Applications of Mathematics, Vol. 63, No. 4, pp. 455-481, 2018


Reconstruction of map projection, its inverse and re-projection

Tomáš Bayer, Milada Kočandrlová

Received March 31, 2018.   Published online July 20, 2018.

Abstract:  This paper focuses on the automatic recognition of map projection, its inverse and re-projection. Our analysis leads to the unconstrained optimization solved by the hybrid BFGS nonlinear least squares technique. The objective function is represented by the squared sum of the residuals. For the map re-projection the partial differential equations of the inverse transformation are derived. They can be applied to any map projection. Illustrative examples of the stereographic and globular Nicolosi projections frequently used in early maps are involved and their inverse formulas are presented.
Keywords:  mathematical cartography; inverse projection; analysis; nonlinear least squares; partial differential equation; optimization; hybrid BFGS; early map; re-projection
Classification MSC:  34B16, 34C25


References:
[1] M. Al-Baali, R. Fletcher: Variational methods for non-linear least-squares. J. Oper. Res. Soc. 36 (1985), 405-421. DOI 10.2307/2582880 | Zbl 0578.65064
[2] Á. Barancsuk: A semi-automatic approach for determining the projection of small scale maps based on the shape of graticule lines. Progress in Cartography. EuroCarto 2015 (G. Gartner, M. Jobst, H. Huang, eds.). Springer, Cham, 2016, pp. 267-288. DOI 10.1007/978-3-319-19602-2_17
[3] T. Bayer: Estimation of an unknown cartographic projection and its parameters from the map. GeoInformatica 18 (2014), 621-669. DOI 10.1007/s10707-013-0200-4
[4] T. Bayer: Advanced methods for the estimation of an unknown projection from a map. GeoInformatica 20 (2016), 241-284. DOI 10.1007/s10707-015-0234-x
[5] T. Bayer: Detectproj - software for the projection analysis. Available athttp://sourceforge.net/projects/detectproj/ (2017).
[6] T. Bayer: Plotting the map projection graticule involving discontinuities based on combined sampling. Geoinformatics FCE CTU 17 (2018), 31-64. DOI 10.14311/gi.17.2.3
[7] I. O. Bildirici: Numerical inverse transformation for map projections. Computers & Geosciences 29 (2003), 1003-1011. DOI 10.1016/S0098-3004(03)00090-6
[8] I. O. Bildirici: An iterative approach for inverse transformation of map projections. Cartography and Geographic Information Science 44 (2017), 463-471. DOI 10.1080/15230406.2016.1200492
[9] G. I. Evenden: libproj4: A comprehensive library of cartographic projection functions. Falmouth, Massachusetts (2005).
[10] W. Flacke, B. Kraus: Working with Projections and Datum Transformations in ArcGIS: Theory and Practical Examples. Points Verlag, Norden (2005).
[11] R. Fletcher, C. Xu: Hybrid methods for nonlinear least squares. IMA J. Numer. Anal. 7 (1987), 371-389. DOI 10.1093/imanum/7.3.371 | MR 0968531 | Zbl 0648.65051
[12] E. Harrison, A. Mahdavi-Amiri, F. Samavati: Optimization of inverse Snyder polyhedral projection. Cyberworlds (CW), 2011 International Conference on Cyberworlds. IEEE, 2011, pp. 136-143. DOI 10.1109/CW.2011.36
[13] J. Huschens: On the use of product structure in secant methods for nonlinear least squares problems. SIAM J. Optim. 4 (1994), 108-129. DOI 10.1137/0804005 | MR 1260409 | Zbl 0798.65064
[14] C. Ipbüker: Inverse transformation for several pseudo-cylindrical map projections using Jacobian matrix. International Conference on Computational Science and Its Applications (O. Gervasi et al., eds.). Springer, Berlin, 2009, pp. 553-564. DOI 10.1007/978-3-642-02454-2_40
[15] C. Ipbüker, I. Bildirici: A general algorithm for the inverse transformation of map projections using jacobian matrices. Proceedings of the Third International Symposium Mathematical & Computational Applications 2002. Konya, Turkey, 2002, pp. 175-182.
[16] B. Jenny: Map Analyst. Available at http://mapanalyst.org (2011).
[17] M. Lapaine: Mollweide map projection. KoG 15 (2011), 7-16. MR 2951618 | Zbl 1261.51016
[18] L. Lukšan: Computational experience with known variable metric updates. J. Optimization Theory Appl. 83 (1994), 27-47. DOI 10.1007/BF02191760 | MR 1298855 | Zbl 0819.90097
[19] L. Lukšan: Hybrid methods for large sparse nonlinear least squares. J. Optimization Theory Appl. 89 (1996), 575-595. DOI 10.1007/BF02275350 | MR 1393364 | Zbl 0851.90118
[20] L. Lukšan, E. Spedicato: Variable metric methods for unconstrained optimization and nonlinear least squares. J. Comput. Appl. Math. 124 (2000), 61-95. DOI 10.1016/S0377-0427(00)00420-9 | MR 1803294 | Zbl 0985.65066
[21] D. A. Ratner: An implementation of the Robinson map projection based on cubic splines. Cartography and Geographic Information Systems 18 (1991), 104-108. DOI 10.1559/152304091783805536
[22] B. Šavrič, B. Jenny: A new pseudocylindrical equal-area projection for adaptive composite map projections. International Journal of Geographical Information Science 28 (2014), 2373-2389. DOI 10.1080/13658816.2014.924628
[23] W. M. Smart, R. M. Green: Textbook on Spherical Astronomy. Cambridge University Press, Cambridge (1977). DOI 10.1017/cbo9781139167574
[24] J. P. Snyder: Map Projections Used by the US Geological Survey. Technical report, US Government Printing Office, Washington (1982).
[25] J. P. Snyder: Map projections: A working manual. US Government Printing Office, Washington (1987).
[26] W. R. Tobler: Medieval distortions: The projections of ancient maps. Annals of the Association of American Geographers 56 (1966), 351-360. DOI 10.1111/j.1467-8306.1966.tb00562.x
[27] W. R. Tobler: Numerical approaches to map projections. Beitrage zur theoretischen Kartographie, Festschrift für Erik Amberger, hg. Ingrid Kretschmer Franz Deuticke, Wien, 1977, pp. 51-64.
[28] W. R. Tobler: Measuring the similarity of map projections. The American Cartographer 13 (1986), 135-139. DOI 10.1559/152304086783900103
[29] H. Yabe, T. Takahashi: Factorized quasi-Newton methods for nonlinear least squares problems. Math. Program., Ser. A 51 (1991), 75-100. DOI 10.1007/BF01586927 | MR 1119246 | Zbl 0737.90064
[30] Q. H. Yang, J. P. Snyder, W. R. Tobler: Map Projection Transformation. Principles and Applications. Taylor & Francis, London (2000). MR 1832180 | Zbl 0980.86006
[31] W. Zhou, X. Chen: Global convergence of a new hybrid Gauss-Newton structured BFGS method for nonlinear least squares problems. SIAM J. Optim. 20 (2010), 2422-2441. DOI 10.1137/090748470 | MR 2678399 | Zbl 1211.90131

Affiliations:   Tomáš Bayer, Faculty of Science, Charles University, Albertov 6, Praha 2, 120 78, Czech Republic, e-mail: bayertom@natur.cuni.cz; Milada Kočandrlová, Faculty of Finance and Accounting, University of Economics, nám. W. Churchilla 1938/4, Praha 3, 130 67, Czech Republic, e-mail: kocandrlova@hotmail.cz


 
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