Faculty

Research

My research activities are in the areas of computational topology and imaging science. In the domain of computational topology, my research objectives are to develop efficient algorithms for

• the computation of topological invariants and structures such as the homologies and cohomologies of spaces and maps, and
• the computation, the characterization, and the representation of critical points and critical regions in scalar and vector fields by means of Morse theory and Conely index theory.

Topological invariants such as homology have been used recently in a wide variety of applications in domains such as dynamical systems, and image processing and recognition. In dynamics, typical problems are translated into problems in topology where invariants such as the Conley index are computed using homology algorithms. In digital image analysis, topological invariants are useful in shape description, indexation, and classification. Scalar and vector fields are used to represent data in different applications areas like geographic information systems and the charaterization of the critical points of the data constitutes a fundamental technique for the study of the important features of the data and its visualization. Other areas of applications include computer graphics, computer aided-design (CAD) and electrical engineering to name but a few. The necessity of improved algorithms for the computation of the structures mentioned above appears evident as new applications of computational topology arise in research for very large data sets. Although several algorithms and software packages have been developed for this purpose, there is still a lot of room for improvement as processing very large data sets is often very time and memory-consuming.

I currently supervise (and co-supervise with colleagues from the Université de Sherbrooke) a group of MSc and Ph.D. students working on these problems and on other problems directly related to computer vision such as shape description and recognition and segmentation using deformable models techniques. I am interested in expanding my research team and I welcome new students interested in working on these issues. I provide for my students an exciting research environment with lab facilities at Bishop’s University and at Université de Sherbrooke where I am a member of two research groups

• The Imaging and Vision Group – MOIVRE (MOdélisation en Imagerie, Vision et RÉseaux de neurones), and
• The Computational Topology Group – GRTC (Groupe de recherche en topologie computationnelle).

Publications

Computational Topology

• M. Allili, D. Corriveau, S. Derivière, and T. Kaczynski. Detecting Critical Regions in Multidimensional Data Sets via the Conley Index Approach. Submitted (2008, 30 pages) .
• M. Allili, D. Corriveau, S. Derivière, T. Kaczynski and A. Trahan. Discrete dynamical system framework for the topological study of lattice height data. Journal of Math. Imaging and Vision 28 (2) (2007), pp. 99-111.
• M. Allili, T. Kaczynski. Geometric construction of a coboundary of a cycle. Journal of Discrete and Computational Geometry 25 (2001), pp. 125-140.
• M. Allili, T. Kaczynski. An Algorithmic approach to the construction of homomorphisms induced by maps in homology. Journal of Transactions of the American Mathematical Society 352 (2000) , pp. 2261-2281.
• M. Allili. Une approche algorithmique pour le calcul de l’homologie de fonctions continues. Ph.D. Thesis, Université de Sherbrooke, 1999.

Image Processing

• R. Dedic, M. Allili. Adaptive Topology Preserving Deformable Model. Submitted (2008, 16 pages).
• R. Dedic, M. Allili, R. Lecomte, A. Benchakroun. Segmentation of Cardiac Images by the Force Field Driven Speed Term . Accepted in Proceedings of CVISP 2008, Prague, Czech Republic July 25-27, 2008.
• R. Dedic, M. Allili, R. Lecomte, A. Benchakroun. Segmentation of Cardiac Images by the Force Field Driven Speed Term . Accepted in Proceedings of 2008 Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC 2008), Dresden, Germany, October 19-25, 2008.
• D. Li, M. Allili. A Digital Topology-based Method for the Topological Filtering of a Reconstructed Surface . Submitted (2008, 13 pages).
• M. Allili, D. Corriveau. Topological Analysis of Shapes Using Morse Theory. Computer Vision and Image Understanding 105 (2007), pp. 188-199.
• R. Dedic, M. Allili. Intelligent topology preserving geometric deformable model. Proceedings of the 2nd International Conference on Computer Vision Theory and Applications VISAPP(1) (2007), pp 322-327.
• R. Dedic, M. Allili. Complete boundary detection of textured objects via deformable models. Proc. SPIE-IS&T Electronic Imaging , Vision Geometry XV, Vol. 6499 (2007), 64990H.
• S. Derdar, M. Allili, Djemel Ziou. Topological feature extraction using algebraic topology. Proc. SPIE-IS&T Electronic Imaging, Vision Geometry XV, Vol. 6499 (2007), 64990G.
• R. Dedic, M. Allili. A Deformable Model for Complete Boundary Detection. Proceedings of IEEE International Symposium on Industrial Electronics ISIE 2006, pp. 685-690.
• M. Allili, D. Corriveau. Descripteur topologique pour les formes basé sur la théorie de Morse. Proceedings of Congrès Reconnaissance des Formes et Intelligence Artificielle, RFIA 2006. Tours, France, January 2006.
• S. Derdar, M. Allili, D. Ziou. Image matching using algebraic topology. Proc. SPIE-IS&T Electronic Imaging , Vision Geometry XIV, Vol. 6066 (2006), 6066-21.
• M. Allili, B. Yang, L. Bentabet. A deformable model with topology analysis and adaptive clustering for boundary detection. Proc. SPIE-IS&T Electronic Imaging , Vision Geometry XIV, Vol. 6066 (2006), 6066-01.
• D. Corriveau, M. Allili, D. Ziou. Morse Connections Graph for Shape Representation. Proceedings Advanced Concepts for Intelligent Vision Systems ACIVS 2005, Lecture Notes in Computer Science 3708 Springer 2005, pp. 219-226.
• B. Yang, M. Allili. Geometric Deformable Models with Topology Analysis for Shape Modeling. Proceedings of Fifth IASTED International Conference VIIP 2005 (track: 480-253), pp. 660-665.
• M. Allili, D. Corriveau, D. Ziou. Efficient Topological Descriptor for Shape Representation. Proceedings of SPIE, Image Processing: Algorithms and Systems IV, vol. 5672, 2005, pp. 287-296.
• M. Allili, D. Corriveau, D. Ziou. Morse Homology Descriptors for Shape Characterization. International Conference on Pattern Recognition (ICPR 2004), vol. 4, 2004, pp 27-30.
• M.-F. Auclair-Fortier, D. Ziou, M. Allili. Global computational algebraic topology approach for diffusion. Proc. SPIE Int. Soc. Opt. Eng., vol. 5299, 357, 2004, pp. 357-368.
• D. Ziou, M. Allili. Image Modeling: new perspective for image processing and computer vision. Proc. SPIE Int. Soc. Opt. Eng., vol. 5299, 123, 2004, pp. 123-133.
• M. Allili, D. Ziou. Computational Homology Approach for Topology Descriptors and Shape Representation. Proceedings of International Conference on Image and Signal Processing (ICISP’2003), vol. 2, 2003, pp. 508-516.
• M.-F. Auclair-Fortier, D. Ziou, M. Allili. A Global CAT Approach for Graylevel Diffusion. 7th IEEE International Symposium on Signal Processing and its Applications (ISSPA 2003), vol. 1, 2003, pp. 453-456.
• M.-F. Auclair-Fortier, P. Poulin, D. Ziou, M. Allili. A Computational Algebraic Topology Approach for Diffusion Process. Proceedings of 3rd Workshop on Physics in Signal and Image Processing (PSIP 2003), 2003, pp. 77-80.
• D. Ziou, M. Allili. Generating Cubical Complexes from Image Data and Computation of the Euler Number. Journal of Pattern Recognition, 35-12, 2002, pp. 2833-2839.
• M.-F. Auclair-Fortier, P. Poulin, D. Ziou, M. Allili. A Computational Algebraic Topology Model for the Deformation of Curves . Proceedings of 2nd International Workshop on Articulated Motion and Deformable Objects (AMDO 2002), vol. 2492, 2002, pp. 56-67.
• P. Poulin, M.-F. Auclair-Fortier, D. Ziou, M. Allili. A Physics Based Model for the Deformation of Curves: A Computational Algebraic Topology Approach. Joint International Symposium on GeoSpatial Theory, Processing and Applications Symposium of the ISPRS Commission IV, WG5, 2002.
• M.-F. Auclair-Fortier, P. Poulin, D. Ziou, M. Allili. A Computational Algebraic Topology Approach for Optical Flow . International Conference on Pattern Recognition (ICPR 2002), vol. 1, 2002, pp. 347-355.
• M. Allili, D. Ziou. Topological Feature Extraction in Binary Images. 6th IEEE International Symposium on Signal Processing and its Applications (ISSPA 2001), vol. 2, 2001, pp. 651-654.
• M. Allili, K. Mischaikow, A. Tannenbaum. Cubical Homology and the Topological Classification of 2D and 3D Imagery. IEEE International Conference on Image Processing (ICIP 2001), v 2, 2001, pp. 173-176.

Dynamical Systems

• M. Allili, S. Day, O. Junge, and K. Mischaikow. A Rigorous Numerical Method for the Global Analysis of Infinite Dimensional Discrete Dynamical Systems. Preliminary Report, 2001.
• M. Allili, T. Kaczynski. Stability of Index Pairs for Multivalued Flows. Nonlinear Analysis, Vol. 30:7 (1997), pp. 4113-4122.
• M. Allili, T. Kaczynski. Stability of Index Pairs for Flows. Proceedings of the Conference “Topological Methods in Differential Equations and Dynamical Systems” (Krakow, Przegorzaly, 1996), Univ. Iagel. Acta Math. No. 36 (1998), pp. 173-175.