Four-point transformation methods in approximation and the smoothing problems

Abstract

The main goal of the adaptive local strategy consists in reducing the complexity of computational problems. We propose a new approach to curve approximation and smoothing based on 4-point transformations or Discrete Projective Transform (DPT). In the framework of DPT, the variable point is related to three data points (accompanying points). The variable y-ordinate is expressed via the convolution of accompanying y-ordinates and weight functions that are defined as cross-ratio functions of four x-coordinates. DPT has some attractive properties (natural norming, scale invariance, threefold symmetry, and “4-point” orthogonality), which are useful in designing new algorithms. Diverse methods and algorithms based on DPT have been developed.