Asymptotic Analysis and Random Sampling of Digitally Convex Polyominoes

Recent work of Brlek et al. gives a characterization of digitally convex polyominoes using combinatorics on words. From this work, we derive a combinatorial symbolic description of digitally convex polyominoes and use it to analyze their limit properties and build a uniform sampler. Experimentally, our sampler shows a limit shape for large digitally convex polyominoes. Introduction In discrete geometry, a finite set of unit square cells is said to be a digitally convex polyomino if it is exactly the set of unit cells included in a convex region of the plane. We only consider digitally convex polyominoes up to translation. The perimeter of a digitally convex polyomino is that of the smallest rectangular box that contains it. The notion of digitally convex polyominoes arises naturally in the context of curvature estimators and consequently in the important field of pattern recognition [18, 10]. Brlek et al. [9] described a characterization of digitally convex polyominoes, in terms of words coding their contour. In this paper, we reformulate this characterization in the context of constructible combinatorial classes and we use it to build and analyze an algorithm to randomly sample digitally convex polyominoes. LIPN, Université Paris 13, 99, avenue Jean-Baptiste Clément 93430 Villetaneuse, France LaBRI, 351, cours de la Libération F-33405 Talence cedex, France LIPN, Université Paris 13, 99, avenue Jean-Baptiste Clément 93430 Villetaneuse, France Institute of Mathematics and Informatics Acad. Georgi Bonchev Str., Block 8 1113 Sofia, BULGARIA The usual definition of a polyomino requires the set to be connected, whereas digitally convex sets may be disconnected. However, one can coherently define some polygon as the boundary of any digitally convex polyomino; in the case of disconnected sets, this boundary will not be self-avoiding. 1 ha l-0 08 32 01 6, v er si on 1 9 Ju n 20 13 Author manuscript, published in "Discrete Applied Mathematics (2013) 1-23"