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Robust fitting of Zernike polynomials to noisy point clouds defined over connected domains of arbitrary shape

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dc.contributor.authorRodríguez Ibáñez, Diego
dc.contributor.authorGómez Pedrero, José Antonio
dc.contributor.authorAlonso Fernández, José
dc.contributor.authorQuiroga Mellado, Juan Antonio
dc.date.accessioned2023-06-18T06:53:10Z
dc.date.available2023-06-18T06:53:10Z
dc.date.issued2016-03-21
dc.descriptionEn abierto en la web del editor. Received 7 Oct 2015; revised 4 Feb 2016; accepted 9 Feb 2016; published 9 Mar 2016 © 2016 Optical Society of America. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibited.
dc.description.abstractA new method for fitting a series of Zernike polynomials to point clouds defined over connected domains of arbitrary shape defined within the unit circle is presented in this work. The method is based on the application of machine learning fitting techniques by constructing an extended training set in order to ensure the smooth variation of local curvature over the whole domain. Therefore this technique is best suited for fitting points corresponding to ophthalmic lenses surfaces, particularly progressive power ones, in non-regular domains. We have tested our method by fitting numerical and real surfaces reaching an accuracy of 1 micron in elevation and 0.1 D in local curvature in agreement with the customary tolerances in the ophthalmic manufacturing industry.
dc.description.departmentDepto. de Óptica
dc.description.departmentSección Deptal. de Óptica (Óptica)
dc.description.facultyFac. de Ciencias Físicas
dc.description.facultyFac. de Óptica y Optometría
dc.description.refereedTRUE
dc.description.sponsorshipMinisterio de Economía y Competitividad (MINECO)
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/37782
dc.identifier.doi10.1364/OE.24.005918
dc.identifier.issn1094-4087
dc.identifier.officialurlhttps://www.osapublishing.org/oe/abstract.cfm?uri=oe-24-6-5918
dc.identifier.urihttps://hdl.handle.net/20.500.14352/24496
dc.issue.number6
dc.journal.titleOptics Express
dc.language.isoeng
dc.page.final5933
dc.page.initial5918
dc.publisherThe Optical Society Of America
dc.relation.projectIDDPI2012-36103
dc.rights.accessRightsopen access
dc.subject.cdu537.533.3
dc.subject.cdu681.7
dc.subject.keywordSurfaces
dc.subject.keywordAspherics
dc.subject.keywordOphthalmic optics and devices
dc.subject.keywordAlgorithms
dc.subject.keywordArtificial intelligence
dc.subject.keywordLearning systems
dc.subject.keywordNumerical methods
dc.subject.keywordArbitrary shape
dc.subject.keywordConnected domains
dc.subject.keywordFitting techniques
dc.subject.keywordLocal curvature
dc.subject.keywordManufacturing industries
dc.subject.keywordOphthalmic lens
dc.subject.keywordRobust fittings
dc.subject.keywordZernike polynomials
dc.subject.ucmÓptica (Física)
dc.subject.ucmÓptica oftálmica
dc.subject.ucmOptoelectrónica
dc.subject.unesco2209.19 Óptica Física
dc.titleRobust fitting of Zernike polynomials to noisy point clouds defined over connected domains of arbitrary shape
dc.typejournal article
dc.volume.number24
dcterms.references1. W. Wang, H. Pottmann, and Y. Liu, “Fitting B-spline curves to point clouds by curvature-based squared distance minimization,” ACM Trans. Graph. 25(2), 214–238 (2006). 2. S. Flöry, “Fitting curves and surfaces to point clouds in the presence of obstacles,” Comput. Aided Geom. Des. 26(2), 192–202 (2009). 3. F. Remondino, “From point cloud to surface: the modelling and visualization problem,” in International Workshop on Visualization and Animation of Reality-Based 3D Models, http://dx.doi.org/10.3929/ethz-a-004655782 (2003). 4. J. G. Hayes and J. Halliday, “The least-squares fitting of cubic spline surfaces to general data sets,” IMA J. Appl. Math. 14(1), 89–103 (1974). 5. P. Dierkx, Curve and Surface Fitting with Splines (Oxford University, 1993). 6. P. Bo, R. Ling, and W. Wang, “A revisit to fitting parametric surfaces to point cloud,” Comput. Graph. 36(5), 534–540 (2012). 7. M. Ares and S. Royo, “Comparison of cubic B-spline and Zernike-fitting techniques in complex wavefront reconstruction,” Appl. Opt. 45(27), 6954–6964 (2006). 8. D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001). 9. M. K. Smolek and S. D. Klyce, “Zernike polynomial fitting fails to represent all visually significant corneal aberrations,” Invest. Ophthalmol. Vis. Sci. 44(11), 4676–4681 (2003). 10. V. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75–85 (1981). 11. J. A. Díaz and R. Navarro, “Orthonormal polynomials for elliptical wavefronts with an arbitrary orientation,” Appl. Opt. 53(10), 2051–2057 (2014). 12. G. M. Dai, “Wavefront reconstruction from slope data within pupils of arbitrary shapes using iterative Fourier Transform,” Open Opt. J. 1(1), 1–3 (2007). 13. K. P. Murphy, Machine Learning a Probabilistic Perspective (Massachusetts Institute of Technology, 2012). 14. A. E. Hoerl and R. W. Kennard, “Ridge regression: biased estimation for nonorthogonal problems,” Technometrics 12(1), 55–67 (1970). 15. J. Schwiegerling, Field Guide to Visual and Ophthalmic Optics (SPIE, 2004). 16. M. M. Lipschutz, Differential Geometry (McGraw Hill, 1970).
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