Research

18 pages, 1040 KiB

Open AccessArticle

Gaussian Mixture Estimation from Lower-Dimensional Data with Application to PET Imaging

by Azra Tafro and Damir Seršić

Mathematics 2024, 12(5), 764; https://0-doi-org.brum.beds.ac.uk/10.3390/math12050764 - 04 Mar 2024

Viewed by 492

In positron emission tomography (PET), the original points of emission are unknown, and the scanners record pairs of photons emitting from those origins and creating lines of response (LORs) in random directions. This presents a latent variable problem, since at least one dimension [...] Read more.

In positron emission tomography (PET), the original points of emission are unknown, and the scanners record pairs of photons emitting from those origins and creating lines of response (LORs) in random directions. This presents a latent variable problem, since at least one dimension of relevant information is lost. This can be solved by a statistical approach to image reconstruction—modeling the image as a Gaussian mixture model (GMM). This allows us to obtain a high-quality continuous model that is not computationally demanding and does not require postprocessing. In this paper, we propose a novel method of GMM estimation in the PET setting, directly from lines of response. This approach utilizes some well-known and convenient properties of the Gaussian distribution and the fact that the random slopes of the lines are independent from the points of origin. The expectation–maximization (EM) algorithm that is most commonly used to estimate GMMs in the traditional setting here is adapted to lower-dimensional data. The proposed estimation method is unbiased, and simulations and experiments show that accurate reconstruction on synthetic data is possible from relatively small samples. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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37 pages, 2738 KiB

Open AccessArticle

Tomographic Reconstruction: General Approach to Fast Back-Projection Algorithms

by Dmitry Polevoy, Marat Gilmanov, Danil Kazimirov, Marina Chukalina, Anastasia Ingacheva, Petr Kulagin and Dmitry Nikolaev

Mathematics 2023, 11(23), 4759; https://0-doi-org.brum.beds.ac.uk/10.3390/math11234759 - 24 Nov 2023

Viewed by 923

Abstract

Addressing contemporary challenges in computed tomography (CT) demands precise and efficient reconstruction. This necessitates the optimization of CT methods, particularly by improving the algorithmic efficiency of the most computationally demanding operators—forward projection and backprojection. Every measurement setup requires a unique pair of these [...] Read more.

Addressing contemporary challenges in computed tomography (CT) demands precise and efficient reconstruction. This necessitates the optimization of CT methods, particularly by improving the algorithmic efficiency of the most computationally demanding operators—forward projection and backprojection. Every measurement setup requires a unique pair of these operators. While fast algorithms for calculating forward projection operators are adaptable across various setups, they fall short in three-dimensional scanning scenarios. Hence, fast algorithms are imperative for backprojection, an integral aspect of all established reconstruction methods. This paper introduces a general method for the calculation of backprojection operators in any measurement setup. It introduces a versatile method for transposing summation-based algorithms, which rely exclusively on addition operations. The proposed approach allows for the transformation of algorithms designed for forward projection calculation into those suitable for backprojection, with the latter maintaining asymptotic algorithmic complexity. Employing this method, fast algorithms for both forward projection and backprojection have been developed for the 2D few-view parallel-beam CT as well as for the 3D cone-beam CT. The theoretically substantiated complexity values for the proposed algorithms align with their experimentally derived estimates. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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24 pages, 1020 KiB

Open AccessArticle

Numerical Reconstruction of a Space-Dependent Reaction Coefficient and Initial Condition for a Multidimensional Wave Equation with Interior Degeneracy

by Hamed Ould Sidi, Mahmoud A. Zaky, Rob H. De Staelen and Ahmed S. Hendy

Mathematics 2023, 11(14), 3186; https://0-doi-org.brum.beds.ac.uk/10.3390/math11143186 - 20 Jul 2023

Cited by 2 | Viewed by 610

Abstract

A simultaneous reconstruction of the initial condition and the space-dependent reaction coefficient in a multidimensional hyperbolic partial differential equation with interior degeneracy is of concern. A temporal integral observation is utilized to achieve that purpose. The well-posedness, existence, and uniqueness of the inverse [...] Read more.

A simultaneous reconstruction of the initial condition and the space-dependent reaction coefficient in a multidimensional hyperbolic partial differential equation with interior degeneracy is of concern. A temporal integral observation is utilized to achieve that purpose. The well-posedness, existence, and uniqueness of the inverse problem under consideration are discussed. The inverse problem can be reformulated as a least squares minimization and the Fréchet gradients are determined, using the adjoint and sensitivity problems. Finally, an iterative construction procedure is developed by employing the conjugate gradient algorithm while invoking the discrepancy principle as a stopping criterion. Some numerical experiments are given to ensure the performance of the reconstruction scheme in one and two dimensions. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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15 pages, 1180 KiB

Open AccessArticle

Combination of Multigrid with Constraint Data for Inverse Problem of Nonlinear Diffusion Equation

by Tao Liu, Di Ouyang, Lianjun Guo, Ruofeng Qiu, Yunfei Qi, Wu Xie, Qiang Ma and Chao Liu

Mathematics 2023, 11(13), 2887; https://0-doi-org.brum.beds.ac.uk/10.3390/math11132887 - 27 Jun 2023

Cited by 4 | Viewed by 877

Abstract

This paper delves into a rapid and accurate numerical solution for the inverse problem of the nonlinear diffusion equation in the context of multiphase porous media flow. For the realization of this, the combination of the multigrid method with constraint data is utilized [...] Read more.

This paper delves into a rapid and accurate numerical solution for the inverse problem of the nonlinear diffusion equation in the context of multiphase porous media flow. For the realization of this, the combination of the multigrid method with constraint data is utilized and investigated. Additionally, to address the ill-posedness of the inverse problem, the Tikhonov regularization is incorporated. Numerical results demonstrate the computational performance of this method. The proposed combination strategy displays remarkable capabilities in reducing noise, avoiding local minima, and accelerating convergence. Moreover, this combination method performs better than any one method used alone. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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21 pages, 546 KiB

Open AccessArticle

Hybrid Method for Inverse Elastic Obstacle Scattering Problems

by Yuhan Yin and Juan Liu

Mathematics 2023, 11(8), 1939; https://0-doi-org.brum.beds.ac.uk/10.3390/math11081939 - 20 Apr 2023

Viewed by 656

Abstract

The problem of determining the shape of an object from knowledge of the far-field of a single incident wave in two-dimensional elasticity was considered. We applied an iterative hybrid method to tackle this problem. An advantage of this method is that it does [...] Read more.

The problem of determining the shape of an object from knowledge of the far-field of a single incident wave in two-dimensional elasticity was considered. We applied an iterative hybrid method to tackle this problem. An advantage of this method is that it does not need a forward solver, and therefore, the exact boundary condition is not essential. By deriving the Fréchet derivatives of two boundary operators, we established reconstruction algorithms for objects with Dirichlet, Neumann, and Robin boundary conditions; by introducing a general boundary condition, we also established the reconstruction algorithm for objects with unknown physical properties. Numerical experiments showed the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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17 pages, 982 KiB

Open AccessArticle

Image Reconstruction Algorithm Using Weighted Mean of Ordered-Subsets EM and MART for Computed Tomography

by Omar M. Abou Al-Ola, Ryosuke Kasai, Yusaku Yamaguchi, Takeshi Kojima and Tetsuya Yoshinaga

Mathematics 2022, 10(22), 4277; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224277 - 15 Nov 2022

Cited by 2 | Viewed by 1437

Abstract

Iterative image reconstruction algorithms have considerable advantages over transform methods for computed tomography, but they each have their own drawbacks. In particular, the maximum-likelihood expectation-maximization (MLEM) algorithm reconstructs high-quality images even with noisy projection data, but it is slow. On the other hand, [...] Read more.

Iterative image reconstruction algorithms have considerable advantages over transform methods for computed tomography, but they each have their own drawbacks. In particular, the maximum-likelihood expectation-maximization (MLEM) algorithm reconstructs high-quality images even with noisy projection data, but it is slow. On the other hand, the simultaneous multiplicative algebraic reconstruction technique (SMART) converges faster at early iterations but is susceptible to noise. Here, we construct a novel algorithm that has the advantages of these different iterative schemes by combining ordered-subsets EM (OS-EM) and MART (OS-MART) with weighted geometric or hybrid means. It is theoretically shown that the objective function decreases with every iteration and the amount of decrease is greater than the mean between the decreases for OS-EM and OS-MART. We conducted image reconstruction experiments on simulated phantoms and deduced that our algorithm outperforms OS-EM and OS-MART alone. Our algorithm would be effective in practice since it incorporates OS-EM, which is currently the most popular technique of iterative image reconstruction from noisy measured projections. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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22 pages, 1604 KiB

Open AccessArticle

Quality-Enhancing Techniques for Model-Based Reconstruction in Magnetic Particle Imaging

by Vladyslav Gapyak, Thomas März and Andreas Weinmann

Mathematics 2022, 10(18), 3278; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183278 - 09 Sep 2022

Cited by 2 | Viewed by 1467

Abstract

Magnetic Particle Imaging is an imaging modality that exploits the non-linear magnetization response of superparamagnetic nanoparticles to a dynamic magnetic field. In the multivariate case, measurement-based reconstruction approaches are common and involve a system matrix whose acquisition is time consuming and needs to [...] Read more.

Magnetic Particle Imaging is an imaging modality that exploits the non-linear magnetization response of superparamagnetic nanoparticles to a dynamic magnetic field. In the multivariate case, measurement-based reconstruction approaches are common and involve a system matrix whose acquisition is time consuming and needs to be repeated whenever the scanning setup changes. Our approach relies on reconstruction formulae derived from a mathematical model of the MPI signal encoding. A particular feature of the reconstruction formulae and the corresponding algorithms is that these are independent of the particular scanning trajectories. In this paper, we present basic ways of leveraging this independence property to enhance the quality of the reconstruction by merging data from different scans. In particular, we show how to combine scans of the same specimen under different rotation angles. We demonstrate the potential of the proposed techniques with numerical experiments. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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12 pages, 669 KiB

Open AccessArticle

A Nonlinear Multigrid Method for the Parameter Identification Problem of Partial Differential Equations with Constraints

by Tao Liu, Jiayuan Yu, Yuanjin Zheng, Chao Liu, Yanxiong Yang and Yunfei Qi

Mathematics 2022, 10(16), 2938; https://0-doi-org.brum.beds.ac.uk/10.3390/math10162938 - 15 Aug 2022

Cited by 4 | Viewed by 1057

Abstract

In this paper, we consider the parameter identification problem of partial differential equations with constraints. A nonlinear multigrid method is introduced to the process of parameter inversion. By keeping the objective functions on coarse grids consistent with those on fine grids, the proposed [...] Read more.

In this paper, we consider the parameter identification problem of partial differential equations with constraints. A nonlinear multigrid method is introduced to the process of parameter inversion. By keeping the objective functions on coarse grids consistent with those on fine grids, the proposed method reduces the dimensions of objective functions enormously and mitigates the risk of trapping in local minima effectively. Furthermore, constraints significantly improve the convergence ability of the method. We performed the numerical simulation based on the porosity identification of elastic wave equations in the fluid-saturated porous media, which suggests that the nonlinear multigrid method with constraints decreases the computational expenditure, suppresses the noise, and improves the inversion results. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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22 pages, 444 KiB

Open AccessArticle

Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation

by Yilihamujiang Yimamu and Zuicha Deng

Mathematics 2022, 10(15), 2608; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152608 - 26 Jul 2022

Cited by 1 | Viewed by 997

Abstract

Based on the theoretical framework of the Black–Scholes model, the convergence of the inverse volatility problem based on the degenerate parabolic equation is studied. Being different from other inverse volatility problems in classical parabolic equations, we introduce some variable substitutions to convert the [...] Read more.

Based on the theoretical framework of the Black–Scholes model, the convergence of the inverse volatility problem based on the degenerate parabolic equation is studied. Being different from other inverse volatility problems in classical parabolic equations, we introduce some variable substitutions to convert the original problem into an inverse principal coefficient problem in a degenerate parabolic equation on a bounded area, from which an unknown volatility can be recovered and deficiencies caused by artificial truncation can be solved. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established, and a rigorous mathematical proof is given for the convergence of the optimal solution. In the end, the gradient-type iteration method is applied to obtain the numerical solution of the inverse problem, and some numerical experiments are performed. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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17 pages, 1279 KiB

Open AccessArticle

Feature Reconstruction from Incomplete Tomographic Data without Detour

by Simon Göppel, Jürgen Frikel and Markus Haltmeier

Mathematics 2022, 10(8), 1318; https://0-doi-org.brum.beds.ac.uk/10.3390/math10081318 - 15 Apr 2022

Cited by 4 | Viewed by 1450

Abstract

In this paper, we consider the problem of feature reconstruction from incomplete X-ray CT data. Such incomplete data problems occur when the number of measured X-rays is restricted either due to limit radiation exposure or due to practical constraints, making the detection of [...] Read more.

In this paper, we consider the problem of feature reconstruction from incomplete X-ray CT data. Such incomplete data problems occur when the number of measured X-rays is restricted either due to limit radiation exposure or due to practical constraints, making the detection of certain rays challenging. Since image reconstruction from incomplete data is a severely ill-posed (unstable) problem, the reconstructed images may suffer from characteristic artefacts or missing features, thus significantly complicating subsequent image processing tasks (e.g., edge detection or segmentation). In this paper, we introduce a framework for the robust reconstruction of convolutional image features directly from CT data without the need of computing a reconstructed image first. Within our framework, we use non-linear variational regularization methods that can be adapted to a variety of feature reconstruction tasks and to several limited data situations. The proposed variational regularization method minimizes an energy functional being the sum of a feature dependent data-fitting term and an additional penalty accounting for specific properties of the features. In our numerical experiments, we consider instances of edge reconstructions from angular under-sampled data and show that our approach is able to reliably reconstruct feature maps in this case. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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11 pages, 320 KiB

Open AccessArticle

On the Numerical Solution of a Hyperbolic Inverse Boundary Value Problem in Bounded Domains

by Roman Chapko and Leonidas Mindrinos

Mathematics 2022, 10(5), 750; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050750 - 26 Feb 2022

Viewed by 1268

Abstract

We consider the inverse problem of reconstructing the boundary curve of a cavity embedded in a bounded domain. The problem is formulated in two dimensions for the wave equation. We combine the Laguerre transform with the integral equation method and we reduce the [...] Read more.

We consider the inverse problem of reconstructing the boundary curve of a cavity embedded in a bounded domain. The problem is formulated in two dimensions for the wave equation. We combine the Laguerre transform with the integral equation method and we reduce the inverse problem to a system of boundary integral equations. We propose an iterative scheme that linearizes the equation using the Fréchet derivative of the forward operator. The application of special quadrature rules results to an ill-conditioned linear system which we solve using Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions. Full article

(This article belongs to the Special Issue Inverse Problems and Imaging: Theory and Applications)

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