Pareto-based evolutionary algorithms for the calculation of transformation parameters and accuracy assessment of historical maps

Highlights

• Historical map data involves transformation of the coordinates to the current system.

• The main problem is that the quality of historical data is heterogeneous.

• Sometimes it is necessary to discard some of the historical input data.

• The proposed method is Pareto front based on evolutionary genetic algorithms.

• This method reduces linear error by 40% by eliminating only 2% of the points used.

Abstract

When historical map data are compared with modern cartography, the old map coordinates must be transformed to the current system. However, historical data often exhibit heterogeneous quality. In calculating the transformation parameters between the historical and modern maps, it is often necessary to discard highly uncertain data. An optimal balance between the objectives of minimising the transformation error and eliminating as few points as possible can be achieved by generating a Pareto front of solutions using evolutionary genetic algorithms. The aim of this paper is to assess the performance of evolutionary algorithms in determining the accuracy of historical maps in regard to modern cartography. When applied to the 1787 Tomas Lopez map, the use of evolutionary algorithms reduces the linear error by 40% while eliminating only 2% of the data points. The main conclusion of this paper is that evolutionary algorithms provide a promising alternative for the transformation of historical map coordinates and determining the accuracy of historical maps in regard to modern cartography, particularly when the positional quality of the data points used cannot be assured.

Introduction

Historical maps are an important part of our cultural heritage (Jenny and Hurni, 2011). These maps not only represent valuable physical artefacts but also provide an important information source for historians and geographers, who frequently incorporate historical data into Geographical Information System (GIS) (Weir, 1997, Audisio et al., 2009). The scales, coordinate systems, projections, and surveying and mapping techniques used in historical maps vary widely (Podobnikar, 2009). The different reference systems employed in different historical maps often require coordinate transformations between them (Tierra et al., 2008). As historical maps typically exhibit higher degrees of inaccuracy and uncertainty compared to contemporary cartographic databases, it is not surprising that these two issues are of particular concern in historical cartography studies and historical GIS applications (Plewe, 2003). Accuracy analysis of early maps is therefore an important topic in historical cartography (Harley, 1968).

The positional accuracy of a point on a map is defined as the difference between its recorded location on the map and its actual location on the ground or location on a source of known higher accuracy (Tucci and Giordano, 2011).

The coordinate method calculates the correlation between two sets of map coordinates of points identified by modern latitudes and longitudes (Tobler, 1966). The deviation between each computer-generated point (digitised from an early map) and the corresponding point on the modern map can be displayed as a vector indicating the direction and magnitude of the error (Ravenhill and Gilg, 1974).

A related problem is the transformation of coordinates between two different geodetic reference systems. A set of points with known coordinates in both reference systems is used to obtain the transformation parameters. The Helmert transformation is one example of a commonly used method for geodesic transformations between two reference frames (Vanícek and Krakiwsky, 1986, Vanícek and Steeves, 1996). The quality of a transformation between two sets of coordinates depends on the positional quality of the points used to calculate the transformation parameters.

However, a problem arises when there is substantial spatial uncertainty in the historical data points. In this case, certain points with gross errors will not be used in the calculation of the accuracy of the entire map. When the inaccuracy of a measurement is not objectively known, as is frequently the case for historical maps, the measured feature is defined as uncertain (Hunter and Goodchild, 1993). Although uncertainty and inaccuracy are ontologically distinct concepts, it is often difficult to measure the two separately in practice, particularly in the context of historical maps (Hu, 2010).

In previous papers (Manzano-Agugliaro et al., 2012), the coordinates obtained from the historical map were displaced by the average latitude and longitude error to correct the absolute displacement error in the historical map, i.e., to correct the georeferencing error. Any points whose displacements from the corresponding points in the modern map exceeded a specified distance were then eliminated from the analysis. These points were considered to contain gross errors, either because of incorrect identification with the current points or due to a gross error in the historical map. The final estimate of the map accuracy is obtained based on the root mean square (RMS) displacement after the removal of the points with gross errors.

In this historical cartography, we cannot be certain that any given point is more accurate than another. To displace the coordinates on a historical map to positions such that the final (RMS) errors are minimal, many combinations must be performed, each time discarding the points that exceed a fixed maximum RMS for a gross or unacceptable error and then calculating the new errors. A historical map may exhibit rotation (due to projection effects) as well as horizontal and vertical scale errors in addition to latitudinal and longitudinal displacement, making the number of possible combinations that must be considered even higher. If we also aim to discard as few points as possible from the historical map, then the problem becomes multi-objective.

The multi-objective nature of these mapping problems makes the decision-making process complex. Fortunately, the increase in computational resources in recent years has allowed researchers to develop efficient computational algorithms for handling complex optimisation problems. In particular, multi-objective evolutionary algorithms (MOEAs) are known for their ability to optimise several objective functions simultaneously to provide a representative Pareto front, which is a set of problem solutions representing a trade-off between the objectives (Márquez et al., 2011). The aim of this paper is to assess the performance of evolutionary algorithms in determining the accuracy of historical maps in regard to modern cartography.