On the application of Lehmer means in signal and image processing

ABSTRACT

This paper is devoted to the construction and analysis of some new non-linear subdivision and multiresolution schemes using the Lehmer means. Our main objective is to attain adaption close to discontinuities. We present theoretical, numerical results and applications for different schemes. The main theoretical result is related to the four-point interpolatory scheme, that we write as a perturbation of a linear scheme. Our aim is to establish a one-step contraction property that allows to prove the stability of the new scheme. Indeed with a one-step contraction property for the scheme of differences, it is possible to prove the stability of the 2D algorithm constructed using a tensor product approach. In this article, we also consider the associated three points cell-average scheme, that we will use to present some results for image compression, and a non-interpolatory scheme, that we will use to introduce an application to subdivision curves in 2D. These applications show that the use of the Lehmer mean is suitable for the design of subdivision schemes for the generation of curves and for image processing.

KEYWORDS:

• Point-values and cell-average multiresolution scheme
• non-linear subdivision scheme
• stability
• Gibbs phenomenon

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 41A05
• 41A10
• 97N50
• 65D05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was supported in part by Programa de Apoyo a la investigación de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia [grant number 20928/PI/18], by (MINECO/FEDER) [grant number MTM2015-64382-P], by Spanish Ministerio de Ciencia e Innovación (MCINN) [grant numbers MTM 2014-54388P, PGC2018-095896-B-C21 and MTM2017-83942-P].