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

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Abstract

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.

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Notes

  1. Compactness, eccentricity, fractal dimension, roundness, Hu’s moment invariants, affine moment invariants [8], circularity [12] and ellipticity [19, 20]. Several other global shape descriptors were considered, such as elongatedness, rectangularity, and convexity, but were not found to improve classification accuracy.

  2. We have post-processed the original vessel tracings by applying thresholding and thinning.

  3. Some of the images contain small portions of disconnected curves in addition to the main single compound connected curve. Since such curves do not satisfy the requirements of \(\mathcal{L}_{comp}(\mathcal{C})\) then it is possible that \(\mathcal{L}_{comp}(C) > 1\). However, in practice this did not occur for any of the retinal images (the largest computed value is 0.289). In any case, \(\mathcal{L}_{comp}(C) > 1\) can still be used as a shape descriptor that is invariant with respect to similarity transformations.

  4. It is difficult to compare classification accuracy for the ARIA data set against other methods as the majority of the literature using this data set concentrates on vessel segmentation rather than disease classification. Hijazi et al. [13] did perform classification, but only considered two classes (86 AMD and 56 normal images), achieving 75 % accuracy. Moreover, they treated the blood vessels as noise and eliminated them, analyzing instead image intensity histograms. In order to more directly compare our results with theirs, we removed the diabetic cases to create a two class problem, achieving 80.6 % accuracy. Since traced centerlines were not provided for the full image database, we also tested the full set of source images for the two classes (92 AMD and 60 normals), and extracted blood vessel centerlines using a general purpose ridge detector [23]. Although the quality of the vessel detection was not high it captured some important structural information from the images, and the classification accuracy obtained using \(\mathcal{L}_{comp}(\mathcal{C})\) was 82.2 %.

References

  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. 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)

    Article  MathSciNet  Google Scholar 

  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. Chang, C., Lin, C.: LIBSVM: A library for support vector machines. ACM Trans. Intell. Syst. Technol. 2(3), 27:1–27:27 (2011)

    Article  Google Scholar 

  5. Chen, Y., Chen, W.: Morphology of Quanzhou city road network based on space syntax. Trop. Geogr. 6, 014 (2011)

    Google Scholar 

  6. Daliri, M., Torre, V.: Classification of silhouettes using contour fragments. Comput. Vis. Image Underst. 113(9), 1017–1025 (2009)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  8. Flusser, J., Suk, T.: Pattern recognition by affine moment invariants. Pattern Recognit. 26, 167–174 (1993)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  11. Goo, B., Lim, C.: Thermal fatigue of cast iron brake disk materials. J. Mech. Sci. Technol. 26(6), 1719–1724 (2012)

    Article  Google Scholar 

  12. Haralick, R.: A measure for circularity of digital figures. IEEE Trans. Syst. Man Cybernet. 4, 394–396 (1974)

    Article  MATH  Google Scholar 

  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)

  14. Klette, R., Rosenfeld, A.: Digital Geometry. Morgan Kaufmann, San Francisko (2004)

    MATH  Google Scholar 

  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. Matoušek, J., Nešetril, J.: Invitation to Discrete Mathematics. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  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)

  18. Neuhaus, M., Bunke, H.: Edit distance-based kernel functions for structural pattern classification. Pattern Recognit. 39(10), 1852–1863 (2006)

    Article  MATH  Google Scholar 

  19. Peura, M., Iivarinen, J.: Efficiency of simple shape descriptors. In: Aspects of Visual Form Processing, pp. 443–451. World Scientific, Singapore (1997)

  20. Proffitt, D.: The measurement of circularity and ellipticity on a digital grid. Pattern Recognit. 15(5), 383–387 (1982)

    Article  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  23. Steger, C.: An unbiased detector of curvilinear structures. IEEE Trans. Patt. Anal. Mach. Intell. 20(2), 113–125 (1998)

    Article  MathSciNet  Google Scholar 

  24. Stojmenović, M., Nayak, A., Žunić, J.: Measuring linearity of planar point sets. Pattern Recognit. 41(8), 2503–2511 (2008)

    Article  MATH  Google Scholar 

  25. Žunić, J., Martinez-Ortiz, C.: Linearity measure for curve segments. Appl. Math. Comput. 215(8), 3098–3105 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Žunić, J., Rosin, P.L.: Measuring linearity of open planar curve segments. Image Vis. Comput. 29(12), 873–879 (2011)

    Article  Google Scholar 

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Acknowledgments

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).

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Correspondence to Paul L. Rosin.

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Rosin, P.L., Pantović, J. & Žunić, J. Measuring Linearity of Connected Configurations of a Finite Number of \(2D\) and \(3D\) Curves. J Math Imaging Vis 53, 1–11 (2015). https://doi.org/10.1007/s10851-014-0542-z

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