Abstract
This paper presents a formal framework for representing all reversible polygonalizations of a digital contour (i.e. the boundary of a digital object). Within these polygonal approximations, a set of local operations is defined with given properties, e.g., decreasing the total length of the polygon or diminishing the number of quadrant changes. We show that, whatever the starting reversible polygonal approximation, iterating these operations leads to a specific polygon: the Minimum Length Polygon. This object is thus the natural representative for the whole class of reversible polygonal approximations of a digital contour. Since all presented operations are local, we obtain the first dynamic algorithm for computing the MLP. This gives us a sublinear time algorithm for computing the MLP of a contour, when the MLP of a slightly different contour is known.
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References
Berstel, J., Lauve, A., Reutenauer, C., Saliola, F.V.: Combinatorics on words, CRM Monograph Series, vol. 27. American Mathematical Society, Providence (2009); christoffel words and repetitions in words
Borel, J.P., Laubie, F.: Quelques mots sur la droite projective réelle. J. Théor. Nombres Bordeaux 5(1), 23–51 (1993)
Coeurjolly, D., Klette, R.: A comparative evaluation of length estimators of digital curves. IEEE Trans. Pattern Analysis and Machine Intelligence 26(2), 252–258 (2004)
Damiand, G., Dupas, A., Lachaud, J.O.: Combining Topological Maps, Multi-Label Simple Points and Minimum Length Polygon for Efficient Digital Partition Model. In: Aggarwal, J.K., et al. (eds.) IWCIA 2011. LNCS, vol. 6636, pp. 56–69. Springer, Heidelberg (2011)
Feschet, F.: Fast guaranteed polygonal approximations of closed digital curves. In: Kalviainen, H., Parkkinen, J., Kaarna, A. (eds.) SCIA 2005. LNCS, vol. 3540, pp. 910–919. Springer, Heidelberg (2005)
Hobby, J.D.: Polygonal approximations that minimize the number of inflections. In: Proc. SODA, pp. 93–102 (1993)
Klette, R., Kovalevsky, V., Yip, B.: On length estimation of digital curves. In: Vision Geometry VIII, vol. 3811, pp. 117–128 (1999)
Klette, R., Rosenfeld, A.: Digital Geometry - Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)
Klette, R., Yip, B.: The length of digital curves. Machine Graphics Vision 9(3), 673–703 (2000)
Lindenbaum, M., Brückstein, A.: On recursive, O(N) partitioning of a digitized curve into digital straight segments. IEEE Trans. On Pattern Analysis and Machine Intelligence 15(9), 949–953 (1993)
Lothaire, M.: Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)
Montanari, U.: A note on minimal length polygonal approximation to a digitized contour. Communications of the ACM 13(1), 41–47 (1970)
Provençal, X., Lachaud, J.-O.: Two Linear-Time Algorithms for Computing the Minimum Length Polygon of a Digital Contour. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 104–117. Springer, Heidelberg (2009)
Roussillon, T., Tougne, L., Sivignon, I.: What does digital straightness tell about digital convexity? In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 43–55. Springer, Heidelberg (2009)
Said, M., Lachaud, J.-O., Feschet, F.: Multiscale discrete geometry. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 118–131. Springer, Heidelberg (2009)
Sklansky, J., Chazin, R.L., Hansen, B.J.: Minimum perimeter polygons of digitized silhouettes. IEEE Trans. Computers 21(3), 260–268 (1972)
Sloboda, F., Stoer, J.: On piecewise linear approximation of planar Jordan curves. J. Comput. Appl. Math. 55(3), 369–383 (1994)
Sloboda, F., Zat́ko, B.: On one-dimensional grid continua in R2. Tech. rep., Institute of Control Theory and Robotics, Slovak Academy of Sciences, Bratislava, Slovakia (1996)
Sloboda, F., Zat́ko, B.: On approximation of jordan surfaces in 3D. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 365–386. Springer, Heidelberg (2002)
Sloboda, F., Zat́ko, B., Stoer, J.: On approximation of planar one-dimensional continua. In: Klette, R., Rosenfeld, A., Sloboda, F. (eds.) Advances in Digital and Computational Geometry, pp. 113–160 (1998)
Smeulders, A.W.M., Dorst, L.: Decomposition of discrete curves into piecewise straight segments in linear time. In: Melter, R.A., Rosenfeld, A., Bhattacharya, P. (eds.) Vision Geometry. Contempory Mathematics, vol. 119, pp. 169–195. Am. Math. Soc., Providence (1991)
de Vieilleville, F., Lachaud, J.-O.: Digital deformable model simulating active contours. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 203–216. Springer, Heidelberg (2009)
Voss, K.: Discrete images, objects, and functions in \({\bf Z}\sp n\), Algorithms and Combinatorics, vol. 11. Springer, Berlin (1993)
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Lachaud, JO., Provençal, X. (2011). Dynamic Minimum Length Polygon. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds) Combinatorial Image Analysis. IWCIA 2011. Lecture Notes in Computer Science, vol 6636. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21073-0_20
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DOI: https://doi.org/10.1007/978-3-642-21073-0_20
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