Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

Abstract

A nonlinear multiresolution scheme within Harten's framework [12], [13] is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.

The PCHIP subdivision scheme

“…Examples of non-linear, non-oscillatory subdivision schemes that fit this general framework are the ENO-WENO subdivision schemes described in [8,7,13] or the Piecewise Parabolic Hyperbolic (PPH) scheme in [2]. The Power p schemes [3,14,25] and the nonlinear monotonicity preserving schemes developed in [24], which do not fit within this framework, are also examples of non-oscillatory subdivision schemes.…”

“…[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…”

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

“…The order of approximation after one iteration of the interpolatory subdivision schemes described by (2) is given by the error formula of the interpolatory technique I[x, ·].…”

“…In recent years, several nonlinear subdivision schemes have been proposed (see e.g. [2] and references therein) in an attempt to avoid the Gibb's-like oscillatory behavior that occurs when data-independent interpolatory techniques are used for the successive refinement of discrete data that display a nearly discontinuous behavior. On the other hand, the need to obtain subdivision processes that preserve the shape of the original discrete data has been considered in [23? ], from a different perspective.…”