WCSTools is a software suite consisting of both of command line programs and C language utility programs and subroutines. The software has a direct interface with, amongst others, the USNO A2.0 catalogue (see Section 5.9.3), and can also interfaces directly with FITS files, enabling the direct implant of a computed WCS to the image frame. The actual algorithm used to “pattern match” the catalogue to the image has not been explicitly released by Mink, however from the information needed to be supplied its form can be inferred to be similar to that used in version 2. The necessary information are (i) the pixel location of detected stars, (ii) the stars’ associated brightness (iii) the CCD pixel scale and (iv) the telescope pointing (extracted from the FITS header). The list of stars is to be presented in order of descending brightness.
Whilst the algorithm used is not available the process is; first a nominal WCS is identified from the FITS header, the appropriate files of the USNO A2.0 are then searched for all stars within range of the world canvas. Each reference star in the catalogue is then fitted with each image star and an offset computed using the nominal WCS. Every reference star position in the image is then checked for an image star within the specified pixel tolerance. The offset with the most matches is deemed to be the best fits (Mink, 2000). The fitting process is carried out using a multi-dimensional minimisation algorithm, see below.
A simplex is the geometrical figure which in N-dimensions consists of N+1 vertices, if any vertex of the simplex is taken as the origin then the other N vertices define vector directions that span the N-dimensional vector space, through which the search takes place. For the problem of fitting a WCS the simplex is a triangle. The simplex must be non-degenerate, meaning it encloses a finite area, to ensure convergence and the vertices must be non-collinear to ensure true N-dimensionality.
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The initial simplex defines the starting point, the function to be minimised is then calculated at each point of the simplex and ranked. The point with the highest value is moved in one of four predetermined ways (see Figure 5.6); (a) the vertex is reflected its base conserving area, (b) the vertex is reflected through its base and additionally expanded, (c) the vertex is contracted towards its base or (d) there is a contraction along all dimensions towards the low point. Firstly a reflection is tried (a) if the likelihood of this point is better than the best point then also try an expansion (b). If the reflected point is worse than the second-highest then look for an intermediate lower point, i.e., try a one dimensional contraction (c). If the high point remains unchanged by this then try a multidimensional contraction (d). In this way the simplex “oozes” towards the function’s minimum hence its popular name, the amoeba algorithm.
The amoeba algorithm is not resilient against local minima, but the initial guess supplied will be the LT pointing which will be accurate enough to overcome this problem.
However a way must be found to remove any rogue points from the airmass curves generated by the PL. This dilemma has been explored in many ways;
from the line could be removed, this however discards points without
scientific reason.
The way in which the PL constructs airmass curves is detailed below. Firstly the zeropoint mode value is calculated. The mode is a more robust estimator of the nightly zeropoint than e.g., the mean or median as it is less likely to be weighted by rogue points. This is achieved using
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A “tolerance band” is then generated around the zpx in a twofold manner. Primarily the Issac Newton Group, the site of the LT, has over many years recorded the extinction per magnitude expected over wide range of wavelengths (see La Palma Isaac Newton Group, 1995). From this a cone may be generated around the zpx widening towards greater airmass. Since the zeropoint should not increase with extinction the top half of the cone is removed to leave a triangular tolerance band, see Figure 5.7. Secondly, since the values used to formulate this tolerance band are rigidly set a lenience value allows for less than perfect results. If points fall outside the tolerance band then they are removed. A line may then be generated using the remaining points within the tolerance band. The line is generated using a least-squares-fit which minimises the residuals of the points about the line.
Due to the modular nature of the PL neither the CFITSIO or SExtractor upgrades have required recoding. The future-proof nature of the PL infrastructure has been indicated.
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