Understanding the Polynomial math model

Polynomial is a simple math model that uses a first-through-fifth order polynomial transformation, which is calculated based on two-dimensional (2-D) ground control points (GCP). This math model produces the 'best' fit, mathematically, to a set of 2-D GCPs on an image.

The polynomial equations are fitted to the x and y coordinates of the GCPs by using least-squares criteria to model the correction in the image without identifying the source of the distortion. You can select from several polynomial orders depending on the accuracy required and the number of GCPs available.

First-order polynomial transformations can model a rotation, scale, and translation. Because up to 21 additional terms are added, giving a fifth-order polynomial, you can achieve more complex warping. However, using a lower-order transformation reduces the time needed to complete the correction, and less geometric distortion may occur in areas with no GCPs.

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

One GCP produces a translation for x and y only.
Two GCPs produce a translation and a scaling change for x and y, if the pixel geometry is nonlinear in the x or y dimension. If it is linear, meaning that the two GCPs have the same x or y coordinate producing a scaling factor of zero, it produces only a translation. If the scaling factor is greater than zero, it may produce a flip in the x-dimension, y-dimension, or both.
Three or more GCPs produce a translation, scale, and a change, rotation, or both for x and y: a full first-order transformation.

Note: A higher-order polynomial will result in a more accurate fit in the immediate vicinity of the GCPs, but it may introduce new and significant errors in the image away from the GCPs. It is possible that the errors introduced into the imagery may be worse than the original errors that needed correcting.