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Mathematics
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Advances in Inverse Problems and Imaging
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Advances in Inverse Problems and Imaging

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 May 2024) | Viewed by 3140

Special Issue Editors

Dr. Leonidas Mindrinos

E-Mail Website
Guest Editor
Department of Natural Resources Development and Agricultural Engineering, Agricultural University of Athens, 11855 Athens, Greece
Interests: inverse problems; integral equations; linear elasticity; tomography
Dr. Nicholas Polydorides

E-Mail Website
Guest Editor
Computational Engineering, School of Engineering, University of Edinburgh, Edinburgh, UK
Interests: inverse problems; data sketching; randomised computing; digital twins
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Inverse problems are an emerging area of applied mathematics with many applications in various disciplines, such as physics, engineering and medical sciences. Especially in the field of medical imaging, many works, both theoretical and numerical, have been presented in recent years that have attracted much interest. The main objectives are either qualitatively dealing with the improvement of the quality of the tomographic images, or quantitively, where one is interested in recovering material parameters from the measured data. Both cases are important and may lead to the early diagnosis of deceases.

Mathematically speaking, we are interested in modelling the interaction of the sample with the penetrating wave, with it being either acoustic or optical. Once the model is derived, the corresponding inverse problem has to be solved. However, due to limited measurement data, this problem is usually ill-posed and special regularization techniques are needed.

This Special Issue aims to present recent works dealing with (but not limited to) the following topics:

  • Direct and inverse scattering problems;
  • Mathematical modelling of tomographic techniques;
  • Reconstruction methods;
  • Optical tomography;
  • Hybrid imaging;
  • Coupled physics imaging.

Dr. Leonidas Mindrinos
Dr. Nicholas Polydorides
Guest Editors

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
  • scattering theory
  • mathematical modelling
  • medical imaging
  • tomography

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Published Papers (1 paper)

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14 pages, 543 KiB  
Article
Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography
by Vangelis Marinakis, Athanassios S. Fokas, George A. Kastis and Nicholas E. Protonotarios
Mathematics 2023, 11(23), 4757; https://doi.org/10.3390/math11234757 - 24 Nov 2023
Viewed by 2638
Abstract
Since their introduction, Chebyshev polynomials of the first kind have been extensively investigated, especially in the context of approximation and interpolation. Although standard interpolation methods usually employ equally spaced points, this is not the case in Chebyshev interpolation. Instead of equally spaced points [...] Read more.
Since their introduction, Chebyshev polynomials of the first kind have been extensively investigated, especially in the context of approximation and interpolation. Although standard interpolation methods usually employ equally spaced points, this is not the case in Chebyshev interpolation. Instead of equally spaced points along a line, Chebyshev interpolation involves the roots of Chebyshev polynomials, known as Chebyshev nodes, corresponding to equally spaced points along the unit semicircle. By reviewing prior research on the applications of Chebyshev interpolation, it becomes apparent that this interpolation is rather impractical for medical imaging. Especially in clinical positron emission tomography (PET) and in single-photon emission computerized tomography (SPECT), the so-called sinogram is always calculated at equally spaced points, since the detectors are almost always uniformly distributed. We have been able to overcome this difficulty as follows. Suppose that the function to be interpolated has compact support and is known at q equally spaced points in −1,1. We extend the domain to −a,a, a>1, and select a sufficiently large value of a, such that exactlyq Chebyshev nodes are included in −1,1, which are almost equally spaced. This construction provides a generalization of the concept of standard Chebyshev interpolation to almost equally spaced points. Our preliminary results indicate that our modification of the Chebyshev method provides comparable, or, in several cases including Runge’s phenomenon, superior interpolation over the standard Chebyshev interpolation. In terms of the L∞ norm of the interpolation error, a decrease of up to 75% was observed. Furthermore, our approach opens the way for using Chebyshev polynomials in the solution of the inverse problems arising in PET and SPECT image reconstruction. Full article
(This article belongs to the Special Issue Advances in Inverse Problems and Imaging)
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