The thin plate as a regularizer in Bayesian SPECT reconstruction
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Abstract:
Bayesian MAP (maximum a posteriori) methods for SPECT reconstruction can both stabilize reconstructions and lead to better bias and variance relative to ML methods. In previous work, a nonquadratic prior (the weak plate) that imposed piecewise smoothness on the first derivative of the solution led to much improved bias/variance behavior relative to results obtained using a more conventional nonquadratic prior (the weak membrane) that imposed piecewise smoothness of the zeroth derivative. By relaxing the requirement of imposing spatial discontinuities and using instead a quadratic (no discontinuities) smoothing prior, algorithms become easier to analyze, solutions easier to compute, and hyperparameter calculation becomes less of a problem. In this work, we investigated whether the advantages of weak plate relative to weak membrane are retained when non-piecewise quadratic versions-the thin plate and membrane priors-are used. We compared, with three different phantoms, the bias/variance behavior of three approaches: (1) FBP with membrane and thin plate implemented as smoothing filters, (2) ML-EM with two stopping criteria, and (3) MAP with thin plate and membrane priors. In cases (1) and (3), the thin plate always led to better bias behavior at comparable variance relative to membrane priors/filters. Also, approaches (1) and (3) outperformed ML-EM at both stopping criteria. The net conclusion is that, while quadratic smoothing priors are not as good as piecewise versions, the simple modification of the membrane model to the thin plate model leads to improved bias behavior.Keywords:
Smoothing
Classification of discontinuities
Hyperparameter
Smoothness
Maximum a posteriori (MAP) estimation, like all Bayesian methods, depends on prior assumptions. These assumptions are often chosen to promote specific features in the recovered estimate. The form of the chosen prior determines the shape of the posterior distribution, thus the behavior of the estimator and complexity of the associated optimization problem. Here, we consider a family of Gaussian hierarchical models with generalized gamma hyperpriors designed to promote sparsity in linear inverse problems. By varying the hyperparameters, we move continuously between priors that act as smoothed $\ell_p$ penalties with flexible $p$, smoothing, and scale. We then introduce a predictor-corrector method that tracks MAP solution paths as the hyperparameters vary. Path following allows a user to explore the space of possible MAP solutions and to test the sensitivity of solutions to changes in the prior assumptions. By tracing paths from a convex region to a non-convex region, the user can find local minimizers in strongly sparsity promoting regimes that are consistent with a convex relaxation derived using related prior assumptions. We show experimentally that these solutions. are less error prone than direct optimization of the non-convex problem.
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Bayes estimator
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Abstract Discontinuities that have unfavourable orientation and are continuous within overall engineering rock regions can have a dominant effect on the strength, deformability and permeability of the rock mass. The concepts of geometrical parameters of basic discontinuities and engineering discontinuities are proposed in this communication. Further, the engineering discontinuities are divided into key discontinuities and non‐key discontinuities. Within any region of the rock mass, the spacing, trace length and probability of engineering discontinuities can be estimated from the geometrical parameters of the basic discontinuities. In general, the geometrical parameters are different from those of the basic discontinuities. Finally, two examples are given to illustrate how to apply these parameters to rock engineering problems.
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Geomechanics
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