Estimation of the Cartographic Projection and~its Application in Geoinformatics-habilitation thesis presentation

Tomáš Bayer

Abstract


Modern techniques for the map analysis allow for the creation of full or partial geometric reconstruction of its content. The projection is described by the set of estimated constant values: transformed pole position, standard parallel latitude, longitude of the central meridian, and a constant parameter. Analogously the analyzed map is represented by its constant values: auxiliary sphere radius, origin shifts, and angle of rotation. Several new methods denoted as M6-M9 for the estimation of an unknown map projection and its parameters differing in the number of determined parameters, reliability, robustness, and convergence have been developed. However, their computational demands are similar. Instead of directly measuring the dissimilarity of two projections, the analyzed map in an unknown projection and the image of the sphere in the well-known (i.e., analyzed) projection are compared. Several distance functions for the similarity measurements based on the location as well as shape similarity approaches are proposed. An unconstrained global optimization problem poorly scaled, with large residuals, for the vector of unknown parameters is solved by the hybrid BFGS method. To avoid a slower convergence rate for small residual problems, it has the ability to switch between first- and second-order methods. Such an analysis is beneficial and interesting for historic, old, or current maps without information about the projection. Its importance is primarily referred to refinement of spatial georeference for the medium- and small-scale maps, analysis of the knowledge about the former world, analysis of the incorrectly/inaccurately drawn regions, and appropriate cataloging of maps. The proposed algorithms have been implemented in the new version of the detectproj software.

Keywords


Map projection; analysis; detection; early maps; location similarity; optimization; non-linear least squares; BFGS; georeference; M-estimators; Huber function

References


M Al-Baali and R Fletcher. Variational methods for non-linear least-squares. Journal of the Operational

Research Society, pages 405–421, 1985.

Ádám Barancsuk. A Semi-automatic Approach for Determining the Projection of Small Scale Maps

Based on the Shape of Graticule Lines, pages 267–288. Springer International Publishing, Cham, 2016.

Tomas Bayer. Estimation of an unknown cartographic projection and its parameters from the map.

GeoInformatica, 18(3):621–669, 2014.

Tomas Bayer. Advanced methods for the estimation of an unknown projection from a map. GeoInformatica,

(2):241–284, 2016.

Tomas Bayer. detectproj - software for the projection analysis, 2017.

Tomas Bayer. Estimation of an Unknown Cartographic Projection and its Parameters form a Map.

Czech Technical University in Prague, 2017. Habiltation thesis.

Å. Björck. Numerical Methods for Least Squares Problems. SIAM, Philadelphia, 1996.

Chryssoula Boutoura. Assigning map projections to portolan maps. e-Perimetron, 1(1):40–50, 2006.

John E Dennis Jr, Sheng Songbai, and Phuong A Vu. A memoryless augmented gauss-newton method

for nonlinear least-squares problems. Technical report, DTIC Document, 1985.

W. Flacke, B. Kraus, and C. Warcup. Working with projections and datum transformations in ArcGIS:

theory and practical examples. Points Verlag, 2005.

R. Fletcher. Practical methods of optimization (2nd ed.). Wiley-Interscience, New York, NY, USA, 1987.

R. Fletcher and C. Xu. Hybrid methods for nonlinear least squares. IMA Journal of Numerical Analysis,

(3):371–389, 1987.

Gene H. Golub and Charles F. Van Loan. Matrix computations (3rd ed.). Johns Hopkins University

Press, Baltimore, MD, USA, 1996.

Colin R. Goodall. Computation using the qr decomposition. In Computational Statistics, volume 9 of

Handbook of Statistics, pages 467 – 508. Elsevier, 1993.

P. Horata, S. Chiewchanwattana, and K. Sunat. A comparative study of pseudo-inverse computing

for the extreme learning machine classifier. In Data Mining and Intelligent Information Technology

Applications (ICMiA), 2011 3rd International Conference on, pages 40–45, Oct.

Bernhard Jenny. Mapanalyst-a digital tool for the analysis of the planimetric accuracy of historical

maps. e-Perimetron, 1(3):239–245, 2006.

Bernhard Jenny. Map analyst, 2011.

Bernhard Jenny and Lorenz Hurni. Studying cartographic heritage: Analysis and visualization of geometric

distortions. Computers & Graphics, 35(2):402–411, 2011.

Keith D. Lilley, Christopher D Lloyd, and Bruce M. S. Campbell. Mapping the realm: A new look at the

gough map of britain (c.1360). Imago Mundi: The International Journal for the History of Cartography,

(1):1–28, 2009.

Evangelos Livieratos. Graticule versus point positioning in ptolemy cartographies. e-Perimetron,

(1):51–59, 2006.

Christopher D Lloyd and Keith D Lilley. Cartographic veracity in medieval mapping: analyzing geographical

variation in the gough map of great britain. Annals of the Association of American Geographers,

(1):27–48, 2009.

L Lukšan. Computational experience with known variable metric updates. Journal of Optimization

Theory and Applications, 83(1):27–47, 1994.

L. Lukšan and Emilio Spedicato. Variable metric methods for unconstrained optimization and

nonlinear least squares. Journal of Computational and Applied Mathematics, 124(1):61–95, 2000.

L. Lukšan. Hybrid methods for large sparse nonlinear least squares. Journal of Optimization Theory

and Applications, 89(3):575–595, 1996.

William Ravenhill. Projections for the large general maps of britain, 1583-1700. Imago Mundi, 33(1):21–

, 1981.

Alastair Strang. The analysis of ptolemy’s geography. The Cartographic Journal, 35(1):27–47, 1998.

Waldo R. Tobler. Medieval distortions: The projections of ancient maps. Annals of the Association of

American Geographers, 56(2):351–360, 1966.

Waldo R. Tobler. Numerical approaches to map projections. eitrage zur theoretischen Kartographie,

Festschrift für Erik Amberger, hg., (14):51–64, 1977.

Waldo R Tobler. Measuring the similarity of map projections. The American Cartographer, 13(2):135–

, 1986.

Waldo R Tobler. Bidimensional regression. Geographical Analysis, 26(3):187–212, 1994.

Weijun Zhou and Xiaojun Chen. Global convergence of a new hybrid gauss-newton structured bfgs

method for nonlinear least squares problems. SIAM J. on Optimization, 20(5):2422–2441, June 2010.


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