Medial axis computation for planar free-form shapes

We present a simple, efficient, and stable method for computing-with any desired precision-the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the approximation of the boundary spline such that the medial axis is geometrically stable, and (2) the efficient decomposition of the domain into base cases where the medial axis can be computed directly and exactly. We solve these problems via spiral biarc approximation and a randomized divide & conquer algorithm.

[1]  Christoph Burnikel.  Rational points on circles , 1998 .

[2]  Pierre Alliez,et al.  Computational geometry algorithms library , 2008, SIGGRAPH '08.

[3]  Hwan Pyo Moon,et al.  MATHEMATICAL THEORY OF MEDIAL AXIS TRANSFORM , 1997 .

[4]  Bert Jüttler,et al.  Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics , 2006, Comput. Aided Des..

[5]  R. Brubaker.  Models for the perception of speech and visual form: Weiant Wathen-Dunn, ed.: Cambridge, Mass., The M.I.T. Press, I–X, 470 pages , 1968 .

[6]  V. Ralph Algazi,et al.  Continuous skeleton computation by Voronoi diagram , 1991, CVGIP Image Underst..

[7]  D. T. Lee,et al.  Medial Axis Transformation of a Planar Shape , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  Martin Held,et al.  Biarc approximation of polygons within asymmetric tolerance bands , 2005, Comput. Aided Des..

[9]  Alan E. Middleditch,et al.  Stable Computation of the 2D Medial Axis Transform , 1998, Int. J. Comput. Geom. Appl..

[10]  Hyeong In Choi,et al.  The Medial Axis Transform , 2002, Handbook of Computer Aided Geometric Design.

[11]  Francis Y. L. Chin,et al.  Finding the Medial Axis of a Simple Polygon in Linear Time , 1995, ISAAC.

[12]  Helmut Pottmann,et al.  Applications of Laguerre geometry in CAGD , 1998, Comput. Aided Geom. Des..

[13]  Jean-Daniel Boissonnat,et al.  Stability and Computation of Medial Axes - a State-of-the-Art Report , 2009, Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration.

[14]  R. Farouki,et al.  Voronoi diagram and medial axis algorithm for planar domains with curved boundaries — II: Detailed algorithm description , 1999 .

[15]  Martin Held,et al.  VRONI: An engineering approach to the reliable and efficient computation of Voronoi diagrams of points and line segments , 2001, Comput. Geom..

[16]  Jason M. Ribando,et al.  Apollonius' Problem: A Study of Solutions and Their Connections , 2004 .

[17]  Jian Liu,et al.  Computation of medial axis and offset curves of curved boundaries in planar domain , 2008, Comput. Aided Des..

[18]  Hyeong In Choi,et al.  New Algorithm for Medial Axis Transform of Plane Domain , 1997, CVGIP Graph. Model. Image Process..

[19]  B. Gurumoorthy,et al.  Constructing medial axis transform of planar domains with curved boundaries , 2003, Comput. Aided Des..

[20]  Chee-Keng Yap,et al.  AnO(n logn) algorithm for the voronoi diagram of a set of simple curve segments , 1987, Discret. Comput. Geom..

[21]  Deok-Soo Kim,et al.  Representing the Voronoi diagram of a simple polygon using rational quadratic Bézier curves , 1995, Comput. Aided Des..

[22]  Deok-Soo Kim,et al.  A sweepline algorithm for Euclidean Voronoi diagram of circles , 2006, Comput. Aided Des..

[23]  Rida T. Farouki,et al.  Degenerate point/curve and curve/curve bisectors arising in medial axis computations for planar domains with curved boundaries , 1998, Comput. Aided Geom. Des..

[24]  Jin J. Chou.  Voronoi diagrams for planar shapes , 1995, IEEE Computer Graphics and Applications.

[25]  D. Walton,et al.  Approximating smooth planar curves by arc splines , 1995 .

[26]  Gershon Elber,et al.  MATHSM: medial axis transform toward high speed machining of pockets , 2005, Comput. Aided Des..

[27]  Gershon Elber,et al.  Bisector curves of planar rational curves , 1998, Comput. Aided Des..

[28]  Wendelin L. F. Degen.  Exploiting curvatures to compute the medial axis for domains with smooth boundary , 2004, Comput. Aided Geom. Des..

[29]  D. Walton,et al.  Spiral arc spline approximation to a planar spiral , 1999 .

[30]  Josef Hoschek,et al.  Handbook of Computer Aided Geometric Design , 2002 .

[31]  Niklaus Wirth,et al.  Algorithms and Data Structures , 1989, Lecture Notes in Computer Science.

[32]  Otfried Cheong,et al.  The Voronoi Diagram of Curved Objects , 1995, SCG '95.

[33]  Franz Aurenhammer,et al.  Computational and Structural Advantages of Circular Boundary Representation , 2007, Int. J. Comput. Geom. Appl..