Arcs and Curves in Digital Pictures

ABSTRACT

Characterizations of digital “simple arcs” and “simple closed curves” are given. In particular, it is shown that the following are equivalent for sets S having more than four points: (1) S is a simple curve; (2) S is connected and each point of S has exactly two neighbors in S; (3) S is connected, has exactly one hole, and has no deletable points. It follows that if a “shrinking” algorithm is applied to a connected S that has exactly one hole, it shrinks to a simple curve.

REFERENCES

• 1: Azriel Rosenfeld, Connectivity in Digital Pictures, Journal of the ACM (JACM), v.17 n.1, p.146-160, Jan. 1970 [doi>10.1145/321556.321570]
• 2: J. C. Alexander , A. I. Thaler, The Boundary Count of Digital Pictures, Journal of the ACM (JACM), v.18 n.1, p.105-112, Jan. 1971 [doi>10.1145/321623.321634]
• 3: R. Stefanelli , A. Rosenfeld, Some Parallel Thinning Algorithms for Digital Pictures, Journal of the ACM (JACM), v.18 n.2, p.255-264, April 1971 [doi>10.1145/321637.321646]
• 4: J. P. Mylopoulos , T. Pavlidis, On the Topological Properties of Quantized Spaces, I. The Notion of Dimension, Journal of the ACM (JACM), v.18 n.2, p.239-246, April 1971 [doi>10.1145/321637.321644]
• 5: J. P. Mylopoulos , T. Pavlidis, On the Topological Properties of Quantized Spaces, II. Connectivity and Order of Connectivity, Journal of the ACM (JACM), v.18 n.2, p.247-254, April 1971 [doi>10.1145/321637.321645]
• 6: Solutionto Problem 5712. Amer. Math.Mon. 77,12 (Dec. 1970), 1119.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations

Additional Classification:
  I. Computing Methodologies
  I.4 IMAGE PROCESSING AND COMPUTER VISION

General Terms:
Algorithms, Design, Theory, Verification