Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
4.0
2.3
Journals
Mathematics
Special Issues
Inverse Problems and Imaging: Theory and Applications
share announcement

Inverse Problems and Imaging: Theory and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 January 2024) | Viewed by 20040

Special Issue Editor

Prof. Dr. Jürgen Frikel

E-Mail Website
Guest Editor
Department of Computer Science and Mathematics, Ostbayerische Technische Hochschule Regensburg (OTH Regensburg), Regensburg, Germany
Interests: inverse problems in imaging; image reconstruction in tomography; data and image processing; applied harmonic analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear colleagues,

Inverse problems and imaging are two closely related and quickly emerging research fields that play a crucial role in many areas, such as medical imaging, nondestructive testing, remote sensing, or geophysics. Driven by the developments of novel imaging modalities and new application areas, theoretical and practical challenges arise that range from pure to applied mathematics and to engineering. For example, because of dose reduction in CT, only incomplete data might be available for reconstruction leading to a substantial information loss. Additionally, objects might move during examination leading to so-called motion artifacts and corrupting the reconstruction quality. To understand the impact of such (incomplete or corrupted) data sets, new theoretical results have been derived that take into account the specific data acquisition protocols. Moreover, new models are developed that capture the data generation process more reliably. Moreover, novel reconstruction techniques are developed that are tailored to specific imaging scenarios. Recently, data-driven approaches that are based on machine learning techniques have also been employed and analyzed in the context of inverse problems. Another important development is given by so-called multimodality or multispectral imaging. All these examples constitute only a small fraction of modern challenges and research directions in inverse problems and imaging.

This Special Issue aims at bringing together original research articles that reflect recent advances in the field of inverse problems and imaging, including a broad range of imaging modalities, mathematics, and applications.

Prof. Dr. Jürgen Frikel
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Inverse Problems
  • Regularization
  • Imaging
  • Image reconstruction
  • Image analysis
  • Tomography
  • Tomographic reconstruction
  • Feature reconstruction
  • Multimodality imaging
  • Multienergy imaging
  • Ultrasound imaging
  • Optical imaging
  • Photoacoustic tomography
  • Magnetic resonance tomography
  • Magnetic particle imaging
  • Artifact reduction
  • Dynamic Inverse problems
  • Dynamic Tomography
  • Motion Artifacts
  • Deep learning in imaging

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue polices can be found here.

Published Papers (11 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

18 pages, 1040 KiB  
Article
Gaussian Mixture Estimation from Lower-Dimensional Data with Application to PET Imaging
by Azra Tafro and Damir Seršić
Mathematics 2024, 12(5), 764; https://doi.org/10.3390/math12050764 - 4 Mar 2024
Cited by 1 | Viewed by 896
Abstract
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)
►▼ Show Figures

Figure 1

37 pages, 2738 KiB  
Article
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://doi.org/10.3390/math11234759 - 24 Nov 2023
Cited by 3 | Viewed by 1944
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)
►▼ Show Figures

Figure 1

24 pages, 1020 KiB  
Article
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://doi.org/10.3390/math11143186 - 20 Jul 2023
Cited by 2 | Viewed by 902
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)
►▼ Show Figures

Figure 1

15 pages, 1180 KiB  
Article
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://doi.org/10.3390/math11132887 - 27 Jun 2023
Cited by 9 | Viewed by 1287
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)
►▼ Show Figures

Figure 1

21 pages, 546 KiB  
Article
Hybrid Method for Inverse Elastic Obstacle Scattering Problems
by Yuhan Yin and Juan Liu
Mathematics 2023, 11(8), 1939; https://doi.org/10.3390/math11081939 - 20 Apr 2023
Viewed by 987
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)
►▼ Show Figures

Figure 1

17 pages, 982 KiB  
Article
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://doi.org/10.3390/math10224277 - 15 Nov 2022
Cited by 3 | Viewed by 2059
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)
►▼ Show Figures

Figure 1

22 pages, 1604 KiB  
Article
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://doi.org/10.3390/math10183278 - 9 Sep 2022
Cited by 5 | Viewed by 2125
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)
►▼ Show Figures

Figure 1

12 pages, 669 KiB  
Article
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://doi.org/10.3390/math10162938 - 15 Aug 2022
Cited by 4 | Viewed by 1474
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)
►▼ Show Figures

Figure 1

22 pages, 444 KiB  
Article
Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation
by Yilihamujiang Yimamu and Zuicha Deng
Mathematics 2022, 10(15), 2608; https://doi.org/10.3390/math10152608 - 26 Jul 2022
Cited by 2 | Viewed by 1381
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)
►▼ Show Figures

Figure 1

17 pages, 1279 KiB  
Article
Feature Reconstruction from Incomplete Tomographic Data without Detour
by Simon Göppel, Jürgen Frikel and Markus Haltmeier
Mathematics 2022, 10(8), 1318; https://doi.org/10.3390/math10081318 - 15 Apr 2022
Cited by 4 | Viewed by 1915
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)
►▼ Show Figures

Figure 1

11 pages, 320 KiB  
Article
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://doi.org/10.3390/math10050750 - 26 Feb 2022
Cited by 1 | Viewed by 1606
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)
►▼ Show Figures

Figure 1

Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-11
Back to TopTop