On the stability of the PPH nonlinear multiresolution

Abstract

This letter is devoted to the stability of the so-called piecewise polynomial harmonic (PPH) multiresolution transform that belongs to the class of data dependent nonlinear multiresolution algorithms. The presentation of the PPH multiresolution as some specific perturbation of a linear multiresolution allows to establish a two step contraction property that leads first to a convergence result and finally to the stability.

On a nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards $C^{s}$ functions with $s>1$

“…Before making a detailed analysis of the properties of the new scheme S ppha we summarize the most important properties of the harmonic mean in the following proposition (properties 1 to 9 are proved in [5] and property 10 is straightforward).…”

“…For nonlinear subdivision schemes, few general results of convergence or stability are available; see for instance [5], [9], [12], [22], [10], [19] and [17].…”

The PCHIP subdivision scheme

“…In [2,4,1,15], the convergence and stability of the proposed schemes are systematically analyzed by using the following results.…”

“…[24,18] and references therein). The Power p schemes [2,4,14], as well as other related subdivision schemes considered in [1,15], can be written in the following general form…”

Nonlinear thresholding of multiresolution decompositions adapted to the presence of discontinuities

“…Different steps towards adaption near singularities has been proposed involving nonlinear prediction operators [2], [3], [5], [6], [7], [8], [9], [14], [16], [18].…”