Validation of Medial Axis Transform Objects

AbstractMedial Axis Transform (MAT) satisfies most of the demands of present day solid modeling and product development methods as it captures the geometric proximity of the boundary elements in a simple form. However, Euler type topology-verification equations are not available for MAT. Different features of 3D MAT like seam-end points, junction points, seams, etc. are described. Available algorithms are used to generate the 3D MATs. MATs available in the literature are also used. From the generated and available MATs, basic relations between number of seam-end points (NSEP), number of junction points (NJP) and number of seams (NS) are developed experimentally. These equations are generalized in such a way that they are valid for both 2D and 3D MATs. The applications of these equations include MAT validation and validation of MAT generation methods. Limitations of the work is presented. The relationships developed are experimental only and mathematical proof is not yet developed.

[1]  F. Chazal,et al.  Stability and Finiteness Properties of Medial Axis and Skeleton , 2004 .

[2]  David G. Kirkpatrick,et al.  Efficient computation of continuous skeletons , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[3]  Christoph M. Hoffmann,et al.  How to Construct the Skeleton of CSG Objects , 1990 .

[4]  Hong Qin,et al.  Medial axis extraction and shape manipulation of solid objects using parabolic PDEs , 2004, SM '04.

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

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

[7]  Dinesh Manocha,et al.  Accurate computation of the medial axis of a polyhedron , 1999, SMA '99.

[8]  Franco P. Preparata,et al.  The Medial Axis of a Simple Polygon , 1977, MFCS.

[9]  André Lieutier,et al.  Any open bounded subset of Rn has the same homotopy type as its medial axis , 2004, Comput. Aided Des..

[10]  F. Bookstein.  The line-skeleton , 1979 .

[11]  Benjamin B. Kimia,et al.  A formal classification of 3D medial axis points and their local geometry , 2000, Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662).

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

[13]  Harry Blum,et al.  An Associative Machine for Dealing with the Visual Field and Some of Its Biological Implications , 1962 .

[14]  H. Blum.  DISCUSSION PAPER: A GEOMETRY FOR BIOLOGY , 1974, Annals of the New York Academy of Sciences.

[15]  Ibrahim Zeid,et al.  CAD/CAM theory and practice , 1991 .

[16]  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.

[17]  André Lieutier,et al.  Any open bounded subset of Rn has the same homotopy type than its medial axis , 2003, SM '03.

[18]  Fritz B. Prinz,et al.  Skeletons for Representation and Reasoning in Engineering Applications , 2001, Engineering with Computers.

[19]  Nicholas M. Patrikalakis,et al.  Differential and Topological Properties of Medial Axis Transforms , 1996, CVGIP Graph. Model. Image Process..

[20]  H. Blum.  Biological shape and visual science (part I) , 1973 .

[21]  B. Gurumoorthy,et al.  Constructing medial axis transform of extruded and revolved 3D objects with free-form boundaries , 2005, Comput. Aided Des..

[22]  J. Brandt.  Theory and application of the skeleton representation of continuous shapes , 1992 .