Measuring Linearity of Ordered Point Sets

  • Milos Stojmenovic
  • Amiya Nayak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4872)

Abstract

It is often practical to measure how linear a certain ordered set of points is. We are interested in linearity measures which are invariant to rotation, scaling, and translation. These linearity measures should also be calculated very quickly and be resistant to protrusions in the data set. No such measures exist in literature. We propose several such measures here: average sorted orientations, triangle sides ratio, and the product of a monotonicity measure and one of the existing measures for linearity of unordered point sets. The monotonicity measure is also a contribution here. All measures are tested on a set of 25 curves. Although they appear to be conceptually very different approaches, six monotonicity based measures appear mutually highly correlated (all correlations are over .93). Average sorted orientations and triangle side ratio appear as effectively different measures from them (correlations are about .8) and mutually relatively close (correlation .93). When compared to human measurements, the average sorted orientations and triangle side ratio methods prove themselves to be closest. We also apply our linearity measures to design new polygonal approximation algorithms for digital curves. We develop two polygonization algorithms: linear polygonization, and a binary search polygonization. Both methods search for the next break point with respect to a known starting point. The break point is decided by applying threshold tests based on a linearity measure.

Keywords

Linearity ordered point sets polygonization 

References

  1. 1.
    Csetverikov, D.: Basic algorithms for digital image analysis, Course, Institute of Informatics, Eotvos Lorand University, visual.ipan.sztaki.hu Google Scholar
  2. 2.
    Hogg, R.V., Tanis, E.A.: Probability and Statistical Inference. Prentice Hall, Englewood Cliffs (1997)Google Scholar
  3. 3.
    Marji, M., Siy, P.: Polygonal representation of digital planar curves through dominant point detection-a nonparametric algorithm. Pattern Recognition 37, 2113–2130 (2004)CrossRefGoogle Scholar
  4. 4.
    Rosin, P.: Measuring sigmoidality. In: Petkov, N., Westenberg, M.A. (eds.) CAIP 2003. LNCS, vol. 2756, pp. 410–417. Springer, Heidelberg (2003)Google Scholar
  5. 5.
    Rosin, P.: Measuring Rectangularity. Machine Vision and Applications 11, 191–196 (1999)CrossRefGoogle Scholar
  6. 6.
    Rosin, P.: Measuring shape: ellipticity, rectangularity, and triangularity. Machine Vision and Applications 14, 172–184 (2003)Google Scholar
  7. 7.
    Rosin, P.: Techniques for assessing polygonal approximations of curves. IEEE Transactions on Pattern Analysis and Machine Intelligence 19(6), 659–666 (1997)CrossRefGoogle Scholar
  8. 8.
    Rosin, P.: Assessing the behavior of polygonal approximation algorithms. Pattern Recognition 36, 505–518 (2003)CrossRefGoogle Scholar
  9. 9.
    Ray, B., Ray, S.: An algorithm for polygonal approximation of digitized curves. Pattern Recognition Letters 13, 489–496 (1992)CrossRefGoogle Scholar
  10. 10.
    Sarkar, D.: A simple algorithm for detection of significant vertices for polygonal approximation of chain-coded curve. Pattern Recognition Letters 14, 959–964 (1993)CrossRefGoogle Scholar
  11. 11.
    Sarfraz, M., Asim, M.R., Masood, A.: Piecewise polygonal approximation of digital curves. In: Proceedings of the Eigth International Conference on Information Visualization, vol. 14, pp. 991–996 (2004)Google Scholar
  12. 12.
    Sonka, M., Hlavac, V., Boyle, R.: Image Processing, Analysis, and Machine Vision, Chapman & Hall (1993) Google Scholar
  13. 13.
    Stojmenovic, M., Nayak, A., Zunic, J.: Measuring linearity of a finite set of points. In: IEEE International Conference on Cybernetics and Intelligent Systems (CIS), Bangkok, Thailand, June 7-9, pp. 222–227 (2006)Google Scholar
  14. 14.
    Teh, C., Chin, R.: On the detection of dominant points on digital curves. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(8), 859–872 (1989)CrossRefGoogle Scholar
  15. 15.
    Ventura, J., Chen, J.M.: Segmentation of two-dimensional curve contours. Pattern Recognition 25(10), 1129–1140 (1992)CrossRefGoogle Scholar
  16. 16.
    Zunic, J., Rosin, P.: Rectilinearity Measurements for Polygons. IEEE Transactions on Pattern Analysis and Machine Intelligence 25(9), 1193–1200 (2003)CrossRefGoogle Scholar
  17. 17.
    Zunic, J., Rosin, P.: A new convexity measure for polygons. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(7), 923–934 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Milos Stojmenovic
    • 1
  • Amiya Nayak
    • 1
  1. 1.SITE, University of Ottawa, Ottawa, Ontario, K1N 6N5Canada

Personalised recommendations