Dimensional operators for mathematical morphology on simplicial complexes

International audienceIn this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are widely used to represent multidimensional data, such as meshes, that are two dimensional complexes, or graphs, that can be interpreted as one dimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of digital structures, and is heavily used for many applications. However, mathematical morphology operators on simplicial complex spaces is not a concept fully developed in the literature. Specifically, we explore properties of the dimensional operators, small, versatile operators that can be used to define new operators on simplicial complexes, while maintaining properties from mathematical morphology. These operators can also be used to recover many morphological operators from the literature. Matlab code and additional material, including the proofs of the original properties, are freely available at~\url{https://code.google.com/p/math-morpho-simplicial-complexes.

Writing Reusable Digital Geometry Algorithms in a Generic Image Processing Framework

Digital Geometry software should reflect the generality of the underlying mathe- matics: mapping the latter to the former requires genericity. By designing generic solutions, one can effectively reuse digital geometry data structures and algorithms. We propose an image processing framework focused on the Generic Programming paradigm in which an algorithm on the paper can be turned into a single code, written once and usable with various input types. This approach enables users to design and implement new methods at a lower cost, try cross-domain experiments and help generalize resultsComment: Workshop on Applications of Discrete Geometry and Mathematical Morphology, Istanb : France (2010

A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

Homological Spanning Forests for Discrete Objects

Computing and representing topological information form an important part in many applications such as image representation and compression, classification, pattern recognition, geometric modelling, etc. The homology of digital objects is an algebraic notion that provides a concise description of their topology in terms of connected components, tunnels and cavities. The purpose of this work is to develop a theoretical and practical frame- work for efficiently extracting and exploiting useful homological information in the context of nD digital images. To achieve this goal, we intend to combine known techniques in algebraic topology, and image processing. The main notion created for this purpose consists of a combinatorial representation called Homological Spanning Forest (or HSF, for short) of a digital object or a digital image. This new model is composed of a set of directed forests, which can be constructed under an underlying cell complex format of the image. HSF’s are based on the algebraic concept of chain homotopies and they can be considered as a suitable generalization to higher dimensional cell complexes of the topological meaning of a spanning tree of a geometric graph. Based on the HSF representation, we present here a 2D homology-based framework for sequential and parallel digital image processing.Premio Extraordinario de Doctorado U

Morse frames

In the context of discrete Morse theory, we introduce Morse frames, which are maps that associate a set of critical simplexes to all simplexes. The main example of Morse frames are the Morse references. In particular, these Morse references allow computing Morse complexes, an important tool for homology. We highlight the link between Morse references and gradient flows. We also propose a novel presentation of the Annotation algorithm for persistent cohomology, as a variant of a Morse frame. Finally, we propose another construction, that takes advantage of the Morse reference for computing the Betti numbers in mod 2 arithmetic

From music to mathematics and backwards: introducing algebra, topology and category theory into computational musicology

International audienceDespite a long historical relationship between mathematics and music, the interest of mathematicians is a recent phenomenon. In contrast to statistical methods and signal-based approaches currently employed in MIR (Music Information Research), the research project described in this paper stresses the necessity of introducing a structural multidisciplinary approach into computational musicology making use of advanced mathematics. It is based on the interplay between three main mathematical disciplines: algebra, topology and category theory. It therefore opens promising perspectives on important prevailing challenges, such as the automatic classification of musical styles or the solution of open mathematical conjectures, asking for new collaborations between mathematicians, computer scientists, musicologists, and composers. Music can in fact occupy a strategic place in the development of mathematics since music-theoretical constructions can be used to solve open mathematical problems. The SMIR project also differs from traditional applications of mathematics to music in aiming to build bridges between different musical genres, ranging from contemporary art music to popular music, including rock, pop, jazz and chanson. Beyond its academic ambition, the project carries an important societal dimension stressing the cultural component of 'mathemusical' research, that naturally resonates with the underlying philosophy of the “Imagine Maths”conference series. The article describes for a general public some of the most promising interdisciplinary research lines of this project

The 'Life Machine': A Quantum Metaphor for Living Matter

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