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# Matching between an image and a catalog

Matching means to obtain a map function with the two lists of coordinates and brightness: detected stars list from an image and extracted star data list from a catalog. If which data is the counterpart of any star in the image is clear, the map function can be calculated with any two pairs.

But actually the two lists are not related to each other. Therefore the system calculates many map functions with many probable counterparts and tries to find out true map function among them. In this way, it expects that the value of map function calculated with wrong counterparts hardly overlaps and the value of true map function is obtained most frequently. Definitely, the process of matching is as follows.

1. Selects three pairs from the detected stars list from the image and extracted data list from a catalog.
2. If the triangle of the three stars selected from the detected list and the triangle from a catalog are not similar, these pairs are regarded as wrong counterparts. If similar, go to next step.

3. If the brightness of each pair is very different, the pair is regarded as a wrong counterpart. If similar, go to next step.

4. Now that the three stars from the image and those from a catalog are probably counterparts, so the system calculates a set of parameters of map function.

5. For all triangles, repeat the step 1-4.
6. The most overlapping set of parameters is adopted as a true map function.

First of all, I made an experiment with a test data. The data is a 600 400 virtual night sky containing 200 stars in it. The difference of the brightest and the faintest is 10 steps. Then the system makes a copy of a partial rectangular region. It becomes an artificial photo image. The width and height of the image is about of the sky. Because there should be some errors in actual photographs, the system changes the position and brightness slightly when copying: about pixels in position, about steps in brightness. In addition, 1 - 4 new objects are also appended. Then it tried to calculate the map function between the artificial photo image and the sky. In this experiment, the approximate position input beforehand should be in the area of the image, and the approximate width of the image should differ less than 20% errors from the true value. As a result of trials so that the system could tell where the image should be in the virtual sky correctly, the process mentioned before is implemented as follows.

• The two triangles are similar when the three magnification rates calculated from each pair of edges are similar in less than 20% errors and the three rotating angles are similar in deg.
• The three pairs of stars can be really counterparts when the magnitude difference of each pair is in mag.
• It assumes that the magnification rate is between . It means the approximate width of the image must not be less than half of the true value and must not be greater than twice of the true value.
• It does not do the process for all pairs of triangles. It selects only those triangles which the three member stars have about the same brightness in order to reduce the cost. Definitely, the system at first sorts the detected star list in order of brightness, selects all sequential three stars, and finds some candidates of each selected star's counterpart from the catalog. When n stars are detected from the image and m stars extracted from the catalog, the cost of this process should be:

which means the order is reduced from to .

• The process to find out the most overlapping set of parameters should be as follows.
1. The parameters are scattering in a fourth dimension space of . The system divides the space into some small blocks in per 10 deg, and in per 10 pixels. So the space becomes a mesh. Then it counts the number of parameter in each block. Here it does not divide on r axis. That is because the approximate width of the image should not be very different from the true value.

2. It selects only a block which contains most parameters. Except for the block and some adjoining ones, all other blocks are ignored at this step.

3. It calculates the mean value and standard deviation of parameters.

4. If the standard deviations of is less than , the system adopts the mean value of parameters as a true map function. Otherwise, it deletes parameters whose differences from the mean value are greater than the standard deviations and returns to the 3rd step.
In this experiment, the system could obtain true map function in all several dozens of trials. Therefore, the process mentioned above is suitable even if the positions or brightness of stars have some errors, or if there are some objects existing in only one side like a new star.

Figure 2: Virtual night sky

Figure 3: Artificial photo image

Then I made some experiments with real images and star catalogs. Star catalogs generally contain all star data in the whole celestial globe, so the system at first chooses required stars in the neighborhood of the image area, with the approximate position and width of the image input in advance. It needs to extract stars around the position in the four times area of the image, because the approximate width is between half and twice of the true value and the approximate position is surely in the image.

Now that the system deals with real images, it has to convert the brightness of detected stars from an image to magnitude to compare with stars in catalogs in matching. The process is as follows.

1. It sorts the detected star list from an image in order of brightness.
2. It also sorts the extracted star list from a catalog in order of magnitude.
3. It applies the magnitude of the brightest one in the extracted list as that of the brightest one in the detected list. But actually, the area of the catalog should be four times of that of the image, the magnitude of the brightest detected star is the average of the four brightest stars in the extracted list.
4. For each detected star, the system applies the average magnitude of four stars in the extracted list sequentially.

This process does not consider the brightness value of detected stars directly but only considers the sequence and applies their magnitudes. Therefore it causes some errors. In the future, I will calculate the converting function from brightness to magnitude in the method of least squares after applying in this process and re-calculate magnitude of all detected stars.

By the way, real images contain much more stars than the test data and the implementation determined with the test data was found to yield many wrong map functions. So I improves the implementations again as follows.

1. The approvable limit of magnification rate to determine similarity of triangles is reduced from 20% to 10%.
2. The assumption of magnification rate is reduced as .
3. The r axis is also divided per at making the parameter space a mesh.

Here I show you the results of experiments with real images. I use the Tycho Catalog as a real star catalog. The image upper left is the original one. The center in the left side shows the detected star list from the image. The large image in the right is the chart around the approximate position, whose data are from the Tycho Catalog. The bottom one in the left side is the result, the system re-mapped the extracted stars with the obtained map function and made the chart of the same area as the original. When the result chart is quite same as the original, it means the system could obtain the true map function.

• Nova Cas 1995
• Comet Hale-Bopp
Photo: Atsuo Kuboniwa
Feb. 22, 1996 05:05:22 - 05:12:20 JST (6 min 58 sec)
Kukizaki Town, Ibaraki Pref.
BORG 125ED F4 (f=500mm), Takahashi EM-200
Fujichrome 400 Provia

In these experiments, true map function was obtained every time independent on the number of stars. However, there are two big problems actually and they obstructs to open the system in the public.

1. It takes extremely much time.
It took about several hours with PentiumPro 180MHz PC for one experiment. But that is mainly because the system in fully written in Java language. The preliminary test system with a test data is in C++ language and finished only within one minute, though the number of stars are a bit smaller.
2. It cannot tell one map function as a true one.
In the experiments mentioned above, the most overlapping parameter was really the true value every time. But the value was not much more overlapping than other candidates. The difference from the second overlapping candidate is often only . So I can say the true map functions were just barely selected in those experiments. However, when the user also input the approximate rotating angle in advance (for example, when the user always takes images as the north upwards), the system could exactly determine one true map function.

By the way, I have also tried the way to select four stars and check if a pair of quadrangles s similar, but only to fail. The system turned to take enormous time because the exponent increased. In addition, the parameters could not converge on the true value. I have also tried the reversed process. That is, the system actually mapped all detected stars with each candidate of map function, definitely the center of each block in the parameter space, counted how many stars mapped exactly on stars in a catalog, and found the parameter with the most exactly mapped pairs, but only to fail either. It seemed to take extreme time to finish and I interrupted it while running.

 Next: Conclusion and future work Up: Technical Details of PIXY Previous: Star detection from an