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ASTAP, the Astrometric STAcking Program, astrometric solver and FITS viewer

How does ASTAP, Astrometric STAcking Program solving works:

ASTAP is using a linear astrometric solution for both stacking and solving.  The method is based on what traditionally is called "reducing the plate measurements. First step is to find star matches between a test image and a reference image. The reference image is either created from a star database or a reference image. The star positions x, y are to be calculated in standard coordinates which is equivalent to the sensor x,y pixel position. The x,y positions are measured relative to the image center.

The test image center, size and orientation position will be different compared with the reference image. The required conversion from the test image [x,y] star positions to the same stars on the test images can be written as:

Xref : = a*xtest + b*ytest + c
Yref :=  d*xtest + e*ytest + f

The factors, a,b,c,d,e,f are called the six plate constants and will be slightly different different for each star. They describe
the conversion of  the test image standard coordinates to the reference image standard coordinates. Using a least square routine the best solution fit can be calculated if at least three matching star positions are found since there are three unknowns.

With the solution and the equatorial center position of the reference image the test image center equatorial position, α and δ can be calculated. From the test image center using small one pixel steps in x, y, the differences in α, δ can be used to calculate the image scale and orientation.

For astrometric solving (plate solving), this "reducing the plate measurement" is done against star positions extracted from a database. The resulting absolute astrometric solution will allow specification of the α, δ equatorial positions of each pixel.
For star alignment this
"reducing the plate measurement" is done against a reference image. The result six plate constants are a relative astrometric solution. The position of the reference image is not required. Pixels of the solved image can be stacked with reference image using the six plate constants only.

To automate this process
rather then using reference stars the matching reference objects are the center positions of tetrahedrons made of four close stars.  Comparing the length ratios of the sides of the tetrahedrons allows automated matching.

Below a brief flowchart of the ASTAP astrometric solving process:

 ASTAP  astrometric solving method (plate solving) rev 2019-10-28 Image Star database 1 Find background, noise and star level 2 Find stars and their CCD x, y position (standard coordinates) Extract a similar amount of stars for the area of interest from the star database that matches the star density of the image.Convert the α, δ equatorial positions into standard coordinates (CCD pixel x,y coordinates for optical projection) using the rigid method. 3 Use the extracted stars to construct the smallest irregular tetrahedrons of four stars (quads). Calculate the length of the six tetrahedron edges in pixels and the mean x, y position of  tetrahedrons. Use the extracted stars to construct the smallest irregular tetrahedrons of four stars (quads). Calculate the length of the six tetrahedron edges in pixels and the mean x, y position of  tetrahedrons. 4 Sort the six tetrahedron edges on length for each tetrahedron. e1 is the longest and e6 shortest. Sort the six tetrahedron edges on length for each tetrahedron. e1 is the longest and e6 shortest. 5 Scale the tetrahedron edges as (e1, e2/e1,e3/e1,e4/e1,e5/e1,e6/e1) Scale the tetrahedron edges as (e1, e2/e1,e3/e1,e4/e1,e5/e1,e6/e1)) 6 Find tetrahedrons matches where edges e2/e1 to e6/e1 match within a small tolerance. 7 For the matching tetrahedrons, calculate the size ratio e1found/e1reference and find the $\textstyle {\overline {x}}$μ (mean), σ (standard deviation) of these ratios.Remove the outlier tetrahedrons with a ratio above 3*σ. 8 From the remaining matching tetrahedrons, prepare the "A" matrix/array containing the x,y center positions of the test image tetrahedrons in standard coordinates and  the arrays Xref, Yref containing the x, y center positions of the reference image tetrahedrons in standard coordinates.  A:         Sx:    Xref:  [x1 y1  1]     [a]     [X1]  [x2 y2  1]  *  [b]  =  [X2]  [x3 y3  1]     [c]     [X3]  [x4 y4  1]             [X4]  [.. ..  .]             [..]  [xn yn  1]             [Xn]  A:         Sy:    Yref:  [x1 y1  1]     [d]     [Y1]  [x2 y2  1]  *  [e]  =  [Y2]  [x3 y3  1]     [f]     [Y3]  [x4 y4  1]             [Y4]  [.. ..  .]             [..]  [xn yn  1]             [Yn]Find the solution matrices Sx and Sy of  this overdetermined system of linear equations.The solutions Sx and Sy describe the six parameter  plate solution Xref:=a*x + b*y + c and Yref:=d*x + e*y +f. 9 With the solution and the equatorial center position of the reference image the test image center equatorial position, α and δ can be calculated.Make from the test image center small one pixel steps in x, y and use the differences in α, δ to calculate the image scale and orientation.This is the final solution. The solution vector (for position, scale, rotation) can be stored as the FITS keywords crval1, crval2, cd1_1,cd1_2,cd_2_1, cd2_2.

The following image shows the irregular tetrahedrons used in an astronomical image:  Version: 2019-10-28

(c) Han Kleijn, www.hnsky.org, 2018, 2019