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Centroid Noise and Precision Astronomical Photometry
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Centroid noise is a random error, or "noise" component of a photometric
measurement that results from placement of the measuring aperture on the object
being measured. Centroid noise is not widely known as part of the internal
error for a photometric measurement. In this Tech Note, I will examine the
effect of centroid noise on stellar photometry. This study uses a series of
repeated measurements of the same object to show the effect of centroid noise.
The software used was Mira Pro version 6, but the identical algorithms are used
in Mira AP and Mira MX.
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Making a Photometric Measurement
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A photometric measurement involves measuring two quantities:
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The total signal is summed inside an aperture of a specified
size, centered on the object.
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The background signal is estimated underneath the object. In aperture photometry,
it is calculated using the pixels inside a ring (or annulus) concentric with the object.
The brightness of the object is given simply as the difference
"total - sky", or (1) - (2). The trick in doing good photometry is how to
calculate both "object+sky" and "sky" measures robustly in the presence of
background variations, nearby objects, bad pixels, under-sampling, and other
factors. But in addition, the measurement of the object signal depends upon how
the aperture is centered on the object. Especially for faint stars, the centroid
position of the object is itself subject to some uncertainty, which translates
into different parts of the star profile being measured, and thence into a
different magnitude measurement. Even small differences in placement of the
aperture add to the uncertainty of the photometric measurement. This uncertainty
will not be known from a single measurement but will appear when different
measurements are compared using the error estimate from each measurement.
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The Nature of Centroid Noise
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The position of an object is computed using the intensity of pixels.
The centroid position is a weighted average position (i.e., a type of
"center of mass" of the star profile), but the weighting scheme varies among
different techniques; some methods are more accurate than others. Even using a
"good method", nature provides a limiting precision with which a centroid can be
computed. Since the centroid is computed from pixel values and each pixel has
random noise (you do not know its "true" value), this makes the centroid
position uncertain to some degree.
Mira has a very good centroid algorithm, but the centroid value still has
some amount of uncertainty. Moreover, the fainter the star, the less certain
the centroid position. This results because the relative uncertainty in pixel
value increases as the signal decreases. In other words, the contribution of
centroid noise is magnitude dependent and becomes greater for fainter objects.
The crux of the problem is this: How do you get a handle on the
amount of centroid noise inherent in your photometric measurements?
There are 2 ways to do this:
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Pick a star and repeat the magnitude measurement by randomly displacing the
measuring aperture in all directions by some small amount. The small amount
should correspond to what you believe to be the uncertainty in the centroid
position. This is difficult to know and this is a limitation of this method.
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Take many frames of the same star under identical conditions and at the same
airmass (or almost the same airmass), then measure the signal from the star
on each frame. Compare the measurements and their random errors. The errors
will almost always be significantly larger than the internal errors
estimated from measuring on a single image.
Centroid noise can become an issue when apertures are placed manually
onto new objects, as would be done using the "move" mode from the Mira
photometry toolbar. If the target object is a point source, it is usually
better to use the centroid button after manually moving the aperture in
order to minimize the contribution of centroid noise.
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Numerical Model Showing Centroid Noise
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How large an effect is centroiding noise? To address this issue, a simple
numerical simulation was used to answer the following question: If we wobble
around the measuring aperture by a small amount, how much does that translate
into the random error of the magnitude measurement? Using a single image, the
brightness of a star was repeatedly measured using apertures placed at slightly
different positions. This simulates the effect of random deviations in the
centroid positions caused by pixel noise in different images.
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To move the apertures, the auto-centroiding flag was disabled so that the
measurement would be made at the literally marked position rather than at the
centroid position. Then Mira was used in "move mode" by clicking the move button
on the photometry toolbar. Each time the aperture was dropped at a new position,
Mira reported the new measurement. These measurements were tallied and listed in
the table below. The target star has a net count, or "volume", of 6343 counts
above background. This corresponds to an internal error of ~0.01 magnitude. The
figure at left shows a Mira image window centered on the target star.
The image was magnified 16 times to allow the aperture to be placed at fractional
pixel offsets. Note that the contrast was also stretched to show where the star
merges into the background. In typical photometric measurements, the object
aperture would enclose about the amount of the star profile shown here,
or perhaps a bit less.
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While a smaller aperture can give higher S/N on faint stars (because less
sky noise is included in the measuring aperture), a smaller aperture increases
the amount of centroid noise. The two effects compensate each other to some
degree. Also notice the partially sampled pixels around the rim of the star
measuring aperture. Mira uses an exact calculation of partial pixel
contributions. Without that, the centroiding noise would be far higher.
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The second figure shows a horizontal intensity cut through the star as marked by
the orange image cursor in the figure above. The column range of the plot
corresponds to the width of the inner background aperture. The repeat
measurements are shown in the table below. The bottom 2 rows of the table give
the mean and standard deviation of the magnitude, column, and row position. The
variation in centroid position given by the standard deviation is ~0.2 pixels.
As shown in the table, jittering the measuring apertures by this small amount
leads to a variation in the measured brightness of 0.005 magnitude.
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The centroiding noise is approximately 1/2 the computed error of the
measurement! It is important to appreciate that centroiding noise is an external
error and is not calculated as part of the magnitude error, which is purely an
internal error based on photon statistics and camera characteristics. Clearly,
doing high-precision photometry means not only using the best photometric
algorithms but also using the best centroiding algorithms.
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| Trial |
Magnitude |
Column |
Row |
| 1 |
14.937 |
54.658 |
63.850 |
| 2 |
14.932 |
55.031 |
63.844 |
| 3 |
14.948 |
55.094 |
64.281 |
| 4 |
14.937 |
54.656 |
63.969 |
| 5 |
14.932 |
54.969 |
63.969 |
| 6 |
14.937 |
54.719 |
63.594 |
| 7 |
14.939 |
54.594 |
64.094 |
| 8 |
14.943 |
54.844 |
64.281 |
| 9 |
14.939 |
54.906 |
63.969 |
| 10 |
14.930 |
54.906 |
63.906 |
| 11 |
14.938 |
54.594 |
64.031 |
| 12 |
14.936 |
54.656 |
63.719 |
| 13 |
14.937 |
54.719 |
63.531 |
| 14 |
14.936 |
54.406 |
63.656 |
| Mean |
14.937 |
54.768 |
63.907 |
| Std Dev |
0.005 |
0.195 |
0.230 |
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Conclusions
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I have shown that Centroiding Noise is a significant part of
the photometric error budget. Unfortunately it cannot be determined from a
single image. The tests performed here showed a 0.005 magnitude random error
attributable only to random misplacement of the aperture by 0.2 pixels RMS about
the true center of the object. Although the centroid uncertainty is far less
than 0.2 pixels for a bright star, it can easily be this large or larger for
faint stars. Centroid noise would appear as external error which increases the
scatter between measurements relative to the scatter predicted by their error
bars that are based on internal noise sources.
When doing high-precision aperture photometry, seeking to
achieve measurement uncertainties near, or below 0.01 magnitude, one must
concerned about the effects of centroid noise. The only way to minimize centroid
noise is by using a robust, high precision centroiding algorithm for determining
the placement of the object measuring aperture.
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Copyright © 2000. Mirametrics, Inc. All Rights Reserved.
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