Determining a Centroid

The centroid of a physical object is its barycenter, or center of gravity. It is the point at which the object—suspended from a thread or teetering on a fulcrum-balances. In an image, the centroid is the point at which the values of all surrounding pixels are "balanced." What makes the centroid interesting is that even though light from a star is spread over many pixels, the distribution of light among them enables us to recover the exact center of the star image within a small fraction of a pixel. The process is called determining a centroid.

The value of a star's centroid lies not in knowing exactly where on the CCD the light from a particular star happened to fall—which depends on factors that we cannot reproduce—but of determining the relative positions of the stars in the im-

Figure 7.4 Blocky star images contain precise information about the location of the star that formed them. In this star image, pixel values record how much light fell on each photosite. The dot under the "4" in 2204 shows the location of the computed centroid, and its diameter shows the probable error in its position.

Figure 7.4 Blocky star images contain precise information about the location of the star that formed them. In this star image, pixel values record how much light fell on each photosite. The dot under the "4" in 2204 shows the location of the computed centroid, and its diameter shows the probable error in its position.

age, since the star images were formed at the same time. Even though every star image has been enlarged by diffraction, spread by atmospheric turbulence, and smeared by guiding errors, a precise record of their relative positions is stored in the image.

On a CCD image, a star appears as a cluster of pixels having values greater than those of the surrounding sky. Images of bright stars are unmistakable, but that of a faint one may be nearly indistinguishable from random noise in the background sky. The human eye is remarkably good at picking out clusters of pixels, but extracting the centroid of a star image from a CCD image with a computer requires a surprisingly large amount of analysis and number crunching. To accomplish what the eye does with such apparent ease, the computer must:

1. determine which pixels belong to a region of interest centered on the approximate location of the star image;

2. determine the mean brightness within the region of interest;

3. determine (assuming that the pixels that make up the star image are on average brighter than pixels of the surrounding sky) which pixels might belong to the star image;

4. determine which candidate star pixels belong in a cluster, presumably a star image; and

5. compute the centroid of the pixels that belongs to the star image.

Computing a centroid begins when an astronomer selects a star for centroiding. Since the cluster of pixels that make up a star image is roughly circular, the region of interest includes all pixels within a defined radius, typically two or three times larger than the visible disk of a bright star. Given an initial center at (x0, y0), the region of interest includes all pixels yi that satisfy the condition:

where r is the radius of the region of interest in pixels, and aspect ratio is the pixel width divided by the pixel height.

The centroid procedure next determines the mean pixel value of the pixels within the region of interest. Assuming this region actually contains a star image, pixels belonging to it should be brighter than the mean pixel value. If necessary, the program applies a "fudge factor" to the mean pixel value to get a threshold value that distinguishes star pixels from sky pixels.

The procedure now identifies pixels belonging to the star image using a double test: the pixel value must be greater than the threshold, and some number of neighboring pixels must also be star pixels. The neighborhood requirement insures that the procedure rejects individual sky pixels that exceed the threshold because of random noise. (Requiring three neighbors does a good job of selecting between star pixels and random noise.) At the same time, the procedure determines the mean pixel value of non-star pixels (i.e., those that fall below the threshold value) to use as the sky background brightness.

Finally, the procedure computes the centroid using the moment equation for the x-axis and y-axis centroids:

where Pi is the pixel value of the pixel at (xi9 y,), and S is the mean background sky brightness. The moment equation weights each pixel along the two axes by the amount of starlight that has fallen on that pixel. The result is (x, y) —the centroid of the star image.

• Tip: The Star Image Tool gives you direct access to stellar centroids for specialized projects. The Distance Tool, the Astrometry Tool, the Photometry Tools, and the image registration and stacking tools all rely on accurate stellar centroids. The centroid procedure also works remarkably well on galactic centers, comet nuclei, and with a sufficiently large radius setting, on irregularly shaped objects.

Figure 7.5 Knowing their pixel coordinates, you can find the distance between two points in an image. The points can be pixel coordinates, or coordinates found by computing the centroids of two star images. The distance between two star images can be determined to sub-pixel accuracy.

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    What are the centroids of imge?
    1 year ago

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