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Convex Digital Polygons, Maximal Digital Straight Segments and Convergence of Discrete Geometric Estimators

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Abstract

Discrete geometric estimators approach geometric quantities on digitized shapes without any knowledge of the continuous shape. A classical yet difficult problem is to show that an estimator asymptotically converges toward the true geometric quantity as the resolution increases. For estimators of local geometric quantities based on Digital Straight Segment (DSS) recognition this problem is closely linked to the asymptotic growth of maximal DSS for which we show bounds both about their number and sizes on Convex Digital Polygons. These results not only give better insights about digitized curves but indicate that curvature estimators based on local DSS recognition are not likely to converge. We indeed invalidate a conjecture which was essential in the only known convergence theorem of a discrete curvature estimator. The proof involves results from arithmetic properties of digital lines, digital convexity, combinatorics and continued fractions.

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Authors and Affiliations

  1. LaBRI, Univ. Bordeaux 1, 351 cours de la Libération, 33405, Talence, France

    François De Vieilleville & Jacques-Olivier Lachaud

  2. LAIC, Univ. Clermont 1, IUT, Campus des Cézeaux, 63172, Aubière Cedex, France

    Fabien Feschet

Authors
  1. François De Vieilleville
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  2. Jacques-Olivier Lachaud
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  3. Fabien Feschet
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Corresponding author

Correspondence to François De Vieilleville.

Additional information

de Vieilleville François is about to defend his Ph.D. in computer science at the University of Bordeaux I (France) and is currently teacher assistant in the same university. His research interests are directed on theoretical and applied subjects on Image Analysis and Computer Vision. He is particularly interested in the problem of designing estimators based on digital geometry and digital straight segment recognition.

Jacques-Olivier Lachaud graduated from ENSIMAG engineering school in Computer Science in 1994 and received a Ph.D. degree in computer science from Joseph Fourier—Grenoble 1 University in 1998. He is currently an associate Professor in Computer Science at the university Bordeaux 1 (France) and works in the LaBRI laboratory. His research interests are in image segmentation and analysis, more specifically deformable models, energy-minimizing techniques, digital geometry, topological models and invariants. He has written about thirty papers in international journals or conferences on these topics.

Fabien Feschet got a Ph.D. in Computer Science both at the University Claude Bernard (Lyon, France) and at the INSA of Lyon in 1999. He was a professor assistant at the University of Lyon from 1999 to 2002. In 2002, he moved to Clermont-Ferrand at the LAIC laboratory. Since 2006, he has the Habilitation in Computer Science and is an associate professeur at the University of Auvergne. His research interests concern mainly digital geometry, pattern recognition and structural analysis of shapes with an emphasis on mathematical methods. He has currently written around 30 papers in International Journals and Conferences.

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De Vieilleville, F., Lachaud, JO. & Feschet, F. Convex Digital Polygons, Maximal Digital Straight Segments and Convergence of Discrete Geometric Estimators. J Math Imaging Vis 27, 139–156 (2007). https://doi.org/10.1007/s10851-007-0779-x

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