Compensation for Nonuniform Resolution Using Penalized-Likelihood Reconstruction in Space-Variant Imaging Systems

Imaging systems that form estimates using a statistical approach generally yield images with nonuniform resolution properties. That is, the reconstructed images possess resolution properties marked by space-variant and/or anisotropic responses. We have previously developed a space-variant penalty for penalized-likelihood (PL) reconstruction that yields nearly uniform resolution properties . We demonstrated how to calculate this penalty efficiently and apply it to an idealized positron emission tomography (PET) system whose geometric response is space-invariant. In this paper, we demonstrate the efficient calculation and application of this penalty to space-variant systems. (The method is most appropriate when the system matrix has been precalculated.) We apply the penalty to a large field of view PET system where crystal penetration effects make the geometric response space-variant, and to a two-dimensional single photon emission computed tomography system whose detector responses are modeled by a depth-dependent Gaussian with linearly varying full-width at half-maximum. We perform a simulation study comparing reconstructions using our proposed PL approach with other reconstruction methods and demonstrate the relative resolution uniformity, and discuss tradeoffs among estimators that yield nearly uniform resolution. We observe similar noise performance for the PL and post-smoothed maximum-likelihood (ML) approaches with carefully matched resolution, so choosing one estimator over another should be made on other factors like computational complexity and convergence rates of the iterative reconstruction. Additionally, because the postsmoothed ML and the proposed PL approach can outperform one another in terms of resolution uniformity depending on the desired reconstruction resolution, we present and discuss a hybrid approach adopting both a penalty and post-smoothing.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85975/1/Fessler63.pd

Spatially -Variant Roughness Penalty Design for Uniform Resolution in Penalized-Likelihood Image Reconstruction

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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85891/1/Fessler150.pd

Nonnegative Definite Quadratic Penalty Design for Penalized-Likelihood Reconstruction

Likelihood-based estimators with conventional regularization methods generally produces images with nonuniform and anisotropic spatial resolution properties. Previous work on penalty design for penalized-likelihood estimators has led to statistical reconstruction methods that yield approximately uniform "average" resolution. However some asymmetries in the local point-spread functions persist. Such anisotropies result in the elongation of otherwise symmetric features like circular lesions. All previously published penalty functions have used nonnegative values for the weighting coefficients between neighboring voxels. Such nonnegativity provides a sufficient (but not necessary) condition to ensure that the penalty function is convex, which in turn ensures that the objective function has a unique maximizer. This paper describes a novel method for penalty design that allows a subset of the weighting coefficients to take negative values, while still ensuring convexity of the penalty function. We demonstrate that penalties designed under these more flexible constraints yield local point-spread functions that are more isotropic than the previous penalty design methods for 2D PET image reconstruction.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85837/1/Fessler171.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

Efficient Calculation of Resolution and Covariance for Penalized-Likelihood Reconstruction in Fully 3-D SPECT

Resolution and covariance predictors have been derived previously for penalized-likelihood estimators. These predictors can provide accurate approximations to the local resolution properties and covariance functions for tomographic systems given a good estimate of the mean measurements. Although these predictors may be evaluated iteratively, circulant approximations are often made for practical computation times. However, when numerous evaluations are made repeatedly (as in penalty design or calculation of variance images), these predictors still require large amounts of computing time. In Stayman and Fessler (2000), we discussed methods for precomputing a large portion of the predictor for shift-invariant system geometries. In this paper, we generalize the efficient procedure discussed in Stayman and Fessler (2000) to shift-variant single photon emission computed tomography (SPECT) systems. This generalization relies on a new attenuation approximation and several observations on the symmetries in SPECT systems. These new general procedures apply to both two-dimensional and fully three-dimensional (3-D) SPECT models, that may be either precomputed and stored, or written in procedural form. We demonstrate the high accuracy of the predictions based on these methods using a simulated anthropomorphic phantom and fully 3-D SPECT system. The evaluation of these predictors requires significantly less computation time than traditional prediction techniques, once the system geometry specific precomputations have been made.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85992/1/Fessler54.pd

Fast Methods for Approximation of Resolution and Covariance for SPECT

Resolution and covariance predictors have been derived previously for penalized-likelihood estimators. These predictors can provide accurate approximations to the local resolution properties and covariance functions for tomographic systems given a good estimate of the mean measurements. However, when numerous evaluations are made repeatedly (as in penalty design or calculation of variance images), these predictors still require large amounts of computing time. In, we discussed methods for precomputing a large portion of the predictor for shift-invariant system geometries. In this paper, we generalize the efficient procedure discussed in to shift-variant single photon emission computed tomography (SPECT) systems. This generalization relies on a new attenuation approximation and several observations on the symmetries in SPECT systems. These new general procedures apply to both 2D and fully-3D SPECT models, that may be either precomputed and stored, or written in procedural form.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86005/1/Fessler180.pd