Abstract
Multi-resolution analysis and numerical precision problems are very important subjects in fields like image analysis or geometrical modeling. In the continuation of previous works of the authors, we expose in this article a new method called the \(\it \Omega\)-arithmetization. It is a process to obtain a multi-scale discretization of a continuous function that is a solution of a differential equation. The constructive properties of the underlying theory leads to algorithms which can be exactly translated into functional computer programs without uncontrolled numerical errors. An important part of this work is devoted to the definition and the study of the theoretical framework of the method. Some significant examples of applications are described with details.
Partially supported by the PPF GIC.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bridges, D.S.: Constructive mathematics: a foundation for computable analysis. Theoretical Computer Science 219, 95–109 (1999)
Bridges, D., Reeves, S.: Constructive mathematics, in theory and programming practice. Technical Report CDMTCS-068, Centre for Discrete Mathematics and Theorical Computer Science (1997)
Chollet, A., Wallet, G., Fuchs, L., Largeteau-Skapin, G., Andres, E.: Insight in discrete geometry and computational content of a discrete model of the continuum. Pattern Recognition 42, 2220–2228 (2009)
Diener, M.: Application du calcul de Harthong-Reeb aux routines graphiques. In: Salanskis, J.M., Sinaceurs, H. (eds.) Le Labyrinthe du Continu, pp. 424–435. Springer, Heidelberg (1992)
Diener, F., Reeb, G.: Analyse Non Standard. Hermann, Paris (1989)
Fuchs, L., Largeteau-Skapin, G., Wallet, G., Andres, E., Chollet, A.: A first look into a formal and constructive approach for discrete geometry using nonstandard analysis. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) DGCI 2008. LNCS, vol. 4992, pp. 21–32. Springer, Heidelberg (2008)
Harthong, J.: Éléments pour une théorie du continu. Astérisque 109/110, 235–244 (1983)
Harthong, J.: Une théorie du continu. In: Barreau, H., Harthong, J. (eds.) La mathématiques non standard, Éditions du CNRS, pp. 307–329 (1989)
INRIA: The caml language, http://www.ocaml.org
Laugwitz, D.: Leibniz’ principle and omega calculus. In: Salanskis, J., Sinacoeur, H. (eds.) Le Labyrinthe du Continu, pp. 144–155. Springer, France (1992)
Laugwitz, D.: Ω-calculus as a generalization of field extension. In: Hurd, A. (ed.) Nonstandard Analysis - Recent developments. Lecture Notes in Mathematics, pp. 144–155. Springer, Heidelberg (1983)
Laugwitz, D., Schmieden, C.: Eine Erweiterung der Infinitesimalrechnung. Mathematische Zeitschrift 89, 1–39 (1958)
Martin-Löf, P.: Constructive mathematics and computer programming. In: Logic, Methodology and Philosophy of Science VI, pp. 153–175 (1980)
Martin-Löf, P.: Intuitionnistic Type Theory. Bibliopolis, Napoli (1984)
Martin-Löf, P.: Mathematics of infinity. In: Martin-Löf, P., Mints, G. (eds.) COLOG 1988. LNCS, vol. 417, pp. 146–197. Springer, Heidelberg (1990)
Nelson, E.: Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society 83, 1165–1198 (1977)
Reveillès, J.P.: Géométrie discrète, Calcul en nombres entiers et algorithmique. PhD thesis, Université Louis Pasteur, Strasbourg, France (1991)
Reveillès, J.P.: Mathématiques discrètes et analyse non standard. In: Salanskis, J.M., Sinaceurs, H. (eds.) Le Labyrinthe du Continu, pp. 382–390. Springer, Heidelberg (1992)
Reveillès, J.P., Richard, D.: Back and forth between continuous and discrete for the working computer scientist. Annals of Mathematics and Artificial Intelligence, Mathematics and Informatic 16, 89–152 (1996)
Richard, A., Wallet, G., Fuchs, L., Andres, E., Largeteau-Skapin, G.: Arithmetization of a circular arc. In: DGCI 2009, Montreal, Canada (2009) (to be published)
Robinson, A.: Non-standard analysis, 2nd edn. American Elsevier, New York (1974)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chollet, A., Wallet, G., Fuchs, L., Andres, E., Largeteau-Skapin, G. (2009). Ω-Arithmetization: A Discrete Multi-resolution Representation of Real Functions. In: Wiederhold, P., Barneva, R.P. (eds) Combinatorial Image Analysis. IWCIA 2009. Lecture Notes in Computer Science, vol 5852. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10210-3_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-10210-3_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10208-0
Online ISBN: 978-3-642-10210-3
eBook Packages: Computer ScienceComputer Science (R0)