A linear-time combinatorial algorithm to find the orthogonal hull of an object on the digital plane

A combinatorial algorithm to compute the orthogonal hull of a digital object imposed on a background grid is presented in this paper. The resolution and complexity of the orthogonal hull can be controlled by varying the grid size, which may be used for a multiresolution analysis of a given object. Existing algorithms on finding the convex hull are based on divide and conquer strategy, sweepline approach, etc., whereas the proposed algorithm is combinatorial in nature whose time complexity is linear on the object perimeter instead of the object area. For a larger grid size, the perimeter of an object decreases in length in terms of grid units, and hence the runtime of the algorithm reduces significantly. The algorithm uses only comparison and addition in the integer domain, thereby making it amenable to usage in real-world applications where speed is a prime factor. Experimental results including the CPU time demonstrate the elegance and efficacy of the proposed algorithm.

[1]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[2]  Luigi Atzori,et al.  JPEG2000-coded image error concealment exploiting convex sets projections , 2005, IEEE Transactions on Image Processing.

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

[4]  Helman I. Stern,et al.  Polygonal entropy: A convexity measure , 1989, Pattern Recognit. Lett..

[5]  Partha Bhowmick,et al.  TIPS: On Finding a Tight Isothetic Polygonal Shape Covering a 2D Object , 2005, SCIA.

[6]  Derick Wood,et al.  Fundamentals of Restricted-Orientation Convexity , 1996, Inf. Sci..

[7]  David G. Kirkpatrick,et al.  The Ultimate Planar Convex Hull Algorithm? , 1986, SIAM J. Comput..

[8]  David Avis,et al.  How good are convex hull algorithms? , 1995, SCG '95.

[9]  Sung-Jea Ko,et al.  A Novel Multiple Image Deblurring Technique Using Fuzzy Projection onto Convex Sets , 2009, IEEE Signal Processing Letters.

[10]  Ana Colubi,et al.  A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread , 2009, Inf. Sci..

[11]  Péter Balázs,et al.  On the Number of hv-Convex Discrete Sets , 2008, IWCIA.

[12]  Azriel Rosenfeld,et al.  Digital Picture Processing , 1976 .

[13]  Chak-Kuen Wong,et al.  On Some Distance Problems in Fixed Orientations , 1987, SIAM J. Comput..

[14]  Derick Wood,et al.  On the Definition and Computation of Rectlinear Convex Hulls , 1984, Inf. Sci..

[15]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[16]  Pierre Hansen,et al.  Extremal problems for convex polygons , 2005, J. Glob. Optim..

[17]  Paul L. Rosin,et al.  A new convexity measure for polygons , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[18]  GARRET SWART,et al.  Finding the Convex Hull Facet by Facet , 1985, J. Algorithms.

[19]  Mark H. Overmars,et al.  Scanline algorithms on a grid , 1988, BIT.

[20]  Ronald L. Graham,et al.  An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set , 1972, Inf. Process. Lett..

[21]  Milan Sonka,et al.  Image Processing, Analysis and Machine Vision , 1993, Springer US.

[22]  Mireille Bousquet-Mélou,et al.  A method for the enumeration of various classes of column-convex polygons , 1996, Discret. Math..

[23]  Bernard Chazelle,et al.  An optimal convex hull algorithm in any fixed dimension , 1993, Discret. Comput. Geom..

[24]  F. Bookstein,et al.  Morphometric Tools for Landmark Data: Geometry and Biology , 1999 .

[25]  Jie Wu,et al.  On constructing the minimum orthogonal convex polygon for the fault-tolerant routing in 2-D faulty meshes , 2005, IEEE Transactions on Reliability.

[26]  Hong Yan,et al.  Artifact reduction in compressed images based on region homogeneity constraints using the projection onto convex sets algorithm , 2002, IEEE Trans. Circuits Syst. Video Technol..

[27]  Partha Bhowmick,et al.  Finding the Orthogonal Hull of a Digital Object: A Combinatorial Approach , 2008, IWCIA.

[28]  Luciano da Fontoura Costa,et al.  Shape Analysis and Classification: Theory and Practice , 2000 .

[29]  Laurence Boxer.  Computing deviations from convexity in polygons , 1993, Pattern Recognit. Lett..

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

[31]  A. Enis Çetin,et al.  Restoration of error-diffused images using projection onto convex sets , 2001, IEEE Trans. Image Process..

[32]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[33]  Subhas C. Nandy,et al.  Recognition of largest empty orthoconvex polygon in a point set , 2010, Inf. Process. Lett..

[34]  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..