Two linear-time algorithms for computing the minimum length polygon of a digital contour

The Minimum Length Polygon (MLP) is an interesting first order approximation of a digital contour. For instance, the convexity of the MLP is characteristic of the digital convexity of the shape, its perimeter is a good estimate of the perimeter of the digitized shape. We present here two novel equivalent definitions of MLP, one arithmetic, one combinatorial, and both definitions lead to two different linear time algorithms to compute them. This paper extends the work presented in Provencal and Lachaud (2009) [26], by detailing the algorithms and providing full proofs. It includes also a comparative experimental evaluation of both algorithms showing that the combinatorial algorithm is about 5 times faster than the other. We also checked the multigrid convergence of the length estimator based on the MLP.

[1]  John D. Hobby,et al.  Polygonal approximations that minimize the number of inflections , 1993, SODA '93.

[2]  Reinhard Klette,et al.  The Length of Digital Curves , 1999 .

[3]  Jack Sklansky,et al.  Minimum-Perimeter Polygons of Digitized Silhouettes , 1972, IEEE Transactions on Computers.

[4]  Laure Tougne,et al.  What Does Digital Straightness Tell about Digital Convexity? , 2009, IWCIA.

[5]  Avraham A. Melkman,et al.  On-Line Construction of the Convex Hull of a Simple Polyline , 1987, Inf. Process. Lett..

[6]  Fridrich Sloboda,et al.  On Approximation of Jordan Surfaces in 3D , 2000, Digital and Image Geometry.

[7]  Leonidas J. Guibas,et al.  Optimal shortest path queries in a simple polygon , 1987, SCG '87.

[8]  Reinhard Klette,et al.  Length estimation of digital curves , 1999, Optics & Photonics.

[9]  Ugo Montanari,et al.  A note on minimal length polygonal approximation to a digitized contour , 1970, CACM.

[10]  Jean-Pierre Reveillès.  Géométrie discrète, calcul en nombres entiers et algorithmique , 1991 .

[11]  Isabelle Debled-Rennesson,et al.  A Linear Algorithm for Segmentation of Digital Curves , 1995, Int. J. Pattern Recognit. Artif. Intell..

[12]  Harold Fredricksen,et al.  Necklaces of beads in k colors and k-ary de Bruijn sequences , 1978, Discret. Math..

[13]  J. Stoer,et al.  On piecewise linear approximation of planar Jordan curves , 1994 .

[14]  Filippo Mignosi,et al.  Some Combinatorial Properties of Sturmian Words , 1994, Theor. Comput. Sci..

[15]  Jean-Pierre Borel,et al.  Quelques mots sur la droite projective réelle , 1993 .

[16]  H. Dorksen-Reiter,et al.  Convex and Concave Parts of digital Curves , 2006 .

[17]  Laure Tougne,et al.  Optimal Time Computation of the Tangent of a Discrete Curve: Application to the Curvature , 1999, DGCI.

[18]  François de Vieilleville,et al.  Fast, accurate and convergent tangent estimation on digital contours , 2007, Image Vis. Comput..

[19]  M. Lothaire.  Applied Combinatorics on Words (Encyclopedia of Mathematics and its Applications) , 2005 .

[20]  Jean Pierre Duval,et al.  Factorizing Words over an Ordered Alphabet , 1983, J. Algorithms.

[21]  M. Lothaire.  Combinatorics on words: Bibliography , 1997 .

[22]  Azriel Rosenfeld,et al.  Digital geometry - geometric methods for digital picture analysis , 2004 .

[23]  M. Lothaire,et al.  Combinatorics on words: Frontmatter , 1997 .

[24]  Jacques-Olivier Lachaud,et al.  Two linear-time algorithms for computing the minimum length polygon of a digital contour , 2009, Discret. Appl. Math..

[25]  Jacques-Olivier Lachaud,et al.  Lyndon + Christoffel = digitally convex , 2009, Pattern Recognit..

[26]  L. Santaló.  Integral geometry and geometric probability , 1976 .

[27]  François de Vieilleville,et al.  Convex Shapes and Convergence Speed of Discrete Tangent Estimators , 2006, ISVC.

[28]  Aldo de Luca,et al.  Sturmian Words, Lyndon Words and Trees , 1997, Theor. Comput. Sci..

[29]  François de Vieilleville,et al.  Digital Deformable Model Simulating Active Contours , 2009, DGCI.

[30]  M. Lothaire,et al.  Applied Combinatorics on Words , 2005 .

[31]  Reinhard Klette,et al.  A Comparative Evaluation of Length Estimators of Digital Curves , 2004, IEEE Trans. Pattern Anal. Mach. Intell..