1 Introduction to astrometric solving


1.1 How do humans do it:

Humans look for the brightest stars in the sky, determine the shortest distances between the stars (constellation lines) and estimate the angles these lines form. They see imaginary constellation figures from close bright stars and compare them with constellations figures they have in their memory and will recognize a match.
 

1.2 How does the solver do it: 

In ASTAP solver more or less the same process is followed. Select the brightest stars, connect them and measure the distances between them. ASTAP will search for four closest stars and measure the distances between them (the hash code or constellation). This process will be done for the image and repeated for the stars in the star database. These hash codes created from the star database will be compared with the image hash codes till some matches are found. Once some matches are found it is possible to calculate the precise position of the image with the matching database stars.
 

This matching process is identical as used for image aligning routine in a stack program. For aligning a match has to be found between the stars in the reference image and a second image prior to stacking. It is possible to compare the distances between two stars, three stars (triangles using the connecting lines) or four stars (quads or tetrahedrons using the connecting lines). Four stars works best with sufficient uniqueness but not being too complicated. It is not required to measure the angles.
 
Irregular tetrahedron figures formed by the star connection lines:


1.3 So how is this done in more detail:
 
The image is analysed for stars and it maybe finds 2000 stars and their x, y positions. It takes the 500 brightest stars and creates from them maybe 300 tetrahedron figures of four stars. Between each four stars six lines can be drawn forming the tetrahedron. Since the image scale is specified, the distances between the four stars could be (example): 


To make it independent of image size, the distance ratios are calculated by dividing the distances through the largest distance. The longest distance is kept for comparison later. So the hash code or constellation figure will consist of two parts:


The five ratios are sufficient for the search. Rotation, scaling, flipping  of the image has no influence on the "five ratios". The "largest distance" is later used to detect outliers.

From the star database numerous artificial images are created using the same star density per degree following a search pattern around the initial Ra, Dec position. The artificial images hash codes are compared with the hash codes of the image to solve. This is the actual search. Only at about the image location a part of the hash codes will match within a small tolerance. In addition the  "largest distances" should have a fixed ratio between the image and artificial database image.

2 How does ASTAP, Astrometric STAcking Program solving works in even more detail:

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:



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 distance ratios between the four stars allows automated matching.


Below a brief flowchart of the ASTAP astrometric solving process:

		ASTAP  astrometric solving method (plate solving)
	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 figures of four stars called quads. Calculate the six distances between the four stars in pixels and the mean x, y position of the four stars.		Use the extracted stars to construct the smallest irregular tetrahedrons figures of four stars called quads. Calculate the six distances between the four stars in pixels and the mean x, y position of the four stars.
4	Sort the six distances for each quad. Label them all where d1 is the longest and d6 the shortest distance.		Sort the six distances for each quad. Label them all where d1 is the longest and d6 the shortest distance.
5	Scale the six quad star distances as (d1, d2/d1,d3/d1,d4/d1,d5/d1,d6/d1). These are the image hash codes		Scale the six quad star distances as (d1, d2/d1,d3/d1,d4/d1,d5/d1,d6/d1)) These are the database hash codes
6		Find quad hash code matches where the five ratios d2/d1 to d6/d1 match within a small tolerance.
7		For the matching quad hash codes, calculate the longest side ratios d1database/d1image in ["/pixels]. Calculate the median ratio. Compare the quads ratios with the median value and remove quads outside a small tolerance.
8		From the remaining matching quad hash codes, prepare the "A" matrix/array containing the x,y center positions of the test image quads in standard coordinates and  the arrays Xref, Yref containing the x, y center positions of the reference image quads 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. Calculate from the solution the pixel size in x (cdelt1) an y (cdelt2) and at the image center position the rotation of the x-axis (crota1) and y-axis (crota2) relative to the celestial north using goniometric formulas. Convert these to cd1_1,cd1_2,cd_2_1, cd2_2. 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.

Note: For a low faint star count (<30) the star patterns can be slightly different between image and database due to small magnitude differences between the database and measured flux. For these cases it can be beneficial to extract triples (three stars patterns) from the found quads (four star patterns). For triples a stricter tolerance check is required to avoid false detections.