Recursive voids for identifying a nonconvex boundary of a set of points in the plane

We introduce a method that identifies the boundary of a nonconvex shape of a set of points in the plane. The boundary of the shape is explored through finding empty regions recursively within a shell that encapsulates all of the points. Our algorithm is output sensitive and runs in linear O(@?n) time determined by the output parameter @?, which is proportional to the length of the nonconvex boundary measured by a threshold unit distance. The recursive nature of our algorithm allows a tree structure that characterizes the empty regions, and by complementarity, the nonconvex shape itself. We use a distance measure based on lowest common ancestor of a pair of nodes in this tree and define the complexity of a shape as the average of the distances between all pairs. We present computational results on data size, threshold, shape complexity and noise on a set of different nonconvex shapes.

[1]  Harith Alani,et al.  Voronoi-based region approximation for geographical information retrieval with gazetteers , 2001, Int. J. Geogr. Inf. Sci..

[2]  Antony Galton,et al.  Assigning Footprints to Dot Sets: An Analytical Survey , 2009, COSIT.

[3]  C. A. Murthy,et al.  Selection of alpha for alpha-hull in R2 , 1997, Pattern Recognit..

[4]  Robert E. Tarjan,et al.  Fast Algorithms for Finding Nearest Common Ancestors , 1984, SIAM J. Comput..

[5]  Antony Galton,et al.  What Is the Region Occupied by a Set of Points? , 2006, GIScience.

[6]  David G. Kirkpatrick,et al.  On the shape of a set of points in the plane , 1983, IEEE Trans. Inf. Theory.

[7]  Antony Galton,et al.  Efficient generation of simple polygons for characterizing the shape of a set of points in the plane , 2008, Pattern Recognit..

[8]  M. J. van Kreveld.  Finding the wood by the trees , 1998 .

[9]  Ray A. Jarvis,et al.  On the Identification of the Convex Hull of a Finite Set of Points in the Plane , 1973, Inf. Process. Lett..

[10]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

[11]  Maribel Yasmina Santos,et al.  Concave hull: A k-nearest neighbours approach for the computation of the region occupied by a set of points , 2007, GRAPP.

[12]  Jean-Charles Pinoli,et al.  Automatic Recognition of 2D Shapes from a Set of Points , 2011, ICIAR.

[13]  Se-Jong Oh,et al.  A New Concave Hull Algorithm and Concaveness Measure for n-dimensional Datasets , 2012, J. Inf. Sci. Eng..

[14]  Bidyut Baran Chaudhuri,et al.  A Novel Approach to Computation of the Shape of a Dot Pattern and Extraction of Its Perceptual Border , 1997, Comput. Vis. Image Underst..

[15]  Avi Arampatzis,et al.  Web-based delineation of imprecise regions , 2006, Comput. Environ. Urban Syst..

[16]  Mahmoud Melkemi,et al.  Shape Reconstruction from Unorganized Set of Points , 2010, ICIAR.

[17]  Alfred V. Aho,et al.  On Finding Lowest Common Ancestors in Trees , 1976, SIAM J. Comput..

[18]  Timothy M. Chan.  Optimal output-sensitive convex hull algorithms in two and three dimensions , 1996, Discret. Comput. Geom..

[19]  Bidyut Baran Chaudhuri,et al.  A split and merge procedure for polygonal border detection of dot pattern , 1999, Image Vis. Comput..

[20]  Afra Zomorodian,et al.  Conformal alpha shapes , 2005, Proceedings Eurographics/IEEE VGTC Symposium Point-Based Graphics, 2005..

[21]  P. J. Clark,et al.  Distance to Nearest Neighbor as a Measure of Spatial Relationships in Populations , 1954 .