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Research Papers

Data Processing for Medial Axis Computation Using B-Spline Smoothing

[+] Author and Article Information
Les A. Piegl

Department of Computer Science
and Engineering,
University of South Florida,
Tampa, FL 33620
e-mail: lpiegl@gmail.com

Parikshit Kulkarni

Synopsys, Inc.,
Mountain View, CA 94043
e-mail: Parikshit.Kulkarni@synopsys.com

Khairan Rajab

College of Computer Science
and Information Systems,
Najran University,
Najran 61441, Saudi Arabia
e-mail: khairanr@gmail.com

1Corresponding author.

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received January 9, 2014; final manuscript received June 25, 2014; published online September 1, 2014. Assoc. Editor: Vijay Srinivasan.

J. Comput. Inf. Sci. Eng 14(4), 041002 (Sep 01, 2014) (11 pages) Paper No: JCISE-14-1011; doi: 10.1115/1.4027991 History: Received January 09, 2014; Revised June 25, 2014

There has been much attention on sophisticated algorithm design to compute geometric arrangements with both time and space efficiency. The issue of robustness and reliability has also been the subject of some interest, although mostly at the level of theory rather than practice and commercial grade implementation. What seems to have received very little attention is the need to prepare the data for successful processing. It is almost universally assumed that the data are valid and well presented and the only real challenge is to come up with a clever way of computing the results with progressively smaller time and space bounds. The aim of this paper is to narrow this gap by focusing entirely on input data anomalies, how to prepare the data for error free computation and how to post process the results for dowstream computing. The medial axis computation, using VRONI (Held, 2001, “VRONI: An Engineering Approach to the Reliable and Efficient Computation of Voronoi Diagram of Points and Line Segments,” Comput. Geom.—Theory Appl., 18, pp. 95–123), is singled out as an example and it is shown that based on how the data are prepared, the results can be vastly different. We argue in this paper that the success of geometric computing depends equally on algorithm design as well as on data processing. VRONI (and most geometric algorithms) does not understand the concept of noise, gaps, or aliasing. It only sees a polygon and generates the medial axis accordingly. It is the job of the applications engineer to prepare the data so that the output is acceptable.

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References

Held, M., 2001, “VRONI: An Engineering Approach to the Reliable and Efficient Computation of Voronoi Diagram of Points and Line Segments,” Comput. Geom.—Theory Appl., 18(2), pp. 95–123. [CrossRef]
Piegl, L., 2005, “Knowledge-Guided Computation for Robust CAD,” Comput-Aided Des. Appl., 2(5), pp. 685–695. [CrossRef]
Blum, H., 1967, “A Transformation for Extracting New Descriptors of Shape,” Models for Perception of Speech and Visual Form, W.Dunn, ed., MIT, Cambridge, MA, pp. 362–380.
Au, C., 2013, “A Simple Algorithm for the Medial Axis Transform Computation,” Eng. Comput., 29(2), pp. 139–149. [CrossRef]
Chin, F., Snoeyink, J., and Wang, C.-A., 1995, “Finding the Medial Axis of a Simple Polygon in Linear Time,” Proceedings of the 6th Annual International Symposium Algorithms Computational, Lecture Notes in Computer Science, Springer-Verlag, Vol. 1004, pp. 382–391.
Lee, D. T., 1982, “Medial Axis Transformation of a Plane Shape,” IEEE Trans. Pattern Anal. Mach. Intell., PAMI-4(4), pp. 363–369. [CrossRef]
Preparata, F. P., 1977, “The Medial Axis of a Simple Polygon,” Proceedings of the 6th Mathematics Foundation of Computer Science, Lecture Notes in Computer Science, Springer-Verlag, Vol. 53, pp. 443–450.
Attali, D., and Montanvert, A., 1997, “Computing and Simplifying 2D and 2D Continuous Skeletons,” Comput. Vision Image Understanding, 67(3), pp. 261–273. [CrossRef]
Brandt, J. W., and Algazi, V. R., 1992, “Continuous Skeleton Computation by Voronoi Diagram,” CVGIP: Image Understanding, 55(3), pp. 329–338. [CrossRef]
Brandt, J. W., 1994, “Convergence and Continuity Criteria for Discrete Approximation of the Continuous Planar Skeletons,” CVGIP: Image Understanding, 59(1), pp. 116–124. [CrossRef]
Dey, T. K., and Zhao, W., 2003, “Approximating the Medial Axis From the Voronoi Diagram With a Convergence Guarantee,” Algorithmica, 38(1), pp. 179–200. [CrossRef]
Dey, T. K., and Zhao, W., 2004, “Approximate Medial Axis as a Voronoi Subcomplex,” Comput.-Aided Des., 36(2), pp. 195–202. [CrossRef]
Fabri, R., Estrozi, L. F., and Costa, L. F., 2002, “On Voronoi Diagrams and Medial Axes,” J. Math. Imaging Vision, 17(1), pp. 27–40. [CrossRef]
Yao, C., and Rokne, J., 1991, “A Straightforward Algorithm for Computing the Medial Axis of a Simple Polygon,” Int. J. Comput. Math., 39(1–2), pp. 51–60. [CrossRef]
Sheehy, D., Armstrong, C., and Robinson, D., 1996, “Shape Description by Medial Axis Construction,” IEEE Trans. Visualization Comput. Graph., 2(1), pp. 62–72. [CrossRef]
Dorado, R., 2009, “Medial Axis of a Planar Shape by Offset Self-Intersection,” Comput.-Aided Des., 41(12), pp. 1050–1059. [CrossRef]
Culver, T., Keyser, J., and Manocha, D., 1999, “Accurate Computation of the Medial Axis of a Polyhedron,” Proceedings of the Solid Modeling’99, pp. 179–190.
Hoffman, C., 1990, “How to Construct the Skeleton of CSG Objects,” Proceedings of Fourth IMA Conference the Mathematics of Surfaces, A.Bowyer and J.Davenport, eds., University of Bath, UK.
Aichholzer, O., Aigner, W., Aurenhammer, F., Hackl, T., Juttler, B., and Rabl, M., 2009, “Medial Axis Computation for Planar Free-Form Shapes,” Comput.-Aided Des., 41(5), pp. 339–349. [CrossRef]
Elber, G., Cohen, E., and Drake, S., 2005, “MATHSM: Medial Axis Transform Toward High Speed Machining of Pockets,” Comput.-Aided Des., 37(2), pp. 241–250. [CrossRef]
Choi, H. I., Han, C. Y., Moon, H. P., and Wee, N. S., 1997, “A New Algorithm for Medial Axis Transform of Plane Domain,” Graph. Models Image Process., 59(6), pp. 463–483. [CrossRef]
Choi, H. I., Choi, S. W., and Moon, H. P., 1997, “Mathematical Theory of the Medial Axis Transform,” Pac. J. Math., 181(1), pp. 57–88. [CrossRef]
Evans, G., Middleditch, A., and Miles, N., 1998, “Stable Computation of the 2D Medial Axis Transform,” Int. J. Comput. Geom. Appl., 8(5–6), pp. 577–598. [CrossRef]
Foskey, M., Lin, M. C., and Manocha, D., 2003, “Efficient Computation of a Simplified Medial Axis,” ASME J. Comput. Inf. Sci. Eng., 3(4), pp. 274–285. [CrossRef]
Degen, W. L. F., 2004, “Exploiting Curvatures to Compute the Medial Axis for Domains With Smooth Boundary,” Comput. Aided Geom. Des., 21(7), pp. 641–660. [CrossRef]
Cao, L., and Liu, J., 2008, “Computation of Medial Axis and Offset Curves of Curved Boundaries in Planar Domains,” Comput.-Aided Des., 40(4), pp. 465–475. [CrossRef]
Cao, L., Jia, Z., and Liu, J., 2009, “Computation of Medial Axis and Offset Curves of Curved Boundaries in Planar Domains Based on the Cesaro's Approach,” Comput. Aided Geom. Des., 26(4), pp. 444–454. [CrossRef]
Cao, L., Ba, W., and Liu, J., 2011, “Computation of the Medial Axis of Planar Domains Based on Saddle Point Programming,” Comput.-Aided Des., 43(8), pp. 979–988. [CrossRef]
Ramanathan, M., and Gurumoorthy, B., 2002, “Constructing Medial Axis Transform of Planar Domains With Curved Boundaries,” Comput.-Aided Des., 35(7), pp. 619–632. [CrossRef]
Vermeer, P. J., 1994, “Medial Axis Transform to Boundary Representation Conversion,” Ph.D. thesis, Purdue University, West Lafayette, IN.
Farouki, R. T., and Ramamurthy, R., 1998, “Degenerate Point/Curve and Curve/Curve Bisectors Arising in Medial Axis Computations for Planar Domains With Curved Boundaries,” Comput.-Aided Geom. Des., 15(6), pp. 615–635. [CrossRef]
Ramamurthy, R., and Farouki, R. T., 1999, “Voronoi Diagram and Medial Axis Algorithm for Planar Domains With Curved Boundaries—I: Theoretical Foundations,” J. Comput. Appl. Math., 102(1), pp. 119–141. [CrossRef]
Ramamurthy, R., and Farouki, R. T., 1999, “Voronoi Diagram and Medial Axis Algorithm for Planar Domains With Curved Boundaries—II: Detailed Algorithm Description,” J. Comput. Appl. Math., 102(2), pp. 253–277. [CrossRef]
Piegl, L., and Tiller, W., 1997, The NURBS Book, Springer-Verlag, New York.
Piegl, L., and Tiller, W., 2000, “Least-Squares NURBS Curve Fitting With Arbitrary End Derivatives,” Eng. Comput., 16(2), pp. 109–116. [CrossRef]
Artaechevarria, X., Muñoz-Barrutia, A., and Ortiz de Solorzano, C., 2007, “Restoration of Biomedical Images Using Locally Adaptive B-Spline Smoothing,” IEEE International Conference on Image Processing, ICIP 2007, San Antonio, TX, Sept. 16–19, pp. 425–428. [CrossRef]
Xu, G., Mourrain, B., Duvigneau, R., and Galligo, A., 2011, “Parameterization of Computational Domain in Isogeometric Analysis: Methods and Comparison,” Comput. Methods Appl. Mech. Eng., 200(23–24), pp. 2021–2031. [CrossRef]
Held, M., and Huber, S., 2009, “Topology-Oriented Incremental Computation of Voronoi Diagrams of Circular Arcs and Straight-Line Segments,” Comput.-Aided Des., 41(5), pp. 327–338. [CrossRef]
Smogavec, G., and Zalik, B., 2012, “A Fast Algorithm for Constructing Approximate Medial Axis of Polygons Using Steiner Points,” Adv. Eng. Software, 52, pp. 1–9. [CrossRef]

Figures

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Fig. 1

Systemic approach to robustness

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Fig. 2

Sparse data with lots of stair casing

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Fig. 3

Dense gridded data with aliasing

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Fig. 4

Noisy data with gaps and near overlaps

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Fig. 5

Good point stream with missing segments

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Fig. 6

Reconstruction of the vascular tree using commercial software (note the gaps throughout the STL model). Image courtesy of William L. Mondy.

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Fig. 7

Original data set of Fig. 2 with smoothed point overlaid (left) and smoothed data set (right)

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Fig. 8

Data in Fig. 4 superimposed with smoothed points (left) and smoothed points alone (right)

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Fig. 9

Data in Fig. 5 superimposed with smoothed points (left) and smoothed points alone (right)

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Fig. 10

B-spline curves approximating the smoothed data point to a high level of accuracy

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Fig. 15

Medial axis to the data in Fig. 5: original data (left), smoothed data (middle), and sampled data (right)

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Fig. 14

Medial axis to the data in Fig. 4: original data (left), smoothed data (middle), and sampled data (right)

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Fig. 13

Medial axis to the data in Fig. 3: original data (left), smoothed data (middle), and sampled data (right)

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Fig. 12

Medial axis to the data in Fig. 2: original data (left), smoothed data (middle), and sampled data (right)

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Fig. 11

Sampled points from the B-spline fit compared to the original data

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