Official presentation of the project

Mathematical Methods in Image Processing under Uncertainty

in the frame of the project cycle PRIZMA

Science fund of the Republic of Serbia

Project team members

1) Nebojša M. Ralević (project manager), Faculty of Technical Sciences, University of Novi Sad.

PI. Nebojša Ralević is a full professor at the University of Novi Sad, Faculty of Technical Sciences, Novi Sad. His primary areas of research are: applied mathematics, fuzzy mathematics, numerical mathematics, machine learning, pattern recognition, image processing, shape analysis, measure theory, fixed point theory, computational geometry and application of deep learning methods. His recent researches are focused on different aspects of image processing especially image processing under uncertainty.

His scientific and educational contribution is verified through 44 books, over 50 papers published in the leading international, peerreviewed journals, numerous papers presented at scientific conferences and mentoring work on PhD thesis. He was a mentor of 13 defended doctoral dissertations and is currently leading several more dissertations on the project topic. Some members of the project team have worked with or collaborated with PI. PI is a reviewer in several national and international highly ranked scientific journals as well as scientific conferences.

Nebojša Ralević was also a researcher in many projects funded by the Ministry of Science and Technology Development of the Republic of Serbia, at least 5 projects related to proposed project topic, as well as in several national (regional) scientific projects related to his primary area of research.

The experience, expertise and competencies of PI in image processing will facilitate the project coordination and the coordination of proposed activities to achieve the set goals. PI has also been the PhD supervisor of some of the project participants, and has had a significant scientific cooperation with them.

2) Tatjana Došenović, Faculty of Technology, University of Novi Sad.

TM1. Tatjana Došenović is a full-time professor at the Faculty of Technology Novi Sad, University of Novi Sad, where she teaches two mathematics subjects to first-year undergraduate students, as well as a subject in doctoral studies. She is a member of the Serbian Scientific Mathematical Society and the European Women in Mathematics Association. She defended her doctoral dissertation entitled "The existence of a fixed point in fuzzy structures" in 2002. So far, she is the author or co-author of more than 80 scientific papers, of which 44 have been published in international and domestic scientific journals from the SCI list. The field of her scientific research is nonlinear analysis, fuzzy mathematics specially the theory of fixed points in fuzzy metric spaces, but also in metric spaces, probabilistic metric spaces, b-metric spaces, fuzzy b-metric spaces, b-metric like spaces and many others. The fixed point theory is interdisciplinary theory and its development was initiated and later branched out by a wide array application in different areas such as numerical, classical and functional analysis, topology, the existence theory of differential and integral equations, fractal and chaos theory, dynamic programming, game theory, approximation theory and other diverse disciplines of mathematical sciences including application of fixed point theory for compression of digital images. According to SCOPUS she is cited by 508 documents and her h-index is 11. She collaborates with over 70 scientists from the country and abroad. She is a reviewer in many national and international high ranked scientific journals.

3) Lidija Čomić, Faculty of Technical Sciences, University of Novi Sad.

TM2. Lidija Čomić is an associate professor at the Faculty of Technical Sciences, University of Novi Sad. Her primary areas of research are: applied mathematics, topological data analysis, image analysis and processing, computational and discrete geometry and topology. Currently, she is involved in research of application of computational and digital geometry and topology to image analysis.

In the period 2004–2015, she visited the Department of Informatics (DISI), University of Genova multiple times, where she worked with prof. Leila De Floriani and her geometric modeling group on applications of Morse and Forman theory to multiresolution and topological shape and image analysis.

She is an author or co-author of 13 scientific papers in high ranked international peer-reviewed journals, 25 international and 8 national conference papers, of which a considerable number is related to computational and digital geometry and topology in image analysis and processing. She has been involved in one national and one international project, related to her area of research.

Her competencies will be the basis for developing novel tools that will be applicable in various image processing tasks, based mostly on computational topology and geometry, as primary objective of the WP1. It is planned that a doctoral thesis of the TM9 will emerge from the direct research of WP1 and the project as a whole. Through the research, TM9 will be able to critically examine the problem from all aspects and successfully complete the work on the preparation of the doctoral dissertation.

4) Bratislav Iričanin, School of Electrical Engineering, University of Belgrade.

TM3. Bratislav Iričanin is an associate professor at the University of Belgrade, Faculty of Electrical Engineering, Department for Applied Mathematics, where he is presently professor in charge of teaching various Mathematical courses at all studying levels. His research field and areas are applied mathematics, difference equations, discrete dynamic systems, numerical analysis, image processing, fuzzy metric spaces. His recent interests are in difference equations, numerical mathematics and image processing.

He participated in International scientific collaboration and mobility, Department for Mathematics, at the Faculty of Electrical Engineering and Communication, at the Brno University of Technology, Czech Republic. He serves as President of National Committee for High School Students Competitions in Mathematics. TM3 is a reviewer in several (more than 40) international high ranked science journals.

He is an author of more than 140 scientific papers, including 60 ISI-SCI list journal papers, that are cited more than 1400 times. He participated in a few national and foreign scientific and research projects in basic and interdisciplinary research.

Together with TM5 and TM8 his role in WP2 will be modelling based on learning techniques for clustering problem and numerical performances.

5) Marija Paunović, Faculty of Hotel Management and Tourism, University of Kragujevac.

TM4. Marija Paunović is an assistant professor at the Faculty of Hotel Management and Tourism Vrnjačka Banja, University of Kragujevac. Her research field and areas are: applied mathematics, fuzzy mathematics, measure theory, uncertainty theory, fixed point. Her primary areaS of research are fuzzy systems and uncertainty measures. She has defended two doctoral theses, and become PhD of computer science and PhD of applied mathematics (under the supervision of PI). She has authored over 59 scientific papers, of which 15 are from the field of fuzzy mathematics and theory of uncertainty related to project topic, published in high ranked international peer-reviewed journals. Her scientific contribution is verified also through 6 books/workbooks and 3 chapters. She is a reviewer in several national and international high ranked scientific journals.

In addition to her work on faculty, she has extensive work experience as actuary consultant for many companies, as well as an adviser in the Federal Ministry for Science and later head of the Department for Project Planning at the Secretariat for Development and Science, where she supervised over 270 scientific projects. She participated in several projects.

She has proposed a new fuzzy measure, called c-credibility measure, and with PI she will continue working on its further development. In accordance with the research interest and expertise of TM4, her main role in the project will be application, implementation and experiments related to the project topic and WP4.

6) Nataša Milosavljević, Faculty of Agriculture, University of Belgrade.

TM5. Nataša Milosavljević is an assistant professor at the Faculty of Agriculture, University of Belgrade. Her research areas are: applied mathematics, optimization, metaheuristicS, metrics and classification. Her primary research is focused on the optimization and development of metrics based on formal and fuzzy logic. Her recent research is focused on work on hybrid methods that incorporate clustering techniques, metric and meta-heuristics to deal with the problem of optimization of neural networks without loss of generality.

She has authored a number of articles, 5 of which are closely related to the project topic (one of which is coauthored by PI), published in high ranked international peer-reviewed journals. She participated in several national and foreign scientific and research projects, including project No. III 44006 since 2011, CA19124 CIRCUL-A-BILITY, Rethinking Packaging for Circular and Sustainable Food Supply Chains of the Future, 2020, University Partnership Program, 2021 and head of national project Improvement of higher education, 2021. She has developed software solutions for different optimization problems, such as clustering problems when data is missing.

Her role is to coordinate activities in WP2 and to provide an expertise with solving clustering problem and big data. Together with TM3 and TM8, she will work on improving existing and suggesting new metrics, the development of software solution based on proposed hybrid model with the goal to deal with fuzzy metrics and numerical performances.

7) Ljubo Nedović, Faculty of Technical Sciences, University of Novi Sad.

TM6. Ljubo Nedović is an assistant professor at the University of Novi Sad, Faculty of Technical Sciences, Novi Sad. His research field and areas are applied mathematics, decision theory, measures and integrals, classification and image segmentation. His recent research is focused on aggregated distance functions and their application in image processing. He is an author and co-author of a number of scientific papers, some of which are coauthored with PI, and 5 of which are closely related to the project topic, published in highly ranked international peer-reviewed journals. He participated in several projects from his area of research, such as “Mathematical models for decision making under uncertain conditions and their application” from the Academy of Sciences and Arts of Vojvodina (marked 629/2005-01).

His competencies and expertise will be the basis for the research envisaged within the WP3. He is also the coordinator of WP3.

It is planned that a doctoral thesis of the TM10 will emerge from the direct research of WP3 and the project as a whole. By means of the research, to TM10 will be offered academic training for writing papers, presenting at conferences and publishing in journals. Mentoring TM10, his involvement in all phases of the research will enable TM10 to critically examine the problem of aggregated distance functions and their application in image processing and successfully complete the work on the preparation of the doctoral dissertation.

8) Vladimir Ilić, Faculty of Technical Sciences, University of Novi Sad.

TM7. Vladimir Ilić has received the Ph.D. degree in Applied Mathematics from the University of Novi Sad, Serbia, under the supervision of PI. He is currently working as assistant professor in mathematics at the Faculty of Technical Sciences, University of Novi Sad, Serbia. His research interests include applied mathematics, digital image processing, shape analysis, pattern recognition, fuzzy shape analysis, invariant features and digital shape representation. He has authored several scientific papers closely associated to the proposed project topic, published in high-ranked international peer-reviewed journals as well as numerous papers presented at scientific conferences. He has been involved as a member in national project “Numerical linear algebra and discrete structure”, Ministry of Education, Science and Technological Development, Number ON174019 (2011-2019).

His competencies will be of particular interest for studying new fuzzy shape descriptors and designing associated fuzzy shape measures closely related to the a task of WP4 and other work packages in providing support for devising experiments in diverse image processing and computer vision tasks. In accordance with the research interest and area of application, his main role within the project will be definition of the new fuzzy shape measure, their theoretical foundation and properties, designing and performing experiments related to the project topic, and in the narrower sense to WP4.

His PhD studies were completed under the supervision of PI, which serves as another indicator of the team’s compactness and unity.

9) Dejan Ćebić, Faculty of Mining and Geology, University of Belgrade.

TM8. Dejan Ćebić is an assistant professor at the Faculty of Mining and Geology, University of Belgrade. As a member of the Chair of Applied Mathematics and Informatics, he is responsible for several courses on the basic and master studies such as Mathematics 1, Mathematics 2, Probability and Statistics, Numerical Analysis, Selected Topics in Mathematics. He came to the Faculty of Mining and Geology after fourteen years of teaching experience at the Grammar School “Svetozar Marković“ in Novi Sad, where he was also partly employed as an assistant managing director. At the Faculty of Technical Sciences, University of Novi Sad, he obtained his Ph.D. in 2018 in applied mathematics. His professional interests are related (but not limited) to numerical analysis, mathematical modeling, fuzzy mathematics, image processing, artificial intelligence, etc.

He is an author and co-author of at least 5 scientific papers closely related to the project topic. Beside his scientific work he has strong skills in Artificial Intelligence and programming skills.

His main role in WP2 will be modelling based on learning techniques for clustering problem and numerical performances.

His PhD studies were completed under the supervision of PI, which serves as another indicator of the team’s compactness and unity.

10) Andrija Blesić, Faculty of Technical Sciences, University of Novi Sad.

TM9. Andrija Blesić is a teaching assistant at the Faculty of Technical Sciences, University of Novi Sad, and has been employed as a junior teaching assistant and teaching assistant since 2015. In 2017 he enrolled PhD studies of Applied Mathematics at the Faculty of Technical Sciences, under the supervision of TM2. During his studies, he successfully completed several exams where PI was the lecturing professor. His research interests are applied mathematics, topological data analysis and computational and digital geometry. He is a co-author of 6 conference papers with TM2, five of which are national, and one international. He has also been involved in 1 national project, “Advanced Techniques of Cryptology, Image Processing and Computational Topology for Information Security“, related to his area of research. His doctoral theses is expected to be largely based on his work with TM1 on research topics related to WP2, as well as the entire project.

11) Đorđe Dragić, Faculty of Technical Sciences, University of Novi Sad.

TM10. Đorđe Dragić is a teaching assistant at the Faculty of Technical Sciences, University of Novi Sad, and has been employed as a junior teaching assistant since 2017 and teaching assistant since 2019. In 2018 he enrolled PhD studies of Applied Mathematics at the Faculty of Technical Sciences, under the supervision of TM6. During his studies, he successfully completed several exams where PI was the lecturing professor. His research interests are applied mathematics, decision theory, measures and integrals, classification and image segmentation. He is a co-author of 5 national conference papers with TM6, one international conference paper and one scientific paper. He has also been involved in 1 national project, “Characterization of the kinetics and impact of the highly hazardous (emerging) pollutants of the waste streams of the graphic industry“, related to his area of research. His doctoral theses is expected to be largely based on his work with TM6 on research topics related to WP1, as well as the entire project.

Objectives of the project

The project "Mathematical Methods in Image Processing under Uncertainty" consists of three main parts.

The first part is related to computational and digital geometry and topology in image analysis and processing context. It discusses the following: Novel algorithms for moments computation; Detecting and drawing not self-crossing digital curves; Improvement of the performance of the existing 2D image repairing algorithm; Computation of the orthogonal hull of a 2D image; Applying persistent homology on related tasks; Studying the fixed point property (FPP) and its role and contribution in digital geometry and digital topology.

The second part of the project consists of: Examining the properties of distances (especially fuzzy metrics) constructed from initial distances using aggregation functions; Defining and constructing new classes of fuzzy shapes, appropriate fuzzy shape descriptors and associated measures, together with theoretical and empirical illustrations of the properties. All this new knowledge is used in image processing tasks, such as image compression through fixed point theory in fuzzy metric spaces.

The third part of the project consists of: Constructing hybrid models to support the decision uncertainty in image segmentation; Design and implementation of the Bee Colony Optimization (BCO) algorithm in software with hybrid model (based on learning techniques) for clustering problem; Optimizing Neural Networks on bioinformatics data. The examination and verification of the results will require the usage of digital images obtained from a number of databases (some of which are online and publicly available, and some of which are obtained with permission).

Members and objectives of the project team sections – work packages (WP)

1) WP1: Preparatory activities, project management and infrastructure setup

WP Coordinator: PI

Members: TM1-TM10

Objectivities

The first goal of this work package is to manage administrative aspects of the MaMIPU project. The management of the project is largely centralized at the lead SRO (FTNUNS), with support from all of the project team members. The objectives of this work package are to organize a kick-off meeting to prepare collaboration among partners.

The second goal of this work package is to manage administrative aspects of the MaMIPU project. The management of the project is largely centralized at the lead SRO (FTNUNS), with support from all of the project team members. The objectives of this work package are:

- Set up the management infrastructure and processes for the MaMIPU project and maintain them throughout the projects duration.

- Monitor the progress of the MaMIPU project and take appropriate actions if needed.

- Communicate with the governing body and prepare all project management documentation.

2) WP2: Computational and digital geometry and topology in image analysis and processing contexts

WP Coordinator: TM2

Members: TM1, TM9.

Objectivities

We plan to develop novel tools that will be applicable in various image processing tasks, based mostly on computational topology and geometry.

3) WP3: Improved metaheuristic through novel computer-aided mathematical models

WP Coordinator: TM5

Members: TM3, TM4, TM8.

Objectivities

Our objective is to further develop applications of the BCO algorithm, in this case, for segmentation images problems addressed in the project.

The second objective is to implement the BCO method for optimization of the segmentation images problems. One of the main goals is to find new distances based on fuzzy S and T metrics that will enable better performance in the problem of image segmentation. Improve existing algorithms by using the results obtained by this project from the fields of metrics, fixed points, computational topology and geometry.

4) WP4: Distance functions and fuzzy metrics in image processing

WP Coordinator: TM6

Members: PI, TM1, TM4, TM7, TM10.

Objectivities

Using appropriate classes of aggregation functions (e.g. average-based, product-based functions, etc.), the construction of new distance functions (using pre-existing ones as well as others derived by us) will be performed with particular reference to fuzzy metrics and measures with appropriate experimental confirmation of the properties of the metrics thus obtained. Experimental confirmation will be performed in the tasks of image segmentation, image denoising, shape recognition and other image processing-based applications such as computer vision and machine learning. Besides, it is also planned to define new classes of fuzzy shapes, appropriate pixel and shape descriptors and associated measures from the theoretical stand point, as well as to examine their properties with an emphasis on application in various tasks of image processing and analysis challenges. It is expected that such defined descriptors will significantly contribute to improve performance of diverse image processing and computer vision algorithms using machine learning routines such as recognition, matching, registration and classification of images (i.e., objects in the image).

Banach Fixed Point Theorem (BCP), as one of the most important theorems in fixed point theory is a very important test for solving some problems in mathematics and engineering. One of the most recent applications of BCP is in image processing using digital metric spaces. One of the main goals is to investigate the possibility of generalizing the digital metric space using uncertainty, where different fuzzy metrics and aggregation functions especially triangular norms, would be used. There are many FPT results in the literature and many of them generalize and extend the BCP (for example the Wardovski, Ćirić, Suzuki contraction, etc.) and it will be very useful to focus our research on the possibilities of applying the above mentioned theorems in digital metrics with application in image processing.

5) WP5: Implementation and experiments

WP Coordinator: PI

Members: PI, TM1-TM10.

Objectivities

Implementing new algorithms for the classification and segmentation of medical images.

6) WP6: Dissemination and exploitation of results

WP Coordinator: TM1

Members: PI, TM1-TM10.

Objectivities

The goal of the work package is to promote the project and increase its visibility among the scientific community and industry in areas where decision-making and image processing is used, especially in relation to medical and other digital images. The main objectives of the dissemination and exploitation of project results are defined in line with the overall project objectives. Additional, scientific cooperation is coordinated and organized and publication is regulated.

Activities and results