Ω-Arithmetization: A Discrete Multi-resolution Representation of Real Functions

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.