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

Sergio Amat, Alberto Magreñan, Juan Ruiz, Juan Carlos Trillo and Dionisio F. Yañez
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Sergio Amat: Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain
Alberto Magreñan: Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain
Juan Ruiz: Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain
Juan Carlos Trillo: Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain
Dionisio F. Yañez: Departamento de Matemáticas, Universidad de Valencia, 46100 Valencia, Spain

Mathematics, 2021, vol. 9, issue 5, 1-15

Abstract: 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.

Keywords: nonlinear; means; multiresolution; subdivision scheme; data compression; stability analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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