Piecewise Linear Approximations of Digitized Space Curves with Applications

Generating piecewise linear approximations of digitized or "densely sampled" curves is an important problem in image processing, pattern recognition, geometric modeling, and computer graphics. Even though much attention bas been paid to the planar curve case, little work has addressed space curve approximation. In this paper, we consider how to approximate an arbitrary digitized 3-D space CUnlC, made of n+l points, with m line segments. First, we present an algorithm which finds, using the dynamic programming technique, an optimal approximation that. minimizes t.he maximum distance error of each line segment. While it computes an optimal solution, the algorithm consumes 0(n ·logm) time and 0(n2 ·logm) space, which is excessive. We then introduce a heuristic algorithm which quickly computes a good approximation of an arbitrary space curve in O(N;/er .n) time and O(n) space. This heuristic algorithm is based upon the notions of curve length and spherical image which are the fundamental concepts describing intrinsic properties of space curves. Our heuristic algorithm consists of two parts, computation oj an initial approximation and iterative refinement of the approximation. The performances of the heuristic algorithm for selected test cases are examined. We apply this fast heuristic algorithm to adaptively linearize implicit space curve segments formed by the intersection of two algebraic surfaces and to adaptively polygonize implicit surface patches. We then use the approximations to adaptively construct binary space partitioning trees for a class of objects made by revolution. We show that the linear approximation of a curve can be naturally extended to linearly approximate some class of curved 3D objects in bsp trees wit.h well-balanced structures. The resulting trees appear to be near optimal in the light of e.xpected time for insertion into the tree.