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.

Copyright © 2014 by ASME
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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. 11

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

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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. 13

Medial axis to the data in Fig. 3: 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. 15

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




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