Parent Topic: Theory

Polynomial Transformations

The transformation model creates a new geocoded image space where interpolated pixel values will later be placed during resampling. The procedure requires that polynomial equations be fitted to the GCPs using least squares criteria to model the correction in the image domain without identifying the source of the distortion. One of several polynomial orders may be chosen based on the desired accuracy and the available number of GCPs.

The polynomial transformation is a 1st to 5th order polynomial which mathematically describes how the uncorrected image has to be warped to make it register (fit) over the georeferenced image. There are a number of georeferenced data types available in GCPWorks for image registration. These can be chosen from the main GCPWorks panel:

Geocoded Image
Hardcopy Map on Digitizing Table
Vectors
User Entered Coordinates
Chip Database

In general, polynomial transformations with terms up to the first order can model a rotation, a scale, and a translation, and are computationally economical. As additional terms are added, (up to 21 in all, giving a 5th order polynomial), more complex warpings can be achieved. If a lower order transformation will suffice, there are two important reasons for using it:

the correction program will run faster
there is less chance of geometric distortion in areas of no GCPs

The number of required GCPs depends on the order of the polynomial. The following chart lists the minimum required number of GCPs.

         Required GCPs                Order
               7                       2nd
              11                       3rd
              16                       4th
              22                       5th

In practice, more than the required number of GCPs should be collected, so that any GCPs which contain significant positional errors on either the georeferenced map or the uncorrected image will have their errors reduced by averaging.

The result of a first order transformation depends on the number of GCPs:

One GCP will produce a translation for only X and Y.
Two GCPs will produce a translation and a scaling change for X and Y if the pixel geometry is not linear in the X or Y dimension. If it is linear, (that is, the two have the same X or Y coordinate, producing a scaling factor of zero), only a translation will be produced. If a scale change is produced, this set-up may be used to produce a flip in the X and/or Y dimension.
Three or more GCPs produce a translation, scale change, and/or rotation for X and Y (a full first order transformation).

It is important to note that while a higher order polynomial will result in a more accurate fit in the immediate vicinity of the GCPs, it may introduce new significant errors in those parts of the image away from the GCPs. Therefore, worse errors may be introduced into the imagery than were to be corrected.

Polynomial Equations

The following equations show the polynomials used for orders one, two, and three. Orders four and five are extrapolations from these with four and five order terms added.

The current polynomial equation being used can be determined by selecting ``Model Coefficients'' from the ``Reports'' menu on the GCP Selection and Editing panel.

Determining the Accuracy of the Polynomial Fit

In order to determine the accuracy of the derived coefficients of the GCPs, examine the results of the least squares regression of the initial GCPs:

The GCP scatter plot report, which can be activated by selecting ``GCP Scatter Plot'' from the ``Reports'' menu on the GCP Selection and Editing panel, shows the X and Y residual errors for each GCP point on a cross hair graph.
The RMS error for each GCP is reported in the lists of Accepted and Check GCPs on the ``GCP Selection and Editing'' panel. A total RMS error is also reported on this panel. The RMS error is calculated in pixel units by the following equation:

By computing the RMS errors for all of the GCPs, it is possible to see which GCPs exhibit the greatest error, and to sum the RMS errors. If a given set of control points produces a total RMS error which exceeds your acceptable limit (a limit of less than the dimension of one pixel is suggested), you should consider deleting those GCPs which have the greatest errors.

The Image Fit Report is also available to help you assess the fit of the regression model. This report can be activated by selecting ``Image Fit'' from the ``Reports'' menu on the GCP Selection and Editing panel. It shows a graphical representation of the outline of the georeferenced image area, uncorrected image area, mosaic cut line, and GCP points. The uncorrected image outline is transformed according to the current GCP model. This preview of how the uncorrected image would map onto the georeferenced data set with the current set of GCPs allows the registration to be visually assessed before it is performed.

Parent Topic: Theory
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