pixels in the images based on row and column number) to an output coordinate system, a spatial reference system of the user's choice, such as the geographic coordinate system or a particular Universal Transverse Mercator zone. It is thus the extension of the typical task of curve fitting a relationship between two variables to four dimensions. The goal is to have a pair of functions of the form: Such that for every pixel in the image ( being its column and row number, respectively), a corresponding real-world coordinate can be calculated. Several types of functions are available in most GIS and remote sensing software for georeferencing. As the simplest type of two-dimensional curve is a straight line, so the simplest form of coordinate transformation is a linear transformation, the most common type being the affine transformation: Where A-F are constant coefficients set for the entire image. These formulas allow an image to be moved (the C and F coefficients specify the desired location of the top left corner of the image), scaled (without rotation, the A and E coefficients specify the size of each cell or spatial resolution), and rotated. In the last case, if the cell size is r in both the x and y directions, and the image is to be rotated α degrees counter-clockwise, then . The world file developed by Esri is a commonly used sidecar file that specifies these six coefficients for image georeferencing. Higher order polynomial transformations are also commonly used. For example, a Second-order polynomial transformation would be: The second-order terms (and third-order terms in a third-order polynomial) allow for the variable warping of the image, which is especially useful for removing the inherent distortion in aerial photographs. In addition to global parametric formulas, piecewise formulas can also be used, which transform different parts of the image in different ways. A common example is a Thin plate spline transformation. The GCP method It is very rare that a user would specify the parameters for the transformation directly. Instead, most GIS and remote sensing software provides an interactive environment for visually aligning the image to the destination coordinate system. The most common method for doing this is to create a series of ground control points (GCP). A ground control point is a location that can be identified on both the image and the ground, so that it has precise coordinates in both the image