On the Application of the Generalized Means to Construct Multiresolution Schemes Satisfying Certain Inequalities Proving Stability

• Published in: Mathematics (Basel) Vol. 9; no. 5; p. 533
• Main Authors: Amat, Sergio, Magreñan, Alberto, Ruiz, Juan, Trillo, Juan Carlos, Yañez, Dionisio F.
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-03-2021
• Subjects: Algorithms; Approximation; Data analysis; Data compression; Dimensional stability; Inequalities; Mathematical analysis; Mathematics; means; multiresolution; nonlinear; Operators; Reconstruction; stability analysis; subdivision scheme; Tensors
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math9050533

Multiresolution representations of data are known to be powerful tools in data analysis and processing, and they are particularly interesting for data compression. In order to obtain a proper definition of the edges, a good option is to use nonlinear reconstructions. These nonlinear reconstruction are the heart of the prediction processes which appear in the definition of the nonlinear subdivision and multiresolution schemes. We define and study some nonlinear reconstructions based on the use of nonlinear means, more in concrete the so-called Generalized means. These means have two interesting properties that will allow us to get associated reconstruction operators adapted to the presence of discontinuities, and having the maximum possible order of approximation in smooth areas. Once we have these nonlinear reconstruction operators defined, we can build the related nonlinear subdivision and multiresolution schemes and prove more accurate inequalities regarding the contractivity of the scheme for the first differences and in turn the results about stability. In this paper, we also define a new nonlinear two-dimensional multiresolution scheme as non-separable, i.e., not based on tensor product. We then present the study of the stability issues for the scheme and numerical experiments reinforcing the proven theoretical results and showing the usefulness of the algorithm.