Reproducing 2D Implicit Curves with Sharp Features

Abstract

Implicit curves play an essential role in the societies of medicine, meteorology, geology, geo-physics, visualization and so on. In this paper, we propose an algorithm to visualize implicit curves and reproduce their sharp features in 2D plane. To access the subdivision cells of a user-defined 2D domain, our algorithm first creates a quadtree by using a top-down and adaptive quad-tree construction technique. In each cell, the method locates exact one feature point of the numerical field defined by the implicit function defining an implicit curve. A discrete optimization technique is employed to calculate the feature points. A dual mesh is subsequently constructed for the quadtree by taking the feature points as its vertices. Our algorithm approximates local part of the implicit curve in each cell of the dual mesh with a modified version of the marching squares method. Collecting all the approximations in the cells, our method finally reproduces the implicit curve with sharp features. Experiments show that our method can efficiently extract the sharp features of implicit curves, and it can work with various implicit curves with or without sharp features robustly.