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The dispersion relation is defined by the following equations:
The first of the equations defines the position of the spectral orders, m, in the raw image, while the second equation gives, for each order, the dispersion relation in one dimension. The mapping between the spaces and is separated into two different equations; the first one will be discussed in this Section, while the description of the second equation will be postponed to Section .
The function is approximated by a polynomial of the form
where the coefficients are computed using least squares techniques on a grid , i.e. sample number and line number of points located within the spectral orders of the image. These points in the grid are found automatically by an order--following algorithm, normally using the FLAT or STD image.
This set of points forms the basic grid with the geometric positions of the orders. Typical values of the standard deviation of the residuals of this approximation are about 0.3 to 0.1 pixel.
It is worth mentioning here that the order following algorithm finds the center of the orders by taking the middle point with respect to the edges of the orders. The edges of the orders are detected automatically by thresholding the order profiles, perpendicular to the dispersion direction; the level of the threshold is a function of the signal in the order. The command DEFINE/ECHELLE performs the automatic order detection.
An alternative method is available, based on the Hough transform to perform the order detection and involving a tracing algorithm able to estimate an optimal threshold for each order independently. The order definition is performed as follows:
This algorithm is implemented in the command DEFINE/HOUGH. The algorithm can run in a fully automatic mode (no parameters are required apart from the name of the input frame). It is also possible to set the following parameters to enforce a given solution: