GeoInformatica

, Volume 20, Issue 2, pp 241–284 | Cite as

Advanced methods for the estimation of an unknown projection from a map

Article

Abstract

This article presents three new methods (M5, M6, M7) for the estimation of an unknown map projection and its parameters. Such an analysis is beneficial and interesting for historic, old, or current maps without information about the map projection; it could improve their georeference. The location similarity approach takes into account the residuals on the corresponding features; the minimum is found using the non-linear least squares. For the shape similarity approach, the minimized objective function ϕ takes into account the spatial distribution of the features, together with the shapes of the meridians, parallels and other 0D-2D elements. Due to the non-convexity and discontinuity, its global minimum is determined using the global optimization, represented by the differential evolution. The constant values of projection φk, λk, φ1, λ0, and map constants RXY, α (in relation to which the methods are invariant) are estimated. All methods are compared and the results are presented for the synthetic data as well as for 8 early maps from the Map Collection of the Charles University and the David Rumsay Map Collection. The proposed algorithms have been implemented in the new version of the detectproj software.

Keywords

Digital cartography Map projection Optimizing Differential evolution Least squares BFGS Early maps Analysis Georeferencing Cartographic heritage History of cartography 

References

  1. 1.
    Al-Baali M, Fletcher R (1985) Variational methods for non-linear least-squares. J Oper Res Soc:405–421Google Scholar
  2. 2.
    Arkin EM, Chew LP, Huttenlocher DP, Kedem K, Mitchell JSB (1991) An efficiently computable metric for comparing polygonal shapes. IEEE J PAMI 13 (3):209–216CrossRefGoogle Scholar
  3. 3.
    Balletti C (2000) Analytical and quantitative methods for the analysis of the geometrical content of historical cartography. International Archives of Photogrammetry and Remote Sensing 33(B5/1; PART 5):30–37Google Scholar
  4. 4.
    Balletti C (2006) Georeference in the analysis of the geometric content of early maps. e-Perimetron 1(1):32–42Google Scholar
  5. 5.
    Bayer T (2014) Estimation of an unknown cartographic projection and its parameters from the map. GeoInformatica 18(3):621–669. doi:10.1007/s10707-013-0200-4 CrossRefGoogle Scholar
  6. 6.
    Bayer T, Potůčková M, Čábelka M (2010) Cartometric Analysis of Old Maps on the Example of Vogt’s Map. In: Gartner G, Ortag F (eds) Cartography in Central and Eastern Europe. Lecture Notes in Geoinformation and Cartography, DOI 10.1007/978-3-642-03294-3_33, (to appear in print)
  7. 7.
    Björck Å (1996) Numerical Methods for Least Squares Problems. SIAM, PhiladelphiaCrossRefGoogle Scholar
  8. 8.
    Boutoura C, Livieratos E (2006) Some fundamentals for the study of the geometry of early maps by comparative methods. e-Perimetron 1(1):60–70Google Scholar
  9. 9.
    Brest J, Greiner S, Boskovic B, Mernik M, Zumer V (2006) Self-adapting control parameters in differential evolution: A comparative study on numerical benchmark problems. Trans Evol Comp 10(6):646–657. doi:10.1109/TEVC.2006.872133 CrossRefGoogle Scholar
  10. 10.
    Broyden CG (1970) The convergence of a class of double rank minimization algorithms: 2. the new algorithm. http://www.mendeley.com/research/the-convergence-of-a-class-of-doublerank-minimization-algorithms-parts-i-and-ii/
  11. 11.
    Broyden CG, DENNIS JE, More JJ (1973) On the local and superlinear convergence of quasi-newton methods. IMA J Appl Math 12(3):223–245. doi:10.1093/imamat/12.3.223 CrossRefGoogle Scholar
  12. 12.
    Chien CW, Hsu ZR, Lee WP et al (2009) Improving the performance of differential evolution algorithm with modified mutation factor. In: Proceedings of International Conference on Machine Learning and Computing (ICMLC 2009)Google Scholar
  13. 13.
    Corne D, Dorigo M, Glover F (1999) New Ideas in Optimization. Advanced Topics in Computer Science Series, McGraw-Hill CompaniesGoogle Scholar
  14. 14.
    Crǎciunescu V, Constantinescu S (2006) Eharta. http://earth.unibuc.ro/articole/eHarta?lang=en
  15. 15.
    Dai YH (2002) Convergence properties of the bfgs algoritm. SIAM J Optim 13 (3):693–701. doi:10.1137/S1052623401383455 CrossRefGoogle Scholar
  16. 16.
    Das S, Konar A, Chakraborty UK (2005) Two improved differential evolution schemes for faster global search. In: Proceedings of the 2005 conference on Genetic and evolutionary computation, GECCO ’05. doi:10.1145/1068009.1068177. NY, USA, pp 991–998
  17. 17.
    Dennis Jr JE, Songbai S, Vu PA (1985) A memoryless augmented gauss-newton method for nonlinear least-squares problems, Technical Report, DTIC DocumentGoogle Scholar
  18. 18.
    Erle S, Krishnan S, Waters T (2009) World map warp. http://warp.worldmap.harvard.edu/
  19. 19.
    Esri (2003) Identify an unknown projected coordinate system using arcmapGoogle Scholar
  20. 20.
    Esri (2005) Identify an unknown projected coordinate system using arcmapGoogle Scholar
  21. 21.
    Fan HY, Lampinen J (2003) A trigonometric mutation operation to differential evolution. J Glob Optim 27(1):105–129. doi:10.1023/A:1024653025686 CrossRefGoogle Scholar
  22. 22.
    Feoktistov V, Janaqi S (2004) New strategies in differential evolution. In: Parmee I (ed) Adaptive Computing in Design and Manufacture VI. Springer, London, pp 335–346CrossRefGoogle Scholar
  23. 23.
    Flacke W, Kraus B, Warcup C (2005) Working with projections and datum transformations in ArcGIS: theory and practical examples. Points Verlag. http://books.google.cz/books?id=PfEsAQAAMAAJ
  24. 24.
    Fletcher R (1970) A new approach to variable metric algorithms. Comput J 13 (3):317–322CrossRefGoogle Scholar
  25. 25.
    Fletcher R (1987) Practical methods of optimization, 2nd. Wiley-Interscience, NY, USAGoogle Scholar
  26. 26.
    Fletcher R, Xu C (1987) Hybrid methods for nonlinear least squares. IMA J Numer Anal 7(3):371–389. doi:10.1093/imanum/7.3.371. http://imajna.oxfordjournals.org/content/7/3/371.abstract CrossRefGoogle Scholar
  27. 27.
    Fraley C (1987) Solution of nonlinear least-squares problems. Ph.D. thesis, CA, USA. AAI8800933Google Scholar
  28. 28.
    Fraley C, Laboratory SUSO (1988) Algorithms for nonlinear least-squares problems. Technical report (Stanford University. Systems Optimization Laboratory). Stanford University, Department of Operations Research, Systems Optimization Laboratory. http://books.google.cz/books?id=dPgEAAAAIAAJ
  29. 29.
    Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], LondonGoogle Scholar
  30. 30.
    Gämperle R, Müller S.D, Koumoutsakos P (2002) A parameter study for differential evolution. In: WSEAS International Conference on Advances in Intelligent Systems, Fuzzy Systems, Evolutionary Computation. Press, pp 293–298Google Scholar
  31. 31.
    Goldfarb D (1970) A family of variable metric methods derived by variational means. Maths Comput 24:23–26CrossRefGoogle Scholar
  32. 32.
    Gong W, Cai Z, Jiang L (2008) Enhancing the performance of differential evolution using orthogonal design method. Appl Math Comput 206(1):56–69. doi:10.1016/j.amc.2008.08.053 Google Scholar
  33. 33.
    Harley JB (1968) The evaluation of early maps: Towards a methodology. Imago Mundi 22:62–74Google Scholar
  34. 34.
    Hooke J, Perry R (1976) The planimetric accuracy of tithe maps. The Cartographic Journal 13(2):177–183CrossRefGoogle Scholar
  35. 35.
    Hui-rong L, Yue-lin G, Chao L, Peng-jun Z (2011) Improved differential evolution algorithm with adaptive mutation and control parameters. In: Proceedings of the 2011 Seventh International Conference on Computational Intelligence and Security, CIS ’11. IEEE Computer Society, DC, USA, pp 81–85. doi:10.1109/CIS.2011.26 CrossRefGoogle Scholar
  36. 36.
    Jenny B (2006) Mapanalyst-a digital tool for the analysis of the planimetric accuracy of historical maps. e-Perimetron 1(3):239–245Google Scholar
  37. 37.
    Jenny B, Hurni L (2011) Cultural heritage: Studying cartographic heritage: Analysis and visualization of geometric distortions. Comput Graph 35(2):402–411. doi:10.1016/j.cag.2011.01.005 CrossRefGoogle Scholar
  38. 38.
    Jenny B, Weber A, Hurni L (2007) Visualizing the planimetric accuracy of historical maps with mapanalyst. Cartographica: The International Journal for Geographic Information and Geovisualization 42(1):89–94CrossRefGoogle Scholar
  39. 39.
    Jiang S, Cai Z (2010) Faster convergence and higher hypervolume for multi-objective evolutionary algorithms by orthogonal and uniform design. In: Cai Z, Hu C, Kang Z, Liu Y (eds) Advances in Computation and Intelligence, Lecture Notes in Computer Science, vol 6382. Springer, Berlin Heidelberg, pp 312–328CrossRefGoogle Scholar
  40. 40.
    Kelley CT (1995) Iterative Methods for Linear and Nonlinear Equations. No. 16 in Frontiers in Applied Mathematics. SIAMGoogle Scholar
  41. 41.
    Kowal KC, Přidal P (2012) Online georeferencing for libraries: The british library implementation of georeferencer for spatial metadata enhancement and public engagement. J Map Geogr Libr 8(3):276–289. doi:10.1080/15420353.2012.700914 CrossRefGoogle Scholar
  42. 42.
    Krejic N, Luzanin Z, Stojkovska I (2009) Gauss-newton-based bfgs method with filter for unconstrained minimization. Appl Math Comput 211(2):354–362Google Scholar
  43. 43.
    Labs M (2009) Map rectifier. http://labs.metacarta.com/rectifier/
  44. 44.
    Laxton P (1976) The geodetic and topographical evaluation of english county maps, 1740–1840. The Cartographic Journal 13(1):37–54CrossRefGoogle Scholar
  45. 45.
    Li D, Fukushima M (1999) A globally and superlinearly convergent gauss–newton-based bfgs method for symmetric nonlinear equations. SIAM J Numer Anal 37(1):152–172. doi:10.1137/S0036142998335704 CrossRefGoogle Scholar
  46. 46.
    Li DH, Fukushima M (2001) A modified BFGS method and its global convergence in nonconvex minimization. J Comput Appl Math 129:15–35CrossRefGoogle Scholar
  47. 47.
    Lilley K, Lloyd C, Campbell B (2009) Mapping 763 the realm: A new look at the gough map of britain (c.1360) 61(1):1–28. doi:10.1080/03085690802456228
  48. 48.
    Liu J, Lampinen J (2005) A fuzzy adaptive differential evolution algorithm. Soft Comput 9(6):448–462. doi:10.1007/s00500-004-0363-x CrossRefGoogle Scholar
  49. 49.
    Livieratos E (2006) Graticule versus point positioning in ptolemy cartographies. e-Perimetron 1(1): 51–59Google Scholar
  50. 50.
    Livieratos E (2006) On the study of the geometric properties of historical cartographic representations. Cartographica: The International Journal for Geographic Information and Geovisualization 41(2):165–176CrossRefGoogle Scholar
  51. 51.
    Lloyd CD, Lilley KD (2009) Cartographic veracity in medieval mapping: analyzing geographical variation in the gough map of great britain. Annals of the Association of American Geographers 99(1):27–48CrossRefGoogle Scholar
  52. 52.
    Lukšan L (1994) Computational experience with known variable metric updates. Journal of Optimization Theory and Applications 83(1):27–47CrossRefGoogle Scholar
  53. 53.
    Lukšan L (1996) Hybrid methods for large sparse nonlinear least squares. J Optim Theory Appl 89(3):575–595. doi:10.1007/BF02275350 CrossRefGoogle Scholar
  54. 54.
    Mascarenhas WF (2004) The bfgs method with exact line searches fails for non-convex objective functions. Math Program 99(1):49–61CrossRefGoogle Scholar
  55. 55.
    Murphy J (1978) Measures of map accuracy assessment and some early ulster maps. Irish Geography 11(1):88-101CrossRefGoogle Scholar
  56. 56.
    Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7(4):308–313. doi:10.1093/comjnl/7.4.308. http://comjnl.oxfordjournals.org/content/7/4/308.abstract CrossRefGoogle Scholar
  57. 57.
    Nemirovski A (2013) Optimization ii: Standard numerical methods for nonlinear continuous optimizationGoogle Scholar
  58. 58.
    Nocedal J, Wright SJ (2006) Numerical Optimization, 2nd. Springer, New YorkGoogle Scholar
  59. 59.
    Page YP (2006) Assigning map projections to portolan maps. e-Perimetron 1 (1):40–50Google Scholar
  60. 60.
    Pedersen MEH (2010) Good parameters for differential evolution. Technical Report, Hvaas LaboratoriesGoogle Scholar
  61. 61.
    Price K, Storn RM, Lampinen JA (2005) Differential Evolution: A Practical Approach to Global Optimization (Natural Computing Series). Springer-Verlag New York, Inc, NJ, USAGoogle Scholar
  62. 62.
    Price KV (1999) New ideas in optimization. Chapter An introduction to differential evolution. McGraw-Hill Ltd, UK, England, pp 79–108. http://dl.acm.org/citation.cfm?id=329055.329069 Google Scholar
  63. 63.
    Přidal MP, Zǎbička P (2008) Tiles as an approach to on-line publishing of scanned old maps, vedute and other historical documents. e-Perimetron 3(1):10–21Google Scholar
  64. 64.
    Pridal P (2011) Georeferencer. http://www.georeferencer.org
  65. 65.
    Qin AK, Huang VL, Suganthan PN (2009) Differential evolution algorithm with strategy adaptation for global numerical optimization. Trans Evol Comp 13(2):398–417. doi:10.1109/TEVC.2008.927706 CrossRefGoogle Scholar
  66. 66.
    Rajaković M, Kljajić I, Lapaine M (2014) Map projection reconstruction of a map by mercator. In: Buchroithner M, Prechtel N, Burghardt D (eds) Cartography from Pole to Pole, Lecture Notes in Geoinformation and Cartography. Springer, Berlin Heidelberg, pp 31–44CrossRefGoogle Scholar
  67. 67.
    Ravenhill W (1981) Projections for the large general maps of britain, 1583-1700. Imago Mundi 33(1):21–32. doi:10.1080/03085698108592512 Google Scholar
  68. 68.
    Ravenhill W, Gilg A (1974) The accuracy of early maps? Towards a computer aided method. The Cartographic Journal 11(1):48–52CrossRefGoogle Scholar
  69. 69.
    Shanno DF (1970) Conditioning of quasi-newton methods for function minimization. Math Comput 24(111):647–656. doi:10.2307/2004840 CrossRefGoogle Scholar
  70. 70.
    Storn R, Price K (1995) Differential Evolution- A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces, Technical Report. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.1.9696
  71. 71.
    Storn R, Price K (1996) Minimizing the real functions of the icec’96 contest by differential evolution. In: Evolutionary Computation, 1996., Proceedings of IEEE International Conference on. IEEE, pp 842–844Google Scholar
  72. 72.
    Strang A (1998) The analysis of ptolemys geography. The Cartographic Journal 35(1):27–47CrossRefGoogle Scholar
  73. 73.
    Tobler WR (1965) Computation of the correspondence of geographical patterns. In: Papers of the Regional Science Association, vol 15. Springer, pp 131–139Google Scholar
  74. 74.
    Tobler WR (1966) Medieval distortions: The projections of ancient maps. Ann Assoc Am Geogr 56(2):351–360. doi:10.1111/j.1467-8306.1966.tb00562.x CrossRefGoogle Scholar
  75. 75.
    Tobler WR (1977) Numerical approaches to map projections. Beitr zur theoretischen Kartographie, Festschrift fur Erik Amberger, hg 1(14):51–64Google Scholar
  76. 76.
    Tobler WR (1986) Measuring the similarity of map projections. Am Cartogr 13 (2):135–139CrossRefGoogle Scholar
  77. 77.
    Tobler WR (1994) Bidimensional regression. Geographical Analysis 26(3):187–212CrossRefGoogle Scholar
  78. 78.
    Yerci M (1989) The accuracy of the first world map drawn by piri reis. The Cartographic Journal 26(2):154–155CrossRefGoogle Scholar
  79. 79.
    Zaharie D (2003) Control of Population Diversity and Adaptation in Differential Evolution Algorithms. In: Matousek R, Osmera P (eds) Proceedings of Mendel 2003, 9th International Conference on Soft Computing, Brno, Czech Republic, pp 41–46Google Scholar
  80. 80.
    Zhou W, Chen X (2010) Global convergence of a new hybrid gauss-newton structured bfgs method for nonlinear least squares problems. SIAM J Optim 20 (5):2422–2441. doi:10.1137/090748470 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Applied Geoinformatics and Cartography, Faculty of SciencesCharles University in PraguePragueCzech Republic

Personalised recommendations