Extended grassfire transform on medial axes of 2D shapes

The medial axis is an important shape descriptor first introduced by Blum (1967)?1] via a grassfire burning analogy. However, the medial axes are sensitive to boundary perturbations, which calls for global shape measures to identify meaningful parts of a medial axis. On the other hand, a more compact shape representation than the medial axis, such as a "center point", is needed in various applications ranging from shape alignment to geography. In this paper, we present a uniform approach to define a global shape measure (called extended distance function, or EDF) along the 2D medial axis as well as the center of a 2D shape (called extended medial axis, or EMA). We reveal a number of properties of the EDF and EMA that resemble those of the boundary distance function and the medial axis, and show that EDF and EMA can be generated using a fire propagation process similar to Blum's grassfire analogy, which we call the extended grassfire transform. The EDF and EMA are demonstrated on many 2D examples, and are related to and compared with existing formulations. Finally, we demonstrate the utility of EDF and EMA in pruning medial axes, aligning shapes, and shape description.

[1]  Tamal K. Dey,et al.  Defining and computing curve-skeletons with medial geodesic function , 2006, SGP '06.

[2]  Markus Ilg,et al.  Voronoi skeletons: theory and applications , 1992, Proceedings 1992 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[3]  Franz-Erich Wolter.  Cut Locus and Medial Axis in Global Shape Interrogation and Representation , 1995 .

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

[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]  Micha Sharir,et al.  Computing the geodesic center of a simple polygon , 1989, Discret. Comput. Geom..

[7]  Micha Sharir,et al.  Computing the link center of a simple polygon , 1987, SCG '87.

[8]  Dinesh Manocha,et al.  Efficient computation of a simplified medial axis , 2003, SM '03.

[9]  Alfred M. Bruckstein,et al.  Pruning Medial Axes , 1998, Comput. Vis. Image Underst..

[10]  J. Brandt.  Convergence and continuity criteria for discrete approximations of the continuous planar skeleton , 1994 .

[11]  Sunghee Choi,et al.  The power crust , 2001, SMA '01.

[12]  Kaleem Siddiqi,et al.  Flux invariants for shape , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[13]  Joseph S. B. Mitchell,et al.  On the Continuous Fermat-Weber Problem , 2005, Oper. Res..

[14]  Alexandru Telea,et al.  Computing Multiscale Curve and Surface Skeletons of Genus 0 Shapes Using a Global Importance Measure , 2008, IEEE Transactions on Visualization and Computer Graphics.

[15]  Dinesh Manocha,et al.  Homotopy-preserving medial axis simplification , 2005, SPM '05.

[16]  Erin W. Chambers,et al.  A simple and robust thinning algorithm on cell complexes , 2010, Comput. Graph. Forum.

[17]  Gabriella Sanniti di Baja,et al.  On Medial Representations , 2008, CIARP.

[18]  Tamal K. Dey,et al.  Approximate medial axis as a voronoi subcomplex , 2002, SMA '02.

[19]  Dinesh Manocha,et al.  Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling, Beijing, China, June 4-6, 2007 , 2007, Symposium on Solid and Physical Modeling.