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Certain topics in ellipsometric data modeling with splines: a review of recent developments

  • Dmitriy V. Likhachev

    Dmitriy V. Likhachev began his studies in Technical Physics at the S.M. Kirov Ural Polytechnic Institute (Sverdlovsk, U.S.S.R.) in 1980 and received his Ph.D. in Solid State Physics (Candidate of Science) from the Ural State Technical University (Ekaterinburg, Russia) in 1993. He is currently a Member of Technical Staff with GlobalFoundries (Dresden, Germany). His present interests include applications of spectroscopic ellipsometry to material characterization and process control in semiconductor development and manufacturing.

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Published/Copyright: July 4, 2022
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Abstract

Dielectric function representation by a variety of polynomial spline functions provides a consistent and efficient method for accurate modeling of the material optical properties in the context of spectroscopic ellipsometry data interpretation. Splines as an elegant and purely mathematical way for such modeling task were introduced about three decades ago. In the following years the use of splines in the area of ellipsometric data analysis became widely utilized. The goal of this review is to provide a self-contained presentation on the current status of the dielectric function modeling by splines for advanced industrial ellipsometry users but, hopefully, it can be useful for some scholarly users as well. It is also intended to promote more extended recognition of the spline-based modeling among optical metrology professionals. Here, a brief description of different ways, – ordinary polynomials, piecewise polynomials (splines), and B(asis)-spline functions, – is presented to parameterize an arbitrary function which can be used as an analytic representation of the dielectric-function curves. A number of particular polynomial-based models for the optical functions of materials and how they may be used in applications are also discussed. Particular attention is paid to different concepts of the efficient and optimal spline construction.


Corresponding author: Dmitriy V. Likhachev, GlobalFoundries Dresden Module One LLC & Co. KG, Wilschdorfer Landstr. 101, D-01109 Dresden, Germany, E-mail:

About the author

Dmitriy V. Likhachev

Dmitriy V. Likhachev began his studies in Technical Physics at the S.M. Kirov Ural Polytechnic Institute (Sverdlovsk, U.S.S.R.) in 1980 and received his Ph.D. in Solid State Physics (Candidate of Science) from the Ural State Technical University (Ekaterinburg, Russia) in 1993. He is currently a Member of Technical Staff with GlobalFoundries (Dresden, Germany). His present interests include applications of spectroscopic ellipsometry to material characterization and process control in semiconductor development and manufacturing.

Acknowledgement

The author wishes to acknowledge the colleagues at GlobalFoundries Fab1 (Dresden, Germany) for their generous and constant support and succor. In particular, the assistance provided by Steffen Brunner, Sven Bürgel, Göran Fleischer, Thomas Kache, Ulf Peter Müller, and Pavel Prunici is appreciated.

  1. Author contributions: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The author declare no conflicts of interest regarding this article.

References

[1] H. G. Tompkins and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry: A User’s Gide, New York, John Wiley & Sons Inc., 1999.Search in Google Scholar

[2] H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, Chichester, West Sussex, U.K., John Wiley & Sons Ltd., 2007.10.1002/9780470060193Search in Google Scholar

[3] H. G. Tompkins and J. N. Hilfiker, Spectroscopic Ellipsometry: Practical Application to Thin Film Characterization, New York, Momentum Press, 2016.Search in Google Scholar

[4] R. W. Collins and A. S. Ferlauto, “Optical physics of materials,” in Handbook of Ellipsometry, Norwich, NY, USA, William Andrew Publishing/Noyes, 2005, pp. 93–235.10.1016/B978-081551499-2.50004-6Search in Google Scholar

[5] J. N. Hilfiker and T. Tiwald, “Dielectric function modeling,” in Spectroscopic Ellipsometry for Photovoltaics: Fundamental Principles and Solar cell Characterization, vol. 1, Cham, Switzerland, Springer-Verlag, 2018, pp. 115–153.10.1007/978-3-319-75377-5_5Search in Google Scholar

[6] I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions.–Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae,” Q. Appl. Math., vol. 4, pp. 45–99, 1946, https://doi.org/10.1090/qam/15914.Search in Google Scholar

[7] I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions.–Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae,” Q. Appl. Math., vol. 4, pp. 112–141, 1946, https://doi.org/10.1090/qam/16705.Search in Google Scholar

[8] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. Amsterdam, The Netherlands, Elsevier Butterworth-Heinemann, 1984.Search in Google Scholar

[9] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Berlin, Wiley-VCH, 1989.10.1002/9783527617210Search in Google Scholar

[10] J. K. Jerome, Three Men in a Boat and Three Men On The Bummel (Oxford World’s Classics), New York, Oxford University Press, 1998.Search in Google Scholar

[11] C. Runge, “Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten,” Z. Math. Phys., vol. 46, pp. 224–243, 1901.Search in Google Scholar

[12] H. A. Buchdahl, Optical Aberration Coefficients, Mineola, NY, U.S.A., Dover Publications, Inc., 1968.Search in Google Scholar

[13] J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications, New York, Academic Press, Inc., 1967.Search in Google Scholar

[14] L. L. Schumaker, Spline Functions: Basic Theory, 3rd ed. Cambridge, U.K., Cambridge University Press, 2007.10.1017/CBO9780511618994Search in Google Scholar

[15] P. Dierckx, Curve and Surface Fitting with Splines, New York, Oxford University Press, 1993.10.1093/oso/9780198534419.001.0001Search in Google Scholar

[16] J. D. Brown, Advanced Statistics for The Behavioral Sciences: A Computational Approach with R, Cham, Switzerland, Springer, 2018.10.1007/978-3-319-93549-2Search in Google Scholar

[17] I. J. Schoenberg, “On interpolation by spline functions and its minimal properties,” in On Approximation Theory/über Approximationstheorie: Proceedings of The Conference Held in the Mathematical Research Institute, P. L. Butzer, and J. Korevaar, Eds., Basel, Springer-Verlag, 1964, pp. 109–129. Oberwolfach, Black Forest, August 4–10, 1963.10.1007/978-3-0348-4131-3_12Search in Google Scholar

[18] H. Akima, “A new method of interpolation and smooth curve fitting based on local procedures,” J. ACM, vol. 17, pp. 589–602, 1970, https://doi.org/10.1145/321607.321609.Search in Google Scholar

[19] J. Fried and S. Zietz, “Curve fitting by spline and Akima methods: Possibility of interpolation error and its suppression,” Phys. Med. Biol., vol. 18, pp. 550–558, 1973, https://doi.org/10.1088/0031-9155/18/4/306.Search in Google Scholar

[20] M. Steffen, “A simple method for monotonic interpolation in one dimension,” Astron. Astrophys., vol. 239, pp. 443–450, 1990.Search in Google Scholar

[21] C. Moler, “Makima piecewise cubic interpolation, cleve’s corner: Cleve moler on mathematics and computing,” Available at: https://blogs.mathworks.com/cleve/2019/04/29/makima-piecewise-cubic-interpolation/#d9a97978-0b09-4a1f-a6a5-504d088631d0 [accessed: Feb. 22, 2022].Search in Google Scholar

[22] C. de Boor, A Practical Guide to Splines, Revised Edition, New York, Springer-Verlag, 2001.Search in Google Scholar

[23] C. de Boor, “On calculating with B-splines,” J. Approx. Theor., vol. 6, pp. 50–62, 1972, https://doi.org/10.1016/0021-9045(72)90080-9.Search in Google Scholar

[24] M. G. Cox, “The numerical evaluation of B-splines,” IMA J. Appl. Math., vol. 10, pp. 134–149, 1972, https://doi.org/10.1093/imamat/10.2.134.Search in Google Scholar

[25] C. de Boor and A. Pinkus, “The B-spline recurrence relations of Chakalov and of Popoviciu,” J. Approx. Theor., vol. 124, pp. 115–123, 2003, https://doi.org/10.1016/s0021-9045(03)00117-5.Search in Google Scholar

[26] T. Lyche, C. Manni, and H. Speleers, “Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement,” in Splines and PDEs: From Approximation Theory to Numerical Linear Algebra, Lecture Notes in Mathematics, vol. 2219, Cham, Switzerland, Springer, 2018, pp. 1–76.10.1007/978-3-319-94911-6_1Search in Google Scholar

[27] J. R. Rice, Numerical Methods in Software and Analysis, 2nd ed. San Diego, Academic Press, 1993.Search in Google Scholar

[28] B. Johs and J. S. Hale, “Dielectric function representation by B-splines,” Phys. Status Solidi, vol. 205, pp. 715–719, 2008, https://doi.org/10.1002/pssa.200777754.Search in Google Scholar

[29] I. Failes, “The tech of ‘Terminator 2’ – an oral history, October 21, 2019, befores & afters visual effects and animation online magazine,” Available at: https://beforesandafters.com/2019/10/21/the-tech-of-terminator-2-an-oral-history/ [accessed: Feb. 22, 2022].Search in Google Scholar

[30] J.-T. Zettler, M. Weidner, and A. Röseler, “On the characterization of silicon dioxide and silicon nitride by spectroscopic ellipsometry in the VIS and IR regions,” Phys. Status Solidi, vol. 124, pp. 547–555, 1991, https://doi.org/10.1002/pssa.2211240222.Search in Google Scholar

[31] J. Vanhellemont and H. E. Maes, “Spectroscopic ellipsometry characterization of silicon-on-insulator materials,” Mater. Sci. Eng. B, vol. 5, pp. 301–307, 1990, https://doi.org/10.1016/0921-5107(90)90073-k.Search in Google Scholar

[32] J. Vanhellemont, P. Roussel, and H. E. Maes, “Spectroscopic ellipsometry for depth profiling of ion implanted materials,” Nucl. Instrum. Methods Phys. Res. B, vol. 55, pp. 183–187, 1991, https://doi.org/10.1016/0168-583x(91)96158-h.Search in Google Scholar

[33] J. Vanhellemont and P. Roussel, “Characterization by spectroscopic ellipsometry of buried layer structures in silicon formed by ion beam synthesis,” Mater. Sci. Eng. B, vol. 12, pp. 165–172, 1992, https://doi.org/10.1016/0921-5107(92)90280-m.Search in Google Scholar

[34] Y. Z. Hu, J.-T. Zettler, S. Chongsawangvirod, Y. Q. Wang, and E. A. Irene, “Spectroscopic ellipsometric measurements of the dielectric function of germanium dioxide films on crystal germanium,” Appl. Phys. Lett., vol. 61, pp. 1098–1100, 1992, https://doi.org/10.1063/1.107680.Search in Google Scholar

[35] J.-T. Zettler, T. Trepk, L. Spanos, Y.-Z. Hu, and W. Richter, “High precision UV-visible-near-IR Stokes vector spectroscopy,” Thin Solid Films, vol. 234, pp. 402–407, 1993, https://doi.org/10.1016/0040-6090(93)90295-z.Search in Google Scholar

[36] D. Zwillinger, Handbook of Integration, Boston, London, Jones and Bartlett Publishers, 1992.10.1201/9781439865842Search in Google Scholar

[37] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichevin, Integrals and Series: Elementary Functions, vol. 1, New York, London, Gordon & Breach Science Publishers, 1986.Search in Google Scholar

[38] M. Zorn, T. Trepk, J.-T. Zettler, et al.., “Temperature dependence of the InP(001) bulk and surface dielectric function,” Appl. Phys. A, vol. 65, pp. 333–339, 1997, https://doi.org/10.1007/s003390050588.Search in Google Scholar

[39] D. De Sousa Meneses, B. Rousseau, P. Echegut, and G. Matzen, “Piecewise polynomial dielectric function model and its application for the retrieval of optical functions,” Appl. Spectrosc., vol. 61, pp. 644–648, 2007, https://doi.org/10.1366/000370207781269710.Search in Google Scholar PubMed

[40] M. Gilliot, A. Hadjadj, and M. Stchakovsky, “Spectroscopic ellipsometry data inversion using constrained splines and application to characterization of ZnO with various morphologies,” Appl. Surf. Sci., vol. 421, pp. 453–459, 2017, https://doi.org/10.1016/j.apsusc.2016.09.106.Search in Google Scholar

[41] M. Gilliot, “Inversion of ellipsometry data using constrained spline analysis,” Appl. Opt., vol. 56, pp. 1173–1182, 2017, https://doi.org/10.1364/ao.56.001173.Search in Google Scholar

[42] C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B, vol. 45, pp. 11749–11767, 1992, https://doi.org/10.1103/physrevb.45.11749.Search in Google Scholar PubMed

[43] C. C. Kim, J. W. Garland, and P. M. Raccah, “Modeling the optical dielectric function of the alloy system AlxGa1−xAs,” Phys. Rev. B, vol. 47, pp. 1876–1888, 1993, https://doi.org/10.1103/physrevb.47.1876.Search in Google Scholar PubMed

[44] S. H. Han, S. Yoo, B. Kippelen, and D. Levi, “Precise determination of optical properties of pentacene thin films grown on various substrates: Gauss–Lorentz model with effective medium approach,” Appl. Phys. B, vol. 104, pp. 139–144, 2011, https://doi.org/10.1007/s00340-011-4383-9.Search in Google Scholar

[45] B. Johs, C. M. Herzinger, J. H. Dinan, A. Cornfeld, and J. D. Benson, “Development of a parametric optical constant model for Hg1−xCdxTe for control of composition by spectroscopic ellipsometry during MBE growth,” Thin Solid Films, vols. 313–314, pp. 137–142, 1998, https://doi.org/10.1016/s0040-6090(97)00800-6.Search in Google Scholar

[46] C. M. Herzinger, B. Johs, W. A. McGahan, and J. A. Woollam, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys., vol. 83, pp. 3323–3336, 1998, https://doi.org/10.1063/1.367101.Search in Google Scholar

[47] E. Franke, C. L. Trimble, M. J. DeVries, J. A. Woollam, M. Schubert, and F. Frost, “Dielectric function of amorphous tantalum oxide from the far infrared to the deep ultraviolet spectral region measured by spectroscopic ellipsometry,” J. Appl. Phys., vol. 88, pp. 5166–5174, 2000, https://doi.org/10.1063/1.1313784.Search in Google Scholar

[48] L. Yan, J. A. Woollam, and E. Franke, “Oxygen plasma effects on optical properties of ZnSe films,” J. Vac. Sci. Technol. A, vol. 20, pp. 693–701, 2002, https://doi.org/10.1116/1.1463085.Search in Google Scholar

[49] T. J. Kim, T. H. Ghong, Y. D. Kim, et al., “Dielectric functions of InxGa1−x As alloys,” Phys. Rev. B, vol. 68, p. 115323, 2003, https://doi.org/10.1103/physrevb.68.115323.Search in Google Scholar

[50] V. R. D’Costa, C. S. Cook, A. G. Birdwell, et al.., “Optical critical points of thin-film Ge1−ySny alloys: a comparative Ge1−ySny/Ge1−xSix study,” Phys. Rev. B, vol. 73, p. 125207, 2006.10.1103/PhysRevB.73.125207Search in Google Scholar

[51] E. Agocs, B. Fodor, et al.., “Approaches to calculate the dielectric function of ZnO around the band gap,” Thin Solid Films, vol. 571, pp. 684–688, 2014, https://doi.org/10.1016/j.tsf.2014.03.028.Search in Google Scholar

[52] C. Xu, J. D. Gallagher, C. L. Senaratne, J. Menéndez, and J. Kouvetakis, “Optical properties of Ge-rich Ge1−xSix alloys: Compositional dependence of the lowest direct and indirect gaps,” Phys. Rev. B, vol. 93, p. 125206, 2016, https://doi.org/10.1103/physrevb.93.125206.Search in Google Scholar

[53] V. L. Le, T. J. Kim, H. G. Park, H. T. Nguyen, X. A. Nguyen, and Y. D. Kim, “Temperature dependence of the dielectric function of monolayer MoS2,” Curr. Appl. Phys., vol. 19, pp. 182–187, 2019, https://doi.org/10.1016/j.cap.2018.10.007.Search in Google Scholar

[54] A. B. Kuzmenko, “Kramers–Kronig constrained variational analysis of optical spectra,” Rev. Sci. Instrum., vol. 76, p. 083108, 2005, https://doi.org/10.1063/1.1979470.Search in Google Scholar

[55] J. Levallois, I. O. Nedoliuk, I. Crassee, and A. B. Kuzmenko, “Magneto-optical Kramers–Kronig analysis,” Rev. Sci. Instrum., vol. 86, p. 033906, 2015, https://doi.org/10.1063/1.4914846.Search in Google Scholar PubMed

[56] J. W. Weber, T. A. R. Hansen, M. C. M. van de Sanden, and R. Engeln, “B-spline parametrization of the dielectric function applied to spectroscopic ellipsometry on amorphous carbon,” J. Appl. Phys., vol. 106, p. 123503, 2009, https://doi.org/10.1063/1.3257237.Search in Google Scholar

[57] L. S. Abdallah, S. Zollner, C. Lavoie, A. Ozcan, and M. Raymond, “Compositional dependence of the optical conductivity of Ni1−xPtx alloys (0 < x < 0.25) determined by spectroscopic ellipsometry,” Thin Solid Films, vol. 571, pp. 484–489, 2014, https://doi.org/10.1016/j.tsf.2013.11.022.Search in Google Scholar

[58] R. Schmidt-Grund, C. Kranert, H. von Wenckstern, V. Zviagin, M. Lorenz, and M. Grundmann, “Dielectric function in the spectral range (0.5–8.5) eV of an (AlxGa1−x)2O3 thin film with continuous composition spread,” J. Appl. Phys., vol. 117, p. 165307, 2015, https://doi.org/10.1063/1.4919088.Search in Google Scholar

[59] V. Zviagin, P. Richter, T. Böntgen, et al.., “Comparative study of optical and magneto-optical properties of normal, disordered, and inverse spinel-type oxides,” Phys. Status Solidi B, vol. 253, pp. 429–436, 2016, https://doi.org/10.1002/pssb.201552361.Search in Google Scholar

[60] P. Petrik, A. Sulyok, T. Novotny, et al.., “Optical properties of Zr and ZrO2,” Appl. Surf. Sci., vol. 421, pp. 744–747, 2017, https://doi.org/10.1016/j.apsusc.2016.11.072.Search in Google Scholar

[61] S. Schöche, N. Hong, M. Khorasaninejad, et al.., “Optical properties of graphene oxide and reduced graphene oxide determined by spectroscopic ellipsometry,” Appl. Surf. Sci., vol. 421, pp. 778–782, 2017.10.1016/j.apsusc.2017.01.035Search in Google Scholar

[62] J. Sun and G. K. Pribil, “Analyzing optical properties of thin vanadium oxide films through semiconductor-to-metal phase transition using spectroscopic ellipsometry,” Appl. Surf. Sci., vol. 421, pp. 819–823, 2017, https://doi.org/10.1016/j.apsusc.2016.09.125.Search in Google Scholar

[63] S. Prucnal, Y. Berencén, M. Wang, et al.., “Strain and band-gap engineering in Ge-Sn alloys via P doping,” Phys. Rev. Applied, vol. 10, p. 064055, 2018, https://doi.org/10.1103/physrevapplied.10.064055.Search in Google Scholar

[64] S. Prucnal, Y. Berencén, M. Wang, et al.., “Band gap renormalization in n-type GeSn alloys made by ion implantation and flash lamp annealing,” J. Appl. Phys., vol. 125, p. 203105, 2019, https://doi.org/10.1063/1.5082889.Search in Google Scholar

[65] S. Richter, O. Herrfurth, S. Espinoza, et al.., “Ultrafast dynamics of hot charge carriers in an oxide semiconductor probed by femtosecond spectroscopic ellipsometry,” New J. Phys., vol. 22, p. 083066, 2020, https://doi.org/10.1088/1367-2630/aba7f3.Search in Google Scholar

[66] J. Mohrmann, T. E. Tiwald, J. S. Hale, J. N. Hilfiker, and A. C. Martin, “Application of a B-spline model dielectric function to infrared spectroscopic ellipsometry data analysis,” J. Vac. Sci. Technol. B, vol. 38, p. 014001, 2020, https://doi.org/10.1116/1.5126110.Search in Google Scholar

[67] V. Zviagin, M. Grundmann, and R. Schmidt-Grund, “Impact of defects on magnetic properties of spinel zinc ferrite thin films,” Phys. Status Solidi B, vol. 257, p. 1900630, 2020, https://doi.org/10.1002/pssb.201900630.Search in Google Scholar

[68] P. H. C. Eilers and B. D. Marx, Practical Smoothing: The Joys of P-Splines, Cambridge, UK, Cambridge University Press, 2021.10.1017/9781108610247Search in Google Scholar

[69] J. Schelten and F. Hossfeld, “Application of spline functions to the correction of resolution errors in small-angle scattering,” J. Appl. Crystallogr., vol. 4, pp. 210–223, 1971, https://doi.org/10.1107/s0021889871006733.Search in Google Scholar

[70] O. Glatter, “A new method for the evaluation of small-angle scattering data,” J. Appl. Crystallogr., vol. 10, pp. 415–421, 1977, https://doi.org/10.1107/s0021889877013879.Search in Google Scholar

[71] J. S. Pedersen, “Model-independent determination of the surface scattering-length-density profile from specular reflectivity data,” J. Appl. Crystallogr., vol. 25, pp. 129–145, 1992, https://doi.org/10.1107/s0021889891010907.Search in Google Scholar

[72] N. F. Berk and C. F. Majkrzak, “Using parametric B splines to fit specular reflectivities,” Phys. Rev. B, vol. 51, pp. 11296–11309, 1995, https://doi.org/10.1103/physrevb.51.11296.Search in Google Scholar PubMed

[73] D. V. Likhachev, “Selecting the right number of knots for B-spline parameterization of the dielectric functions in spectroscopic ellipsometry data analysis,” Thin Solid Films, vol. 636, pp. 519–526, 2017, https://doi.org/10.1016/j.tsf.2017.06.056.Search in Google Scholar

[74] D. V. Likhachev, “B-spline parameterization of the dielectric function and information criteria: the craft of non-overfitting,” in Modeling Aspects in Optical Metrology VI, B. Bodermann, K. Frenner, and R. M. Silver, Eds., Munich, Germany, June 25–29, 2017, SPIE Proc. 10330, 2017, p. 103300B.10.1117/12.2270249Search in Google Scholar

[75] D. V. Likhachev, “Spectroscopic ellipsometry data analysis using penalized splines representation for the dielectric function,” Thin Solid Films, vol. 669, pp. 174–180, 2019, https://doi.org/10.1016/j.tsf.2018.10.057.Search in Google Scholar

[76] D. V. Likhachev, “A practitioner’s approach to evaluation strategy for ellipsometric measurements of multilayered and multiparametric thin-film structures,” Thin Solid Films, vol. 595, pp. 113–117, 2015, https://doi.org/10.1016/j.tsf.2015.10.078.Search in Google Scholar

[77] D. V. Likhachev, “Model selection in spectroscopic ellipsometry data analysis: combining an information criteria approach with screening sensitivity analysis,” Appl. Surf. Sci., vol. 421, pp. 617–623, 2017, https://doi.org/10.1016/j.apsusc.2016.09.139.Search in Google Scholar

[78] A. Jogalekar, “Derek Lowe to world: “Beware of von Neumann’s elephants,” 2015. Available at: http://wavefunction.fieldofscience.com/2015/02/derek-lowe-to-world-beware-of-von.html [accessed: Jan. 11, 2022].Search in Google Scholar

[79] F. Dyson, “A meeting with Enrico Fermi,” Nature (London), vol. 427, p. 297, 2004, https://doi.org/10.1038/427297a.Search in Google Scholar PubMed

[80] J. Mayer, K. Khairy, and J. Howard, “Drawing an elephant with four complex parameters,” Am. J. Phys., vol. 78, pp. 648–649, 2010, https://doi.org/10.1119/1.3254017.Search in Google Scholar

[81] K. P. Burnham and D. Anderson, Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. New York, Springer-Verlag, 2002.Search in Google Scholar

[82] D. V. Likhachev, “On the optimization of knot allocation for B-spline parameterization of the dielectric function in spectroscopic ellipsometry data analysis,” J. Appl. Phys., vol. 129, p. 034903, 2021, https://doi.org/10.1063/5.0035456.Search in Google Scholar

[83] V. T. Dung and T. Tjahjowidodo, “A direct method to solve optimal knots of B-spline curves: an application for non-uniform B-spline curves fitting,” PLoS ONE, vol. 12, p. e0173857, 2017, https://doi.org/10.1371/journal.pone.0173857.Search in Google Scholar PubMed PubMed Central

[84] P. Laube, M. O. Franz, and G. Umlauf, “Learnt knot placement in B-spline curve approximation using support vector machines,” Comput. Aided Geomet. Des., vol. 62, pp. 104–116, 2018, https://doi.org/10.1016/j.cagd.2018.03.019.Search in Google Scholar

[85] R. Yeh, Y. S. G. Nashed, T. Peterka, and X. Tricoche, “Fast automatic knot placement method for accurate B-spline curve fitting,” Comput. Aided Des., vol. 128, p. 102905, 2020, https://doi.org/10.1016/j.cad.2020.102905.Search in Google Scholar

[86] D. Michel and A. Zidna, “A new deterministic heuristic knots placement for B-spline approximation,” Math. Comput. Simulat., vol. 186, pp. 91–102, 2021, https://doi.org/10.1016/j.matcom.2020.07.021.Search in Google Scholar

[87] D. Lenz, O. Marin, V. Mahadevan, R. Yeh, and T. Peterka, “Fourier-informed knot placement schemes for B-spline approximation”, preprint arXiv: 2012.04123 [math.NA], 2020.Search in Google Scholar

[88] D. L. B. Jupp, “Approximation to data by splines with free knots,” SIAM J. Numer. Anal., vol. 15, pp. 328–343, 1978, https://doi.org/10.1137/0715022.Search in Google Scholar

[89] T. J. Jacobson and M. J. Murphy, “Optimized knot placement for B-splines in deformable image registration,” Med. Phys., vol. 38, pp. 4579–4582, 2011, https://doi.org/10.1118/1.3609416.Search in Google Scholar PubMed PubMed Central

[90] D. R. Forsey and R. H. Bartels, “Hierarchical B-spline refinement,” ACM SIGGRAPH Comput. Graph., vol. 22, pp. 205–212, 1988, https://doi.org/10.1145/378456.378512.Search in Google Scholar

[91] C. Giannelli, B. Jüttler, and H. Speleers, “THB-splines: The truncated basis for hierarchical splines,” Comput. Aided Geomet. Des., vol. 29, pp. 485–498, 2012, https://doi.org/10.1016/j.cagd.2012.03.025.Search in Google Scholar

[92] K. A. Johannessen, F. Remonato, and T. Kvamsdal, “On the similarities and differences between classical hierarchical, truncated hierarchical and LR B-splines,” Comput. Methods Appl. Mech. Eng., vol. 291, pp. 64–101, 2015, https://doi.org/10.1016/j.cma.2015.02.031.Search in Google Scholar

[93] C. Conti, R. Morandi, C. Rabut, and A. Sestini, “Cubic spline data reduction choosing the knots from a third derivative criterion,” Numer. Algorithm., vol. 28, pp. 45–61, 2001, https://doi.org/10.1023/a:1014022210828.10.1023/A:1014022210828Search in Google Scholar

[94] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Washington, D.C., V. H. Winston & Sons, 1977.Search in Google Scholar

[95] A. G. Yagola, “Ill-posed problems and methods for their numerical solution,” in Optimization and Regularization for Computational Inverse Problems and Applications, Berlin, Heidelberg, Springer, 2010, pp. 17–34.10.1007/978-3-642-13742-6_2Search in Google Scholar

[96] F. O’Sullivan, “A statistical perspective on ill-posed inverse problems (with discussion),” Stat. Sci., vol. 1, pp. 502–527, 1986.10.1214/ss/1177013525Search in Google Scholar

[97] P. H. C. Eilers and B. D. Marx, “Flexible smoothing with B-splines and penalties,” Stat. Sci., vol. 11, pp. 89–102, 1996, https://doi.org/10.1214/ss/1038425655.Search in Google Scholar

[98] P. H. C. Eilers, B. D. Marx, and M. Durbán, “Twenty years of P-splines,” SORT-Stat. Oper. Res. Trans., vol. 39, pp. 149–186, 2015.Search in Google Scholar

[99] G. Wahba, Spline Models for Observational Data, Philadelphia, PA, SIAM: Society for Industrial and Applied Mathematics, 1990.10.1137/1.9781611970128Search in Google Scholar

[100] D. Ruppert, M. P. Wand, and R. J. Carroll, Semiparametric Regression, Cambridge, UK, Cambridge University Press, 2003.10.1017/CBO9780511755453Search in Google Scholar

[101] T. C. M. Lee, “Smoothing parameter selection for smoothing splines: A simulation study,” Comput. Stat. Data Anal., vol. 42, pp. 139–148, 2003, https://doi.org/10.1016/s0167-9473(02)00159-7.Search in Google Scholar

[102] L. N. Berry and N. E. Helwig, “Cross-validation, information theory, or maximum likelihood? A comparison of tuning methods for penalized splines,” Stats, vol. 4, no. 2021, pp. 701–724.10.3390/stats4030042Search in Google Scholar

[103] S. Salvador and P. Chan, “Learning states and rules for detecting anomalies in time series,” Appl. Intell., vol. 23, pp. 241–255, 2005, https://doi.org/10.1007/s10489-005-4610-3.Search in Google Scholar

[104] A. A. Urbas and S. J. Choquette, “Automated spectral smoothing with spatially adaptive penalized least squares,” Appl. Spectrosc., vol. 65, pp. 665–677, 2011, https://doi.org/10.1366/10-05971.Search in Google Scholar PubMed

[105] X. Wang, P. Du, and J. Shen, “Smoothing splines with varying smoothing parameter,” Biometrika, vol. 100, pp. 955–970, 2013, https://doi.org/10.1093/biomet/ast031.Search in Google Scholar

[106] K. Oiwake, Y. Nishigaki, S. Fujimoto, S. Maeda, and H. Fujiwara, “Fully automated spectroscopic ellipsometry analyses: application to MoOx thin films,” J. Appl. Phys., vol. 129, p. 243102, 2021, https://doi.org/10.1063/5.0052210.Search in Google Scholar

Received: 2022-03-09
Accepted: 2022-04-22
Published Online: 2022-07-04
Published in Print: 2022-09-27

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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