruction scheme does not use all the information in the transmission and emission scans. Post-injection transmission scans contain emission contamination which includes information about emission parameters. Conversely emission scans contain information about the attenuating medium. To use all the available information, the authors propose a joint estimation approach that estimates the attenuation map and the emission image from these two scans. The penalized-likelihood objective function is nonconvex for this problem. The authors propose an algorithm based on paraboloidal surrogates that alternates between emission and attenuation parameters and is guaranteed to monotonically decrease the objective function.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85807/1/Fessler159.pd

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

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

Emission Image Reconstruction for Randoms-Precorrected PET Allowing Negative Sinogram Values

Most positron emission tomography (PET) emission scans are corrected for accidental coincidence (AC) events by real-time subtraction of delayed-window coincidences, leaving only the randoms-precorrected data available for image reconstruction. The real-time randoms precorrection compensates in mean for AC events but destroys the Poisson statistics. The exact log-likelihood for randoms-precorrected data is inconvenient, so practical approximations are needed for maximum likelihood or penalized-likelihood image reconstruction. Conventional approximations involve setting negative sinogram values to zero, which can induce positive systematic biases, particularly for scans with low counts per ray. We propose new likelihood approximations that allow negative sinogram values without requiring zero-thresholding. With negative sinogram values, the log-likelihood functions can be nonconcave, complicating maximization; nevertheless, we develop monotonic algorithms for the new models by modifying the separable paraboloidal surrogates and the maximum-likelihood expectation-maximization (ML-EM) methods. These algorithms ascend to local maximizers of the objective function. Analysis and simulation results show that the new shifted Poisson (SP) model is nearly free of systematic bias yet keeps low variance. Despite its simpler implementation, the new SP performs comparably to the saddle-point model which has shown the best performance (as to systematic bias and variance) in randoms-precorrected PET emission reconstruction.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85994/1/Fessler61.pd

Algorithms for Joint Estimation of Attenuation and Emission Images in PET

In positron emission tomography (PET), positron emission from radiolabeled compounds yields two high energy photons emitted in opposing directions. However, often the photons are not detected due to attenuation within the patient. This attenuation is nonuniform and must be corrected to obtain quantitatively accurate emission images. To measure attenuation effects, one typically acquires a PET transmission scan before or after the injection of radiotracer. In commercially available PET scanners, image reconstruction is performed sequentially in two steps regardless of the reconstruction method: 1. Attenuation correction factor computation (ACF) from transmission scans, 2. Emission image reconstruction using the computed ACFs. This two-step reconstruction scheme does not use all the information in the transmission and emission scans. Postinjection transmission scans contain emission contamination that includes information about emission parameters. Similarly, emission scans contain information about the attenuating medium. To use all the available information, we propose a joint estimation approach that estimates the attenuation map and the emission image simultaneously from these two scans. The penalized-likelihood objective function is nonconvex for this problem. We propose an algorithm based on paraboloidal surrogates that alternates between updating emission and attenuation parameters and is guaranteed to monotonically decrease the objective function.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85956/1/Fessler160.pd