Are there any general rules of thumb for selecting a transformation for Georeferencing?

Question

There are several transformations that can be used when georeferencing a raster:

• 1st order Polynomial (affine)

• 2nd order Polynomial

• 3rd order Polynomial

• Spline... etc

Are there any specific rules or rules of thumb, about which transformation should be used with a particular raster?

For example a particular transformation should generally be used with aerial photographs & another should be used with satellite imagery while a third should be used while digitizing maps?

Is there any book which explain the mathematics behind this in detail?

Answer 1

Last first: this is the material of regression analysis and statistical modeling. Polynomial regression is discussed in most textbooks. (Draper and Smith treat it in chapter 5.) There really is nothing special (mathematically) about georeferencing compared to Excel's fitting of a straight line to a scatterplot: there are just more variables. (You can actually persuade Excel to georeference for you if you trust it ;-).

Now for some rules of thumb:

1. Use the lowest order that meets your accuracy requirements. (For more about this, see #7 below.)
2. Use a method that can represent the distortions that might have occurred. With paper map scanning, the distortions can be local and irregular, so consider splines. With changes of projection (including those that occur in most aerial and satellite image processing) the proper transformation to use is a projective one. Projective transformations are neither polynomials (in general) nor splines. If your software does not support them, a second-order polynomial often provides a reasonable approximation.
3. Do not overfit. There is a limit to the accuracy of any geodata source. You want the rmse to be approximately equal to that inaccuracy, not significantly lower.
4. Especially when using higher powers (polynomials of order 2 or greater), weed out outlying links. Even a single outlier can grossly distort the transformation. Cross-validate the results by creating some links not used to compute the transformation and checking how well the transformation resolves them.
5. For greatest accuracy, georeference the smallest area you can. There's no point to georeferencing lots of area beyond what you're interested in and doing so can worsen the quality of the map within your study area.
6. For flexibility, georeference a slightly larger area. Sooner or later you might need to bring in data from adjoining regions: it will help immensely to have some common control points near boundaries where they overlap. (This rule conflicts with the preceding one!)
7. Favor creating control points around the boundary of the study area. Geometrically, most of your region lies near its boundary anyway; statistically, you get the best information from these extremal control points. Use a few within the interior for checking the fit and assessing the polynomial order. (If the fit based on the boundary points alone is acceptable but the interior points do not fit well, you might need to increase the polynomial order.)