|ASTAP astrometric solving method (plate solving)||rev 2019-10-28|
|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 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.
the solution and the equatorial center position of the reference image
the test image center equatorial position, α and δ can be
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.
(c) Han Kleijn, www.hnsky.org, 2018, 2019