Coordinate Transformations

Coordinate transformations are methods that either convert the coordinates of locations between coordinate systems or alter locations’ coordinates within the same coordinate system. Coordinate transformations are used commonly in geographic information science to assign or convert a geographic reference for data locations, to correct inaccuracies in remote sensing data pixel locations, and to change the display of geographic data on a computer screen. A coordinate transformation is specified by a mathematical model, or function, that relates locations in a source coordinate system to their corresponding locations in a destination coordinate system. When an appropriate transformation model is not known, the [Page 581]coordinates in each reference system are used to either globally or locally calibrate a model from a set of point locations; then, the calibrated model is used to transform the data set.

Map projections are a commonly used coordinate transformation that apply a mathematical model to, for example, go from spherical coordinates of latitude and longitude, specified in degrees, minutes, and seconds, to planar map coordinates of Easting and Northing, specified in meters. This may involve a transformation between datums as well. Often, mathematical functions for converting between coordinate systems are not known. This is particularly the case for georeferencing remote sensor data or coregistering two geographic data sources to each other. In such cases, the model must be calibrated by establishing relationships between several locations in each coordinate system. These locations are referred to as control points (CP).

Affine Transformation

An affine transform is the most basic linear transformation model and is defined by the following equations:

where (u, v) are coordinates in the target system, (x, y) are the coordinates in the source system, and ß0 through ß5 are the parameters to be estimated. These equations are solved simultaneously using a least squares technique so that the root mean-squared error,

is minimized. The affine transformation requires at least three CPs. Once the parameters have been estimated by solving the affine equations using CP coordinates, the transformation can be applied to the entire data set.

Polynomial Models

Polynomial models are often used to transform coordinates that have greater distortion or geometric error between the input and output coordinate systems that are not accounted for by the affine. The general form of polynomial models is

where ßij and αij are the parameters to be estimated.