The superposition calculus has been formalized in Isabelle/HOL twice before but in both cases without a type system. Nowadays, modern superposition provers support types. We extend an existing Isabelle formalization of untyped superposition with simple monomorphic types, or sorts. This extension is straightforward on paper but surprisingly tricky to implement formally. We also use this opportunity to refactor the proof text to avoid quadruplicated definitions, lemmas, and proofs about terms, atoms, literals, and clauses. The extended formalization and its refactoring benefit from Isabelle's locales, structured Isar proofs, and Sledgehammer proof tool.

Superposition is an efficient proof calculus for reasoning about first-order logic with equality that is implemented in many automatic theorem provers. It works by saturating the given set of clauses and is refutationally complete, meaning that if the set is inconsistent, the saturation will contain a contradiction. In this work, we restructured the completeness proof to cleanly separate the ground (i.e., variable-free) and nonground aspects, and we formalized the result in Isabelle/HOL. We relied on the IsaFoR library for first-order terms and on the Isabelle saturation framework.

Abstract Dynamic tomography reconstructs a time activity curve (TAC) for every voxel assuming that the algebraic form of the function is known a priori. The algebraic form derived from the analysis of compartmental models depends nonlinearly on the nonnegative parameters to be determined. Direct methods apply fitting in every iteration step. Because of the iterative nature of the maximum likelihood–expectation maximization (ML–EM) reconstruction, the fitting result of the previous step can serve as a good starting point in the current step; thus, after the first iteration we have a guess that is not far from the solution, which allows the use of gradient-based local optimization methods. However, finding good initial guesses for the first ML–EM iteration is a critical problem since gradient-based local optimization algorithms do not guarantee convergence to the global optimum if they are started at an inappropriate location. This paper examines the robust solution of the fitting problem both in the initial phase and during the ML–EM iteration. This solution is implemented on GPUs and is built into the 4D reconstruction module of the TeraTomo software.

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This paper proposes the application of multiple importance sampling in fully 3-D positron emission tomography to speed up the iterative reconstruction process. The proposed method combines the results of lines of responses (LOR) driven and voxel driven projections keeping their advantages, like importance sampling, performance and parallel execution on graphics processing units. Voxel driven methods can focus on point like features while LOR driven approaches are efficient in reconstructing homogeneous regions. The theoretical basis of the combination is the application of the mixture of the samples generated by the individual importance sampling methods, emphasizing a particular method where it is better than others. The proposed algorithms are built into the Tera-tomo system.

Iterative positron emission tomography (PET) reconstruction computes projections between the voxel space and the lines of response (LOR) space, which are mathematically equivalent to the evaluation of multi-dimensional integrals. The dimension of the integration domain can be very high if scattering needs to be compensated. Monte Carlo (MC) quadrature is a straightforward method to approximate high-dimensional integrals. As the numbers of voxels and LORs can be in the order of hundred millions and the projection also depends on the measured object, the quadratures cannot be precomputed, but Monte Carlo simulation should take place on-the-fly during the iterative reconstruction process. This paper presents modifications of the maximum likelihood, expectation maximization (ML-EM) iteration scheme to reduce the reconstruction error due to the on-the-fly MC approximations of forward and back projections. If the MC sample locations are the same in every iteration step of the ML-EM scheme, then the approximation error will lead to a modified reconstruction result. However, when random estimates are statistically independent in different iteration steps, then the iteration may either diverge or fluctuate around the solution. Our goal is to increase the accuracy and the stability of the iterative solution while keeping the number of random samples and therefore the reconstruction time low. We first analyze the error behavior of ML-EM iteration with on-the-fly MC projections, then propose two solutions: averaging iteration and Metropolis iteration. Averaging iteration averages forward projection estimates during the iteration sequence. Metropolis iteration rejects those forward projection estimates that would compromise the reconstruction and also guarantees the unbiasedness of the tracer density estimate. We demonstrate that these techniques allow a significant reduction of the required number of samples and thus the reconstruction time. The proposed methods are built into the Teratomo system.

Abstract This paper presents efficient algorithms for free path sampling in heterogeneous participating media defined either by high‐resolution voxel arrays or generated procedurally. The method is based on the concept of mixing ‘virtual’ material or particles to the medium, augmenting the extinction coefficient to a function for which the free path can be sampled in a straightforward way. The virtual material is selected such that it modifies the volume density but does not alter the radiance. We define the total extinction coefficient of the real and virtual particles by a low‐resolution grid of super‐voxels that are much larger than the real voxels defining the medium. The computational complexity of the proposed method depends just on the resolution of the super‐voxel grid and does not grow with the resolution above the scale of super‐voxels. The method is particularly efficient to render large, low‐density, heterogeneous volumes, which should otherwise be defined by enormously high resolution voxel grids and where the average free path length would cross many voxels.

This paper presents a fast parallel method to solve the radiative transport equation in inhomogeneous participating media. We apply a novel approximation scheme to find a good initial guess for both the direct and scattered components. Then, the initial approximation is used to bootstrap an iterative multiple scattering solver, i.e., we let the iteration concentrate just on the residual problem. This kind of bootstrapping makes the volumetric source approximation more uniform, thus it helps to reduce the discretization artifacts and improves the efficiency of the parallel implementation. The iterative refinement is executed on a face-centered cubic grid. The implementation is based on CUDA and runs on the GPU. For large volumes that do not fit into the GPU memory, we also consider the implementation on a GPU cluster, where the volume is decomposed to blocks according to the available GPU nodes. We show how the communication bottleneck can be avoided in the cluster implementation by not exchanging the boundary conditions in every iteration step. In addition to light photons, we also discuss the generalization of the method to γ-photons that are relevant in medical simulation.