3. Morphological Method Based on the Alpha Shape

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3. Morphological Method Based on the Alpha Shape University of Huddersfield Repository Lou, Shan Discrete algorithms for morphological filters in geometrical metrology Original Citation Lou, Shan (2013) Discrete algorithms for morphological filters in geometrical metrology. Doctoral thesis, University of Huddersfield. This version is available at http://eprints.hud.ac.uk/id/eprint/18103/ The University Repository is a digital collection of the research output of the University, available on Open Access. Copyright and Moral Rights for the items on this site are retained by the individual author and/or other copyright owners. Users may access full items free of charge; copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational or not-for-profit purposes without prior permission or charge, provided: • The authors, title and full bibliographic details is credited in any copy; • A hyperlink and/or URL is included for the original metadata page; and • The content is not changed in any way. For more information, including our policy and submission procedure, please contact the Repository Team at: [email protected]. http://eprints.hud.ac.uk/ DISCRETE ALGORITHMS FOR MORPHOLOGICAL FILTERS IN GEOMETRICAL METROLOGY SHAN LOU A thesis submitted to the University of Huddersfield in partial fulfilment of the requirements for the degree of Doctor of Philosophy The University of Huddersfield May 2013 1 COPYRIGHT STATEMENT i. The author of this thesis (including any appendices and/or schedules to this thesis) owns any copyright in it (the “Copyright”) and s/he has given The University of Huddersfield the right to use such Copyright for any administrative, promotional, educational and/or teaching purposes. ii. Copies of this thesis, either in full or in extracts, may be made only in accordance with the regulations of the University Library. Details of these regulations may be obtained from the Librarian. This page must form part of any such copies made. iii. The ownership of any patents, designs, trademarks and any and all other intellectual property rights except for the Copyright (the “Intellectual Property Rights”) and any reproductions of copyright works, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property Rights and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property Rights and/or Reproductions. 2 ABSTRACT In geometrical metrology, morphological filters are useful tools for the surface texture analysis and functional prediction. Although they are generally accepted and regarded as the complement to mean-line based filters, they are not universally adopted in practice due to a number of fatal limitations in their implementations — they are restricted to planar surfaces, uniform sampled surfaces, time-consuming and suffered from end distortions and limited sizes of structuring elements. A novel morphological method is proposed based on the alpha shape with the advantages over traditional methods that it enables arbitrary large ball radii, and applies to freeform surfaces and non-uniform sampled surfaces. A practical algorithm is developed based on the theoretical link between the alpha hull and morphological envelopes. The performance bottleneck due to the costly 3D Delaunay triangulation is solved by the divide-and-conquer optimization. Aiming to overcome the deficits of the alpha shape method that the structuring element has to be circular and the computation relies on the Delaunay triangulation, a set of definitions, propositions and comments for searching contact points is proposed and mathematically proved based on alpha shape theory, followed by the construction of a recursive algorithm. The algorithm could precisely capture contact points without performing the Delaunay triangulation. By correlating the convex hull and morphological envelopes, the Graham scan algorithm, originally developed for the convex hull, is modified to compute morphological profile envelopes with an excellent performance achieved. The three novel methods along with the two traditional methods are compared and analyzed to evaluate their advantages and disadvantages. The end effects of morphological filtration on open surfaces are discussed and four end effect correction methods are explored. Case studies are presented to demonstrate the feasibility and capabilities of using the proposed discrete algorithms. 3 ACKNOWLEDGEMENTS I would like to express my appreciation to all the people who explicitly and implicitly supported me during my doctoral research. First and foremost, I would present my deepest gratitude to my main supervisor Professor Xiangqian (Jane) Jiang, who has given me the precious opportunity to engage in the study of surface filtration techniques with sufficient supports on research and finance. Her proficiency and knowledge on surface metrology, coupled with the practical view on instrument and manufacture, are of tremendous help for this work. Special thanks go to my second supervisor, Professor Paul J. Scott, for providing me with a continuous stream of evaluable guidance and suggestion for my work. His extensive knowledge in mathematics and experiences in practical algorithm development are of paramount contribution to the completion of this thesis. I would also like to express my gratitude to Dr Wenhan Zeng, Dr Xiangchao Zhang, Dr Phillip Cooper, Dr Hussein Abdul-Rahman for guidance on mathematics and surface metrology, Dr Paul J. Bills for instruction on practical applications, and all other colleagues in the EPSRC Centre for advanced metrology. My thanks also go to my former employer, Wilcox Associates, Inc. Hexagon Metrology, especially the development manager of the portable group, Troy Johnson, and the team member, Andy Roberts, who led me into the world of geometrical metrology and raised my interests in algorithm development. Finally, I own my sincere thanks and appreciation to my parents Guangda Lou and Zhihua Wang, and my beloved wife Wenjuan Wu for their loving considerations and great confidence in me all through these years, with whom I can share my joy of success and from whom I can gain motivation, encouragement and comfort. 4 TABLE OF CONTENT COPYRIGHT STATEMENT ..................................................................................... 2 ABSTRACT .................................................................................................................. 3 ACKNOWLEDGEMENTS ........................................................................................ 4 TABLE OF CONTENT ............................................................................................... 5 LIST OF FIGURES ................................................................................................... 10 LIST OF TABLES ..................................................................................................... 15 PUBLICATION LIST ............................................................................................... 16 1. INTRODUCTION .............................................................................................. 17 1.1 Background ...................................................................................................... 17 1.2 Filtration techniques for geometrical metrology .............................................. 19 1.2.1 Motivation ................................................................................................ 19 1.2.2 Historical development ............................................................................ 21 1.2.3 Recent development ................................................................................. 24 1.3 Aims and objectives ......................................................................................... 26 1.4 Structure of thesis ............................................................................................ 27 2. LITERATURE REVIEW .................................................................................. 29 2.1 Mathematical morphology ............................................................................... 29 2.1.1 Morphological operations ........................................................................ 29 2.1.2 Granulometry ........................................................................................... 31 Word count: 33,736. 5 2.1.3 Morphological image processing techniques ........................................... 32 2.1.4 Morphological operations on sets and functions ..................................... 33 2.2 Morphological filters ....................................................................................... 34 2.2.1 Closing and opening filters ...................................................................... 35 2.2.2 Alternating symmetrical filters ................................................................ 39 2.2.3 Scale-space techniques ............................................................................. 40 2.3 Applications of morphological operations ....................................................... 41 2.3.1 Applications of the dilation and erosion operation .................................. 41 2.3.2 Applications of the closing and opening filter ......................................... 45 2.3.3 Applications of the alternating symmetrical
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