Four-Point Transformation Methods in Approximation and the Smoothing Problems
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 Transforms (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, «4-point» orthogonality), which are useful in designing new algorithms. Diverse methods and algorithms based on DPT have been developed.