A rational functions math model is a simple math model that builds a correlation between the pixels and their ground locations.
A rational functions math model can be more accurate than Polynomial or Thin Plate Spline, because it considers elevations.
A rational functions math model is computed separately for each image. A ratio of two polynomial functions is used to compute the image row, and a similar ratio to compute the image column. All four polynomials are functions of three ground coordinates: latitude, longitude, and height or elevation. The polynomials are described by using a set of up to 20 coefficients, although some of the coefficients are often zero. The polynomial coefficients, often called Rapid Positioning Capability (RPC) data, can be obtained by using one of the rational functions math models available in OrthoEngine.
Available with Optical Satellite Modeling and Radar Satellite Modeling, the image-distribution agency of this math model computes the polynomial coefficients for each image and distributes the data with the images.
You can refine the math-model solution built from the coefficients by adding GCPs. If you have one or two GCPs per image, you can perform a zero-order transformation. A zero-order transformation produces a translation for x and y only. If you have at least three GCPs per image, you can perform a first-order transformation. A first-order transformation produces a translation and a rotation.
Typically, performing a first-order transformation is best, except when the GCPs are not well distributed. If your GCPs are clustered together, a first-order transformation can introduce new and significant errors in the image away from the GCPs. If your GCPs are not well distributed, you will probably obtain better results from the zero-order transformation.
IKONOS Ortho Kit imagery is distributed with an auxiliary text file, called an Image Geometry Model (IGM), containing the coefficients. Even if the coefficients are imported automatically into your project, adding GCPs can refine the math model of a project that has IKONOS imagery.
Available only with Optical Satellite Modeling, this math model determines the minimum number of required GCPs by multiplying the number of coefficients by two and then subtracting one. For example, if you want to use 10 coefficients, multiply 10 by two and then subtract one; therefore, you need 19 GCPs per image. CATALYST Professional OrthoEngine can calculate the polynomial coefficients from GCPs.
Using more coefficients will result in a more accurate fit in the immediate vicinity of the GCPs, but it can introduce new and significant errors in the image away from the GCPs. The errors introduced into the imagery may be worse than the original errors that needed correcting.
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