Georeferencing

In the early days of geoinformation science, spatially referenced data usually originated within national boundaries, i.e. these data were derived from printed maps published by national mapping organizations. Nowadays, users of geoinformation are combining spatial data from a given country with global spatial data sets, reconciling spatial data from published maps with coordinates established by satellite positioning techniques, and integrating their spatial data with that from neighbouring countries.To perform these kinds of tasks successfully, we need to understand basic spatial referencing concepts.

The simplest way to link image coordinates to map coordinates is to use a transformation formula. A geometric transformation is a function that relates the coordinates of two systems. A transformation relating (x, y) to (i, j) is commonly defined by linear equations, such as:  x = 3 + 5i, and y = −2 + 2.5j.

Using the above transformation, for example, the image position (i = 3, j = 4) corresponds to map coordinates (x = 18, y = 8). Once the transformation parameters have been determined, the map coordinates for each pixel can be calculated. This implies that we can superimpose data that are given in the map coordinate system on the image vector, or that we can store features by map coordinates when applying on-screen digitizing. Note that the image in the case of georeferencing remains stored in the original (i, j) raster structure and that its geometry is not altered. As we will see in Geocoding, transformations can also be used to change the actual shape of an image and thus make it geometrically equivalent to the map.

The effect of georeferencing is illustrated in the Figure below along with geocoding.  

The process of georeferencing involves two steps: (1) selection of the appropriate type of transformation, and (2) determination of the transformation parameters. The type of transformation depends mainly on the sensor–platform system used. For aerial photographs (of flat terrain) what is known as “projective transformation” models well the effect of pitch and roll. Polynomial transformation, which enables 1^st, 2^nd to n^th order transformations, is a more general type of transformation. In many situations a 1st order transformation is adequate. Such transformation relates map coordinates (x, y) with image coordinates (i, j) as follows:

Equations 1 and 2 require that six parameters (a to f) be determined. The transformation parameters can be determined by means of ground control points (GCPs). GCPs are points that can be clearly identified in the image and on the target map. Once a sufficient number of GCPs have been specified, software is used to determine the parameters a to f of the Equations 1 and 2 and quality indications.

To solve the 1^st order polynomial equations, only three GCPs are required; nevertheless, you should use more points than the strict minimum. Using merely the minimum number of points for solving the system of equations would obviously lead to a wrong transformation if you made an error in one of the measurements, whereas including more points for calculating the transformation parameters enables software to also compute the error of the transformation. Table 1 gives an example of the input and output of a georeferencing computation in which five GCPs have been used. Each GCP is listed with its image coordinates (i, j) and its map coordinates (x, y).

Software performs a “least-squares adjustment” to determine the transformation parameters. The least squares adjustment ensures an overall best fit of the GCPs. We then use the computed parameter values to calculate coordinates (x_c, y_c) for any image point (pixel) of interest:

and

For example, for the pixel corresponding to GCP 1 (i = 254, j = 68) we can calculate the transformed image coordinates x_c and y_c as 958.552 and 154.935, respectively. These values deviate slightly from the input map coordinates (as measured on the map). Discrepancies between measured and transformed coordinates of GCPs are called residual errors (residuals for short). The residuals are listed in the table as d_x and d_y. Their magnitude is an indicator of the quality of the transformation. Residual errors can be used to analyse whether all GCPs have been correctly determined.

GCP Accuracy

The overall accuracy of a transformation is either stated in the accuracy reportusually provided by software in terms of variances or as Root Mean Square Error (RMSE), which calculates a mean value from the residuals (at check points). The RMSE in the x-direction, m_x, is calculated using the following equation:

For the y-direction, a similar equation can be used to calculate m_y. The overall error, m_p, is calculated by:

For the example data set given in Table 1, the residuals m_x, m_y and m_p are 1.159, 0.752 and 1.381, respectively. The RMSE is a convenient measure of overall accuracy, but it does not tell us which parts of the image are accurately transformed and which parts are not. Note also that the RMSE is only valid for the area that is bounded by the GCPs. In the selection of GCPs, therefore, points should be well distributed and include locations near the edges of the image.