Research - Dr. Madjid Allili
Dr. Madjid Allili
Department of Mathematics
Dr. Allili's research focuses on the development of algebraic topological tools to analyze data sets, structures, and systems in computer science and mathematics. His aim is to produce highly efficient algorithms to be used in shape recognition and representation, image enhancement and segmentation. His work, focusing on digital image processing and dynamical systems, applies to an array of technical fields from medicine to forensic science to digital entertainment. His research can be used to help doctors recognize the size and shape of tumors through medical image data, to help police recognize and match crime-scene fingerprints, and to allow satellite television companies to improve image clarity. It also applies to computer graphics, robotics, military target detection, engineering design, and astronomical observation.
Topology theory is an area of mathematics that studies shapes and continuous deformations of shapes. Mathematicians classify and categorize shapes by identifying their special properties, called topological invariants. As the name suggests, topological invariants remain constant even when an object's appearance changes due to orientation, change of position, noise, and other distortions. The relative stability of topological invariants makes them useful in image processing - including pattern and object recognition, document analysis, and defects inspection - and in dynamical systems that may appear chaotic in numerical experiments. Furthermore, topology-based methods allow for less data to characterize and represent objects.
Dr. Allili's specific research objectives are to develop algorithms for the computations of algebraic topological structures such as the (co)homologies of spaces and maps and to use these structures to tackle problems in image processing and dynamical systems. Within the area of image processing Dr. Allili will combine classical methods to obtain segmented and registered image data then analyze their topological properties using recently introduced cubical homology methods. He aims to provide a complete image model based on algebraic topology. His study of dynamical systems is connected with the Conley index theory, a theory defined through algebraic topology structures that can prove the existence of specific solutions and analyze their long-term behaviour or formally prove chaotic dynamics in systems that appear chaotic in numerical experiments.
Partners
Groupe de recherche MOIVRE (Modélisation en Imagerie, Vision et Réseaux de neurones)/Computer Vision and Image Processing Research Group, Université de Sherbrooke
Kanstantin Mischaikow, Mathematics, Georgia Institute of Technology
Allen Tannenbaum, Electrical Engineering, Georgia Institute of Technology
Bill Kalies, Mathematics, Florida Atlantic University
Oliver Junge, Applied Mathematics, University of Paderborn, Germany
