Journal of Mathematical Imaging and Vision

, Volume 53, Issue 1, pp 1–11 | Cite as

Measuring Linearity of Connected Configurations of a Finite Number of \(2D\) and \(3D\) Curves

  • Paul L. RosinEmail author
  • Jovanka Pantović
  • Joviša Žunić


We define a new linearity measure for a wide class of objects consisting of a set of of curves, in both \(2D\) and \(3D\). After initially observing closed curves, which can be represented in a parametric form, we extended the method to connected compound curves—i.e. to connected configurations of a number of curves representable in a parametric form. In all cases, the measured linearities range over the interval \((0,1],\) and do not change under translation, rotation and scaling transformations of the considered curve. We prove that the linearity is equal to \(1\) if and only if the measured curve consists of two straight line overlapping segments. The new linearity measure is theoretically well founded and all related statements are supported with rigorous mathematical proofs. The behavior and applicability of the new linearity measure are explained and illustrated by a number of experiments.


Shape Shape descriptors 2D Curves  3D Curves Compound curves Linearity measure Image processing 



J. Pantović and J. Žunić are also with the Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade. This work is partially supported by the Serbian Ministry of Science and Technology/projects OI174026/OI174008. Initial results of this paper were presented in [21]. We would like to thank the following for providing data used in this paper: Zicheng Liu (MSR Action3D Dataset), Andreu-García Gabriela (chicken pieces), St Paul’s Eye Unit, Royal Liverpool University Hospital (ARIA Retinal Image Archive).


  1. 1.
    Andreu, G., Crespo, A., Valiente, J.: Selecting the toroidal self-organizing feature maps (TSOFM) best organized to object recognition. In: Int. Conf. Neural Netw. 2, 1341–1346 (1997)Google Scholar
  2. 2.
    Benhamou, S.: How to reliably estimate the tortuosity of an animal’s path: Straightness, sinuosity, or fractal dimension. J. Theor. Biol. 229(2), 209–220 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Black, B., Perron, J., Burr, D., Drummond, S.: Estimating erosional exhumation on Titan from drainage network morphology. J. Geophys. Res. 117, E08006 (2012)Google Scholar
  4. 4.
    Chang, C., Lin, C.: LIBSVM: A library for support vector machines. ACM Trans. Intell. Syst. Technol. 2(3), 27:1–27:27 (2011)CrossRefGoogle Scholar
  5. 5.
    Chen, Y., Chen, W.: Morphology of Quanzhou city road network based on space syntax. Trop. Geogr. 6, 014 (2011)Google Scholar
  6. 6.
    Daliri, M., Torre, V.: Classification of silhouettes using contour fragments. Comput. Vis. Image Underst. 113(9), 1017–1025 (2009)CrossRefGoogle Scholar
  7. 7.
    DeCarlo, K., Shokri, N.: Effects of substrate on cracking patterns and dynamics in desiccating clay layers. W. Resour. Res. 50(4), 3039–3051 (2014)CrossRefGoogle Scholar
  8. 8.
    Flusser, J., Suk, T.: Pattern recognition by affine moment invariants. Pattern Recognit. 26, 167–174 (1993)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gautama, T., Mandić, D., Van Hulle, M.: Signal nonlinearity in fMRI: a comparison between BOLD and MION. IEEE Trans. Med. Images 22(5), 636–644 (2003)CrossRefGoogle Scholar
  10. 10.
    Gautama, T., Mandić, D., Van Hulle, M.: A novel method for determining the nature of time series. IEEE Trans. Biomed. Eng. 51(5), 728–736 (2004)CrossRefGoogle Scholar
  11. 11.
    Goo, B., Lim, C.: Thermal fatigue of cast iron brake disk materials. J. Mech. Sci. Technol. 26(6), 1719–1724 (2012)CrossRefGoogle Scholar
  12. 12.
    Haralick, R.: A measure for circularity of digital figures. IEEE Trans. Syst. Man Cybernet. 4, 394–396 (1974)zbMATHCrossRefGoogle Scholar
  13. 13.
    Hijazi, M., Coenen, F., Zheng, Y.: Retinal image classification using a histogram based approach. In: International Joint Conference on Neural Networks, pp 1–7 (2010)Google Scholar
  14. 14.
    Klette, R., Rosenfeld, A.: Digital Geometry. Morgan Kaufmann, San Francisko (2004)zbMATHGoogle Scholar
  15. 15.
    Lahlil, S., Li, W., Xu, J.: Crack patterns morphology of ancient Chinese wares. The Old Potter’s Alm. 18(1), 1–9 (2013)Google Scholar
  16. 16.
    Matoušek, J., Nešetril, J.: Invitation to Discrete Mathematics. Clarendon Press, Oxford (1998)zbMATHGoogle Scholar
  17. 17.
    Mollineda, R., Vidal, E., Casacuberta, F.: A windowed weighted approach for approximate cyclic string matching. In: International Conference on Pattern Recognition, vol. 4, pp. 188–191 (2002)Google Scholar
  18. 18.
    Neuhaus, M., Bunke, H.: Edit distance-based kernel functions for structural pattern classification. Pattern Recognit. 39(10), 1852–1863 (2006)zbMATHCrossRefGoogle Scholar
  19. 19.
    Peura, M., Iivarinen, J.: Efficiency of simple shape descriptors. In: Aspects of Visual Form Processing, pp. 443–451. World Scientific, Singapore (1997)Google Scholar
  20. 20.
    Proffitt, D.: The measurement of circularity and ellipticity on a digital grid. Pattern Recognit. 15(5), 383–387 (1982)CrossRefGoogle Scholar
  21. 21.
    Rosin, P., Pantović, J., Žunić, J.: Measuring linearity of closed curves and connected compound curves. In: 11th Asian Conference on Computer Vision, ACCV (3), Lecture Notes in Computer Science, vol. 7726, pp. 310–321. Springer, Brelin (2012)Google Scholar
  22. 22.
    Schmitz, S., Hjorth, J., Joemai, R., Wijntjes, R., et al.: Automated analysis of neuronal morphology, synapse number and synaptic recruitment. J. Neurosci. Methods 195(2), 185–193 (2011)CrossRefGoogle Scholar
  23. 23.
    Steger, C.: An unbiased detector of curvilinear structures. IEEE Trans. Patt. Anal. Mach. Intell. 20(2), 113–125 (1998)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Stojmenović, M., Nayak, A., Žunić, J.: Measuring linearity of planar point sets. Pattern Recognit. 41(8), 2503–2511 (2008)zbMATHCrossRefGoogle Scholar
  25. 25.
    Žunić, J., Martinez-Ortiz, C.: Linearity measure for curve segments. Appl. Math. Comput. 215(8), 3098–3105 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Žunić, J., Rosin, P.L.: Measuring linearity of open planar curve segments. Image Vis. Comput. 29(12), 873–879 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Paul L. Rosin
    • 1
    Email author
  • Jovanka Pantović
    • 2
  • Joviša Žunić
    • 3
  1. 1.School of Computer ScienceCardiff UniversityCardiffWales, UK
  2. 2.Faculty of Technical SciencesUniversity of Novi SadNovi SadSerbia
  3. 3.Computer ScienceUniversity of ExeterExeterUK

Personalised recommendations