Measuring the Related Properties of Linearity and Elongation of Point Sets

  • Milos Stojmenovic
  • Amiya Nayak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5197)


The concept of elongation is generally well understood. However, there is no clear, precise, mathematical definition of elongation in any dictionary we could find. We propose that the definition of elongation should overlap with the definition of linearity since we will show that these two measures produce results that are highly correlated when applied to different types of 2D shapes. Our experiments consist of testing known methods of linearity and elongation on sets of closed shapes contours, shapes whose areas are filled, and shapes with open contours. We tested each algorithm on 25 different shapes in each category. It was found that the Average Orientations linearity measure from [10] best correlates to the elongation measures found in literature. It has a correlation value of above 0.9 with measures of elongation for open and closed curves. Also, we have discovered that the standard measure of elongation, applied to its intended area based shapes, gives almost identical results when it is applied to just the boundary pixels of the same area based shapes. They are over .98 correlated. This leads to a new linearity/elongation measure which is fast, applicable to both open and closed shapes, is given by a closed formula, and highly agrees with existing measures.


Linearity elongation unordered point sets 


  1. 1.
    Freeman, H., Shapira, R.: Determining the Minimum-Area Encasing Rectangle for an Arbitrary Closed Curve. Comm. of the ACM 18, 409–413 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Horn, B.K.P.: Robot Vision. MIT Press, Cambridge (1986)Google Scholar
  3. 3.
    Jain, R., Kasturi, R., Schunck, B.G.: Machine Vision. McGraw-Hill, New York (1995)Google Scholar
  4. 4.
    Klette, R., Rosenfeld, A.: Digital Geometry. Morgan Kaufmann, San Francisco (2004)zbMATHGoogle Scholar
  5. 5.
    Martin, R.R., Stephenson, P.C.: Putting Objects into Boxes. Computer Aided Design 20, 506–514 (1988)CrossRefGoogle Scholar
  6. 6.
    Rosin, P.: Measuring shape: ellipticity, rectangularity, and triangularity. Machine Vision and Applications 14, 172–184 (2003)CrossRefGoogle Scholar
  7. 7.
    Stojmenovic, M., Nayak, A.: Measuring linearity of ordered point sets. In: Mery, D., Rueda, L. (eds.) PSIVT 2007. LNCS, vol. 4872, pp. 274–288. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Stojmenovic, M., Nayak, A.: Direct Ellipse Fitting and Measuring Based on Shape Boundaries. In: Mery, D., Rueda, L. (eds.) PSIVT 2007. LNCS, vol. 4872, pp. 221–235. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Stojmenovic, M., Nayak, A.: Shape based circularity measures of planar point sets. In: IEEE International Conference on Signal Processing and Communications, Dubai, UAE, November 24-27, pp. 1279–1282 (2007)Google Scholar
  10. 10.
    Stojmenovic, M., Nayak, A., Zunic, J.: Measuring linearity of a finite set of points. Pattern Recognition (to appear, 2008)Google Scholar
  11. 11.
    Stojmenovic, M., Zunic, J.: New measure for Shape Elongation. In: Martí, J., Benedí, J.M., Mendonça, A.M., Serrat, J. (eds.) IbPRIA 2007. LNCS, vol. 4478, pp. 572–579. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Weisstein, E.: Eccentricity, MathWorld. CRC Press LLC (1999),
  13. 13.
    Zunic, J., Kopanja, L., Fieldsend, J.E.: Notes on Shape Orientation where the Standard Method Does not Work. Pattern Recognition 39(5), 856–865 (2006)CrossRefzbMATHGoogle Scholar
  14. 14.
    Zunic, J., Stojmenovic, M.: Boundary based shape orientation. Journal of Mathematical Imaging and Vision (to appear, 2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Milos Stojmenovic
    • 1
  • Amiya Nayak
    • 1
  1. 1.SITEUniversity of OttawaOttawaCanada

Personalised recommendations