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Mathematics
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Advances in PDE-Based Methods for Image Processing
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Advances in PDE-Based Methods for Image Processing

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

Deadline for manuscript submissions: 17 July 2024 | Viewed by 8737

Special Issue Editor

Prof. Dr. Tudor Barbu

E-Mail Website
Guest Editor
Institute of Computer Science of the Romanian Academy, Iași Branch, 700481 lasi, Romania
Interests: image processing and analysis; computer vision machine learning; partial differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Partial differential equation (PDE)-based models express continuous change, so they have long been used to formulate dynamical phenomena in a lot of engineering fields, such as image processing and analysis and computer vision. In the last three decades, they have been applied successfully in various subdomains of these areas, such as image denoising and restoration, inpainting, image segmentation, image compression, image decomposition, and image registration.

This proposed Special Issue deals with all these image processing and analysis fields, its main purpose being the dissemination of advanced and original research in these scientific areas and bringing together the researchers working in the PDE-based image processing and analysis, so as to extend the existing knowledge in these domains.

We thus encourage you to send high-quality articles disseminating novel research achievements for this issue.

Prof. Tudor Barbu
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

  • Partial differential equations
  • Diffusion-based models
  • Image denoising and restoration
  • Image interpolation
  • Image segmentation
  • Image registration
  • Image compression
  • Image decomposition
  • Numerical approximation scheme

Published Papers (3 papers)

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Research

20 pages, 2047 KiB  
Article
Reliable Learning with PDE-Based CNNs and DenseNets for Detecting COVID-19, Pneumonia, and Tuberculosis from Chest X-Ray Images
by Anca Nicoleta Marginean, Delia Doris Muntean, George Adrian Muntean, Adelina Priscu, Adrian Groza, Radu Razvan Slavescu, Calin Lucian Timbus, Gabriel Zeno Munteanu, Cezar Octavian Morosanu, Maria Margareta Cosnarovici and Camelia-M. Pintea
Mathematics 2021, 9(4), 434; https://doi.org/10.3390/math9040434 - 22 Feb 2021
Cited by 9 | Viewed by 2700
Abstract
It has recently been shown that the interpretation by partial differential equations (PDEs) of a class of convolutional neural networks (CNNs) supports definition of architectures such as parabolic and hyperbolic networks. These networks have provable properties regarding the stability against the perturbations of [...] Read more.
It has recently been shown that the interpretation by partial differential equations (PDEs) of a class of convolutional neural networks (CNNs) supports definition of architectures such as parabolic and hyperbolic networks. These networks have provable properties regarding the stability against the perturbations of the input features. Aiming for robustness, we tackle the problem of detecting changes in chest X-ray images that may be suggestive of COVID-19 with parabolic and hyperbolic CNNs and with domain-specific transfer learning. To this end, we compile public data on patients diagnosed with COVID-19, pneumonia, and tuberculosis, along with normal chest X-ray images. The negative impact of the small number of COVID-19 images is reduced by applying transfer learning in several ways. For the parabolic and hyperbolic networks, we pretrain the networks on normal and pneumonia images and further use the obtained weights as the initializers for the networks to discriminate between COVID-19, pneumonia, tuberculosis, and normal aspects. For DenseNets, we apply transfer learning twice. First, the ImageNet pretrained weights are used to train on the CheXpert dataset, which includes 14 common radiological observations (e.g., lung opacity, cardiomegaly, fracture, support devices). Then, the weights are used to initialize the network which detects COVID-19 and the three other classes. The resulting networks are compared in terms of how well they adapt to the small number of COVID-19 images. According to our quantitative and qualitative analysis, the resulting networks are more reliable compared to those obtained by direct training on the targeted dataset. Full article
(This article belongs to the Special Issue Advances in PDE-Based Methods for Image Processing)
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26 pages, 3764 KiB  
Article
A pde-Based Analysis of the Spectrogram Image for Instantaneous Frequency Estimation
by Vittoria Bruni, Michela Tartaglione and Domenico Vitulano
Mathematics 2021, 9(3), 247; https://doi.org/10.3390/math9030247 - 27 Jan 2021
Cited by 7 | Viewed by 2374
Abstract
Instantaneous frequency (IF) is a fundamental feature in multicomponent signals analysis and its estimation is required in many practical applications. This goal can be successfully reached for well separated components, while it still is an open problem in case of interfering modes. Most [...] Read more.
Instantaneous frequency (IF) is a fundamental feature in multicomponent signals analysis and its estimation is required in many practical applications. This goal can be successfully reached for well separated components, while it still is an open problem in case of interfering modes. Most of the methods addressing this issue are parametric, that is, they apply to a specific IF class. Alternative approaches consist of non-parametric time filtering-based procedures, which do not show robustness to destructive interference—the most critical scenario in crossing modes. In this paper, a method for IF curves estimation is proposed. The case of amplitude and frequency modulated two-component signals is addressed by introducing a spectrogram time-frequency evolution law, whose coefficients depend on signal IFs time derivatives, that is, the chirp rates. The problem is then turned into the resolution of a two-dimensional linear system which provides signal chirp rates; IF curves are then obtained by a simple integration. The method is non-parametric and it results quite robust to destructive interference. An estimate of the estimation error, as well as a numerical study concerning method sensitivity and robustness to noise are also provided in the paper. Full article
(This article belongs to the Special Issue Advances in PDE-Based Methods for Image Processing)
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16 pages, 2907 KiB  
Article
Feature Keypoint-Based Image Compression Technique Using a Well-Posed Nonlinear Fourth-Order PDE-Based Model
by Tudor Barbu
Mathematics 2020, 8(6), 930; https://doi.org/10.3390/math8060930 - 7 Jun 2020
Cited by 6 | Viewed by 2457
Abstract
A digital image compression framework based on nonlinear partial differential equations (PDEs) is proposed in this research article. First, a feature keypoint-based sparsification algorithm is proposed for the image coding stage. The interest keypoints corresponding to various scale-invariant image feature descriptors, such as [...] Read more.
A digital image compression framework based on nonlinear partial differential equations (PDEs) is proposed in this research article. First, a feature keypoint-based sparsification algorithm is proposed for the image coding stage. The interest keypoints corresponding to various scale-invariant image feature descriptors, such as SIFT, SURF, MSER, ORB, and BRIEF, are extracted, and the points from their neighborhoods are then used as sparse pixels and coded using a lossless encoding scheme. An effective nonlinear fourth-order PDE-based scattered data interpolation is proposed for solving the decompression task. A rigorous mathematical investigation of the considered PDE model is also performed, with the well-posedness of this model being demonstrated. It is then solved numerically by applying a consistent finite difference method-based numerical approximation algorithm that is next successfully applied in the image compression and decompression experiments, which are also discussed in this work. Full article
(This article belongs to the Special Issue Advances in PDE-Based Methods for Image Processing)
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