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1. An Integral Equation Model for PET Imaging

   Tang, Xinhuang; Schmidtlein, Charles Ross; Li, Si; Xu, Yuesheng (January 2021, International journal of numerical analysis and modeling)

   Positron emission tomography (PET) is traditionally modeled as discrete systems. Such models may be viewed as piecewise constant approximations of the underlying continuous model for the physical processes and geometry of the PET imaging. Due to the low accuracy of piecewise constant approximations, discrete models introduce an irreducible modeling error which fundamentally limits the quality of reconstructed images. To address this bottleneck, we propose an integral equation model for the PET imaging based on the physical and geometrical considerations, which describes accurately the true coincidences. We show that the proposed integral equation model is equivalent to the existing idealized model in terms of line integrals which is accurate but not suitable for numerical approximation. The proposed model allows us to discretize it using higher accuracy approximation methods. In particular, we discretize the integral equation by using the collocation principle with piecewise linear polynomials. The discretization leads to new ill-conditioned discrete systems for the PET reconstruction, which are further regularized by a novel wavelet-based regularizer. The resulting non-smooth optimization problem is then solved by a preconditioned proximity fixed-point algorithm. Convergence of the algorithm is established for a range of parameters involved in the algorithm. The proposed integral equation model combined with the discretization, regularization, and optimization algorithm provides a new PET image reconstruction method. Numerical results reveal that the proposed model substantially outperforms the conventional discrete model in terms of the consistency to simulated projection data and reconstructed image quality. This indicates that the proposed integral equation model with appropriate discretization and regularizer can significantly reduce modeling errors and suppress noise, which leads to improved image quality and projection data estimation. 

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2. Inexact rational Krylov subspace methods for approximating the action of functions of matrices

   https://doi.org/10.1553/etna_vol58s538  

   Xu, Shengjie; Xue, Fei (September 2023, ETNA - Electronic Transactions on Numerical Analysis)

   This paper concerns the theory and development of inexact rational Krylov subspace methods for approximating the action of a function of a matrix f(A) to a column vector b. At each step of the rational Krylov subspace methods, a shifted linear system of equations needs to be solved to enlarge the subspace. For large-scale problems, such a linear system is usually solved approximately by an iterative method. The main question is how to relax the accuracy of these linear solves without negatively affecting the convergence of the approximation of f(A)b. Our insight into this issue is obtained by exploring the residual bounds for the rational Krylov subspace approximations of f(A)b, based on the decaying behavior of the entries in the first column of certain matrices of A restricted to the rational Krylov subspaces. The decay bounds for these entries for both analytic functions and Markov functions can be efficiently and accurately evaluated by appropriate quadrature rules. A heuristic based on these bounds is proposed to relax the tolerances of the linear solves arising in each step of the rational Krylov subspace methods. As the algorithm progresses toward convergence, the linear solves can be performed with increasingly lower accuracy and computational cost. Numerical experiments for large nonsymmetric matrices show the effectiveness of the tolerance relaxation strategy for the inexact linear solves of rational Krylov subspace methods. 

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3. Automatic Utility Pole Inclination Angle Measurement Using Unmanned Aerial Vehicle and Deep Learning

   Zhu, Zanbo; Zhang, Jing; Alam, Md Morshedul; Tokgoz, Berna Eren; Hwang, Seokyon (May 2019, Proceedings of the Institute of Industrial and Systems Engineers Annual conference 2019)

   Measuring the inclination angles of utility poles of the electric power distribution lines is critical to maintain power distribution systems and minimize power outages, because the poles are very vulnerable to natural disasters. However, traditional human-based pole inspection methods are very costly and require heavy workloads. In this paper, we propose a novel pole monitoring system to measure the inclination angle of utility poles from images captured by unmanned aerial vehicle (UAV) automatically. A state-of-the-art deep learning neural network is used to detect and segment utility poles from UAV street view images, and computer vision techniques are used to calculate the inclination angles based on the segmented poles. The proposed method was evaluated using 64 images with 84 utility poles taken in different weather conditions. The pole segmentation accuracy is 93.74% and the average inclination angle error is 0.59 degrees, which demonstrate the efficiency of the proposed utility pole monitoring system. 

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4. Low-Memory Krylov Subspace Methods for Optimal Rational Matrix Function Approximation

   https://doi.org/10.1137/22M1479853  

   Chen, Tyler; Greenbaum, Anne; Musco, Cameron; Musco, Christopher (June 2023, SIAM Journal on Matrix Analysis and Applications)

   Abstract. We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal rational matrix function approximation (Lanczos-OR), returns the optimal approximation from a given Krylov subspace in a norm depending on the rational function’s denominator, and it can be computed using the information from a slightly larger Krylov subspace. We also provide a low-memory implementation which only requires storing a number of vectors proportional to the denominator degree of the rational function. Finally, we show that Lanczos-OR can be used to derive algorithms for computing other matrix functions, including the matrix sign function and quadrature-based rational function approximations. In many cases, it improves on the approximation quality of prior approaches, including the standard Lanczos method, with little additional computational overhead. 

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5. Proxima: An Approach for Time or Accuracy Budgeted Collision Proximity Queries

   https://doi.org/10.15607/RSS.2022.XVIII.043  

   Rakita, Daniel; Mutlu, Bilge; Gleicher, Michael (January 2022, Proceedings of Robotics: Science and Systems (RSS))

   Many applications in robotics require computing a robot manipulator's "proximity" to a collision state in a given configuration. This collision proximity is commonly framed as a summation over closest Euclidean distances between many pairs of rigid shapes in a scene. Computing many such pairwise distances is inefficient, while more efficient approximations of this procedure, such as through supervised learning, lack accuracy and robustness. In this work, we present an approach for computing a collision proximity function for robot manipulators that formalizes the trade-off between efficiency and accuracy and provides an algorithm that gives control over it. Our algorithm, called Proxima, works in one of two ways: (1) given a time budget as input, the algorithm returns an as-accurate-as-possible proximity approximation value in this time; or (2) given an accuracy budget, the algorithm returns an as-fast-as-possible proximity approximation value that is within the given accuracy bounds. We show the robustness of our approach through analytical investigation and simulation experiments on a wide set of robot models ranging from 6 to 132 degrees of freedom. We demonstrate that controlling the trade-off between efficiency and accuracy in proximity computations via our approach can enable safe and accurate real-time robot motion-optimization even on high-dimensional robot models. 

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