Abstract
There is an increasing demand for a new measure of convexity for discrete sets for various applications. For example, the well-known measures for h-, v-, and hv-convexity of discrete sets in binary tomography pose rigorous criteria to be satisfied. Currently, there is no commonly accepted, unified view on what type of discrete sets should be considered nearly hv-convex, or to what extent a given discrete set can be considered convex, in case it does not satisfy the strict conditions. We propose a novel directional convexity measure for discrete sets based on various properties of the configuration of 0s and 1s in the set. It can be supported by proper theory, is easy to compute, and according to our experiments, it behaves intuitively. We expect it to become a useful alternative to other convexity measures in situations where the classical definitions cannot be used.
L.G. Nyúl was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. P. Balázs was supported by the OTKA PD100950 grant of the National Scientific Research Fund. The research of P. Balázs and T.S. Tasi was partially supported by the European Union and the State of Hungary co-financed by the European Social Fund under the grant agreement TÁMOP 4.2.4.A/2-11-1-2012-0001 (’National Excellence Program’) and under the grant agreement TÁMOP-4.2.2.A-11/1/KONV-2012-0073 (’Telemedicine-focused research activities on the field of Mathematics, Informatics and Medical sciences’).
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Balázs, P.: A benchmark set for the reconstruction of hv-convex discrete sets from horizontal and vertical projections. Discrete Appl. Math. 157, 3447–3456 (2009)
Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Medians of polyominoes: A property for the reconstruction. Int. J. Imag. Syst. Tech. 9, 69–77 (1998)
Boxter, L.: Computing deviations from convexity in polygons. Pattern Recogn. Lett. 14, 163–167 (1993)
Brunetti, S., Del Lungo, A., Del Ristoro, F., Kuba, A., Nivat, M.: Reconstruction of 4- and 8-connected convex discrete sets from row and column projections. Linear Algebra Appl 339, 37–57 (2001)
Chrobak, M., Dürr, C.: Reconstructing hv-convex polyominoes from orthogonal projections. Inform. Process. Lett. 69(6), 283–289 (1999)
Herman, G.T., Kuba, A. (eds.): Advances in Discrete Tomography and its Applications. Birkhäuser, Boston (2007)
Kuba, A., Nagy, A., Balogh, E.: Reconstruction of hv-convex binary matrices from their absorbed projections. Discrete Appl. Math. 139, 137–148 (2004)
Latecki, L.J., Lakamper, R.: Convexity rule for shape decomposition based on discrete contour evolution. Comput. Vis. Image Und. 73(3), 441–454 (1999)
Rahtu, E., Salo, M., Heikkila, J.: A new convexity measure based on a probabilistic interpretation of images. IEEE T. Pattern Anal. 28(9), 1501–1512 (2006)
Rosin, P.L., Zunic, J.: Probabilistic convexity measure. IET Image Process. 1(2), 182–188 (2007)
Sonka, M., Hlavac, V., Boyle, R.: Image Processing, Analysis, and Machine Vision, 3rd edn. Thomson Learning, Toronto (2008)
Stern, H.: Polygonal entropy: a convexity measure. Pattern Recogn. Lett. 10, 229–235 (1989)
Zunic, J., Rosin, P.L.: A New Convexity Measure for Polygons. IEEE T. Pattern Anal. 26(7), 923–934 (2004)
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Tasi, T.S., Nyúl, L.G., Balázs, P. (2013). Directional Convexity Measure for Binary Tomography. In: Ruiz-Shulcloper, J., Sanniti di Baja, G. (eds) Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications. CIARP 2013. Lecture Notes in Computer Science, vol 8259. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41827-3_2
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DOI: https://doi.org/10.1007/978-3-642-41827-3_2
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