Two Algorithms for Approximation in Highly Complicated Planar Domains

Motivated by an adaptive method for image approximation, which identifies ”smoothness domains” of the image and approximates it there, we developed two algorithms for the approximation, with small encoding budget, of smooth bivariate functions in highly complicated planar domains. The main application of these algorithms is in image compression. The first algorithm partitions a complicated planar domain into simpler subdomains in a recursive binary way. The function is approximated in each subdomain by a low-degree polynomial. The partition is based on both the geometry of the subdomains and the quality of the approximation there. The second algorithm maps continuously a complicated planar domain into a kdimensional domain, where approximation by one k-variate, low-degree polynomial is good enough. The integer k is determined by the geometry of the domain. Both algorithms are based on a proposed measure of domain singularity, and are aimed at decreasing it.