Linear versus Nonlinear Approximation of StepFunctions
Abstract
We investigate, from the point of view of a comparison of linear and nonlinear approximation methods, one specific example, where many of the important general phenomena become fairly apparent. This is the family of the step (or Heaviside) functions. The main problem we have in mind is the compression of images, still and moving. A remarkable fact about the curve of the step functions, which makes it very suitable for our problem, is it's universality: for example, it represents, up to an isomorphism, any nested family of compact subsets in the plane. Therefore, as videosequences are concerned, we can consider our curve as representing the motion in time of the objects boundaries. An accurate estimate of various "complexity measures" of this curve, like Kolmogorov's entropy and nwidth, as well as the "nonlinear width" recently introduced by V. Temlyakov, may be instructive. This is what we do in the present paper.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2007
 arXiv:
 arXiv:math/0701791
 Bibcode:
 2007math......1791E
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Computer Science  Computer Vision and Pattern Recognition;
 41A46 (Primary);
 94A08 (Secondary)
 EPrint:
 22 pages, 3 figures