An Optimal Algorithm for the Straight Segment Approximation of Digital Arcs

In this paper, we define the straight segment approximation problem (SSAP) for a given digital arc as that of locating a minimum subset of vertices on the arc such that they form a connected sequence of digital straight segments. Sharaiha (Ph.D. thesis, Imperial College, London, 1991) introduced the compact chord property, and proved its equivalence to Rosenfeld′s chord property (IEEE Trans. Comput. C-23, 1974, 1264-1269). The SSAP is now constrained by the compact chord property, which offers a more convenient geometric representation than the chord property. We develop an O(n²) optimal algorithm for the solution of the SSAP using integer arithmetic. A relaxation of the problem is also presented such that the optimal number of vectors can be reduced according to a user definition. The original algorithm is adapted for the optimal solution of the relaxed problem. An extension to the relaxed problem is also addressed which finds a minimum level of relaxation such that the optimal number of vectors cannot be reduced.