Step EE-3: Finding the best fit

We now have a way to transform the stars detected in an image into our virtual camera reference frame, but the previous step was just the “rough alignment” based on two bright stars. This is vulnerable to errors in how those two star positions were identified, especially if they were bright enough to saturate the detector, or were influenced by poor atmospheric seeing conditions.

More importantly, we are attempting to measure a tiny change in radial distance from the center of the sun. If we make a small error in where that center point is, by even a fraction of a pixel, it will affect all of our distance measurements to the stars revealed during the eclipse.

This means that it is less important to align the star positions as it is to get their angles with respect to the sun correct. If the angles are correct, then we can measure the radial distance from the image center and be confidant when we compare the “before” distances to the “during” distances that we are measuring from the same center point and are not being fooled by some offset to the actual center. This is particularly important in the before images. In the during image, we can, in principle, locate the center of the sun (though this too has uncertainties since we are seeing the moving moon’s edge, not the sun’s). The collection of observed stars is an unambiguous pointer to the sun’s position at the moment of maximal eclipse.

To this end, the output of step EE-3 is the small additional offset and rotation that places the observed stars in best angular alignment with known angles of the reference stars. This small adjustment will be applied to all of the detected stars to obtain their final position in the reference image plane. Their radial distance from the sun is then easily computed.

Matlab script radialAlign.m was created to do this task. It takes the results from step EE-2, the rough alignment and transform to the reference frame. The output is a set of star positions—the reference star positions in the reference frame, and the slight deviations from those positions as detected in the image.  In an ideal world, the deviations would be zero in the “before” image, and would show some gravitational deflection in the “during” image.

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Step EE-2A: A side process to calibrate (radial) lens distortion

radialFitTestErrorsUnexpected errors while looking for the best angular fit between the imaged stars and their reference locations.


My early efforts to map the imaged stars onto their virtual camera positions showed unexpected errors. They were close, but displayed error amounts that were inconsistent with a simple scale or rotation mismatch.

It seemed possible that the errors came about because of the radial distortion of the lens, which one reference suggested might be as much as 0.05%, a huge amount for this measurement.

To calibrate, a frame with more stars included (via increasing the ISO) were compared with their target (reference) locations and the radial errors were plotted and a correction curve computed to best fit them.

radialCalFig2Reference stars and detected stars. The correlated pairs are used for evaluating radial distortion.


There are reference stars that do not have a corresponding star in the image, but there are many more detected stars without a reference coordinate. These may be spurious noise, or faint stars that were not listed in the reference star database (displayed by Stellarium).

radialCalFig7The correlated pairs of reference and detected image stars with an indication of the position discrepancies. Near-center stars are too close, and farther stars are too far.


The errors range over +/- 15 arcseconds, a few pixels in the image, but ten times the amount of deflection signal we are trying to detect.

To improve on this, we need to perform the rigid transform that finds the image center and angle that best aligns the detected stars with their reference locations.


radialCal6The radial errors, plotted as a function of distance from frame center. A polynomial fit is made to approximate them.


A radial correction function can then be obtained by finding a polynomial fit to the residual errors. This means that a function with factors of radius: r0, r1, r2, r3 etc. is used to approximate the distortion introduced by the lens. Knowing the distance from the center, we can use it to compute how the lens has shifted the star positions toward or away from the center, and then correct them.

I experimented with several forms of the polynomial function and found that the best behaved was among the simplest. It did NOT include an offset term (r0), and the linear term (r1, equivalent to magnification), was minimized by adjusting the overall base image magnification (arcseconds per pixel). This results in the linear term representing the slight differences in magnification between red, green and blue channels, something that apochromatic lenses strive to match but cannot be perfect at. The remaining error was well represented by the r2 term.

Here are the results averaged from multiple image frames of the eclipse target area:

Camera base magnification to reference frame (arcseconds per pixel):
scale = 2.8443;     % average of EOS 6D calibration frames taken 20170601

This value is used in imagePos.m (step EE-2) as the scale factor to bring star positions into the reference frame where one increment (pixel) equals one arcsecond.

Linear (r1) coefficients (x10-3):
R            0.7731
            G            0.5185
            B            1.0245

Squared (r2) coefficients (x10-3)
R            1.1969
            G            1.2564
            B            1.1745

The linear coefficients are the differential magnifications between the red, green, and blue channels of the image. Green has a small (1/2 arcsecond per thousand) magnification difference from the base magnification. Red is a little higher, and the blue correction is about twice that of the green channel. The image would show a star’s image as having a green fringe toward the image center and a blue fringe at the outside. This captures the general chromatic aberration of the Televue-85 telescope I am using. It is a highly corrected lens that offers beautiful visual views; only a high resolution camera can quantify its color mismatch.

The squared coefficients indicate a general behavior of lenses and how ideal they are in projecting their field of view onto the perfect geometric projection we are using as our reference frame. These corrections are also quite small, but over enough radial distance contribute a position error that builds up to many arcseconds.

The lens correction coefficients (the linear and squared terms just described) are applied in step 3, radialAlign.m which provides the final set of detected star positions which have been angularly centered, and radial lens corrected. There are still some errors from the reference positions, but they are much reduced.

The remaining errors, plotted against radial distance from the center are limited to around +/- 5 arcseconds, and do not show any significant trend. If they did, we could identify it and apply additional corrections to remove it.



The residual errors after using the polynomial correction.


The radial correction brings the RMS errors to within 2 arcseconds (red) to 4 arcsec (blue). I’m not sure what the source of this residual error is, but it is of the same magnitude as typical Minnesota astronomical seeing. It may just be the twinkling motion of the stars. We could check this by comparing two separate exposures to see if the errors are correlated. If they are, then it is NOT due to atmospheric turbulence. If they are uncorrelated, then we need to find some way to overcome this variation, because it is larger than the signal we are looking for.



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Step EE-2: Transform to reference coordinates

Stellarium.20170821.totalityThe field of view for my EOS60Da APSC sensor (inner rectangle) and the EOS6D full-frame sensor (middle rectangle). The map overlay is the view from Idaho Falls on the day of the eclipse as displayed by Stellarium (and inverted to negative view).


Once we know where things are on the image, we need to correlate them to their celestial locations (right ascension and declination, or their equivalent). To do this, we need two landmarks in the image, one for position, and the other for rotation. We will position the image so the first landmark aligns with its celestial coordinate, and then rotate the image until the second matches up. This will provide a “rough alignment” so that the next step (point set registration) will have a good starting point for matching all of the stars in the field of view.

The reference coordinate system could, in principle, be any particular view of the sky. Using the pixel coordinates of the “before” image is a reasonable choice (as would be the “during” image). This would have the advantage of not needing to transform that image at all; it is already in the reference system — only the ‘during” image would need to be converted.

But to keep the mapping easy to think about, and utilize multiple before pictures, I chose the reference coordinate system to be that of a camera, aimed at the sun’s center, and oriented to “up” at the location where I will be taking the “during” photos (near Idaho Falls, ID).

To perform this transformation from image coordinates, we need the projected celestial coordinates of our landmark stars. One star that will be certain to be bright enough to record is Regulus, “the heart of the lion”, alpha-Leo, the brightest star in the constellation Leo, in which the sun will be residing in this eclipse (hence the astrological sign of those born during most of August, including me).

For the second landmark, the next brightest star in the field is nu-Leo, considerably dimmer. Its visual brightness is magnitude 5, compared to Regulus at 1.3, roughly 30 times dimmer.

To gain a sense of star positions, it is helpful to use a use a sky map simulator. I discovered a (free) open source program that does a very nice job of simulating the view of the night sky and allowing me to make maps of the area I am exploring. It is Stellarium (, and I have been able to identify and locate the candidate stars that will be near the sun during the eclipse.

Using Stellarium, I can identify the reference locations for my landmark stars:

Regulus:             RA 10h 8m 22.01s, Dec 11o 58’ 2.9”
Nu-Leo:            RA 9h 58m 13.35s, Dec 12o 26’ 41.1”

These are the brightest, but there are a few others that may be useful in case one or both of these are not in the image.

I need to map these angles onto the image plane of the virtual camera in Idaho that is aimed at the sun. This is a basic calculation in computer graphics, a field that I accidentally have some experience in.

Now that we have a reference frame established, we need to transform our before and during images into it. This too is a standard operation in computer graphics. All that is needed is the transformation matrix, a set of six numbers. Given two reference star locations, the six numbers can be calculated and applied to all the other stars in the image to find out where they land in the reference frame.

The virtual camera can have any number of pixels we wish. I decided that it would be convenient if one pixel represented one arc-second in the sky.  The image frame from this virtual camera looks like this; the reference stars are plotted, and the alignment stars are marked. The stars detected from a one-second exposure of this part of the sky have been mapped into this reference frame.

imageViewHeiseA test “before” image transformed to the reference virtual camera view at eclipse time in Idaho.  The location where the sun will be is indicated, centered at coordinate (0,0).

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Step EE-1, Detect the star locations in a photograph

It seems easy to locate stars in an astrophotograph: they are the bright dots.

I am recording images of the sky with my Canon EOS 60Da. It has a resolution of 5184 x 3456 pixels (after demosaic) in each of red, green and blue color channels. This is a wonderfully high resolution that exceeds the resolution of the lens and the atmosphere through which it is seeing. A “pinpoint” star is actually around 10 pixels in diameter, more if it is really bright.

And the background is not black. There is light pollution, and when the moon (or eclipsed sun) is near, there is skyglow, and there is always the background noise of the detector. All of these add up and make it more difficult to find the stars, especially if they are dim.

Here is my strategy to detect them, and then to measure their exact sub-pixel positions.

  1. Establish the local average and local variation of the background sky. This tells me what the average light level is, and the amount of random noise.
  2. Set a threshold above that noise level and detect everything that exceeds it. These will include stars and “hot pixels” (detector noise).
  3. Eliminate the isolated single pixels; they are the sensor hot pixels.
  4. For what remains, these are the “cores” of the star images. Expand the neighborhood around each. Take the weighted average of those pixels and find their geometric center. This is the star’s location in the image.

I wrote a Matlab script to do this, and applied it to various test images I have taken of the eclipse field of stars. It yields (subpixel image) coordinates of the stars it identifies. These are used as inputs to the next step.

starRegionsCompositeChannelsStars detected in each of the three color channels (RGB) for the eclipse field. Not all stars are seen in all channels, and noise creates spurious stars in each channel. This exposure is 1 second at ISO 100, the best signal to noise setting on the camera.


starRegionsCommonChannelsThe detected stars found in all three color channels. The double star to the left is Regulus (alpha Leo) and its companion.


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Measuring star deflections, step by step for the Eddington Experiment.


From “Measuring Starlight Deflection during the 2017 Eclipse: Repeating the Experiment that made Einstein Famous,” Donald Bruns

Even with high quality equipment, it will be challenging to make measurements of the tiny deflections expected. There are some helpful references available on the web; some are encouraging, others are scary. Here are a few:

Measuring Starlight Deflection during the 2017 Eclipse: Repeating the Experiment that made Einstein Famous:
An update from Bruns:

After reviewing them, is not clear to me whether my equipment and technical abilities will be able to find the small signal in a big noisy field, but I will enjoy the attempt.

Here is my star deflection measurement outline. It comprises distinct image and data processing stages, which I demark by the following labels, and will subsequently refer to:

EE-1.      Detect star locations in the “before” and “during” images. These will be pixel locations relative to the image origin (upper left corner).

EE-2.      Transform the image into a standard orientation where the sun is at the exact center, and two reference stars are at the correct relative angles to it. Do this for both before and during images.

EE-3.      Solve the “rigid point set registration” problem that best aligns the stars between the two images. Do not allow scaling, only translation and rotation.

EE-4.      Measure the radial distance from the (sun’s) center for each of the stars in the field and compare their relative values. They are expected to fall on a 1/R curve and at a value that matches Einstein’s predictions. The differences will be small, a pixel or less.


These all sound quite doable, but of course there are details. I will describe some of them for each step in what follows.  If all goes well, perhaps I will end up with a chart like Eddington’s:


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A plan for 2017, but first, an introduction to celestial coordinates



Astrometry is the term used for measuring the positions of the stars. It has a long history, dating from pre-telescope times, and which has introduced units that may seem archaic, but persist to this day (making them seem both arcane and archaic). Because of the Earth’s rotation, just like the sun, stars rise in the east and set in the west. The angular position in the sky depends on how many hours it has been ascending since “star-rise”.   If we pick one point in the sky as a reference, we can count the hours of ascension the star has traveled from that reference point: its “right ascension”. Right ascension (RA) is measured in hours, minutes and seconds, but is easily converted into its sky angle (24 hours equals 360 degrees).

I think of right ascension as the “horizontal” position in the sky (from east to west). The vertical coordinate, declination, is more conventional: it is the north-south angle from the Earth’s equator. These two numbers, right ascension and declination, define a star’s position in the sky.

Before photography, astrometry was conducted by aiming a fixed telescope at a specific declination angle, and the observer would watch the stars drifting through the field of view. When a target star appeared, the observer would mark the time at which it crossed the center of the view. This would be its right ascension, hence the units of time rather than angle.

When photographic emulsions became sensitive enough, the star field could be recorded and then the positions measured from the photographic plate. But because the Earth is rotating, the stars moved during the exposure, blurring their positions. This required that telescopes be equipped with “clock drives” so that they could track the star’s motion across the sky during the length of the exposure. This was the (mechanical, weight-powered) mechanism employed in Eddington’s expedition to track the sun and stars during his exposures, which lasted from 2 to 20 seconds.

As I contemplate the modern reconstruction of this experiment, I have the advantage of telescope mounts with motorized clock drives. I need only to align them with the Earth’s polar axis, and the battery-powered motors will do the rest. I also have the advantage of digital image sensors, but this comes with some tradeoffs as I hope to explain later.

The part of this experiment that will be challenging is to detect the background stars through the sun’s corona and any residual sky brightness. Eddington had the advantage of relatively bright stars, and a long, deep, period of totality. The stars we have available are at least 5 times dimmer, and totality will be two minutes, not six.

In anticipation, I have made plans to record the stars in the sun’s future gravitational neighborhood during the eclipse, and to compare them to the same field without the sun. There are two challenges. One is the practical ability to detect the starlight above the ambient light background of the sky and corona of the eclipse. The second is the technical ability to compare images taken before and during the eclipse, in order to measure the stellar deflections to subpixel accuracy.

I will outline the technical plan first, assuming that the stars will be successfully recorded on eclipse day. The issue of how to best detect them will follow.

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The “Modern Eddington Experiment”

0*K_tYN4NbxjACv4RGImages credit: New York Times, 10 November 1919 (L); Illustrated London News, 22 November 1919 (R).


Notes and comments on reproducing a famous historical event.

Most people today do not know the reference to Sir Arthur Eddington, but he is responsible for making Albert Einstein a household name. In 1919, the English physicist took on an experiment that would test Einstein’s theory of gravity, general relativity, an abstract and mathematically beautiful explanation of space and time that few people at the time fully understood.

The prevailing theory of gravitation was Isaac Newton’s, one that worked to great success in explaining nearly all the motions of the planets. Einstein’s theory was a refinement that would show only tiny differences. But it is important to get this right; modern GPS navigation would not otherwise work.

Both theories predicted that light would be deflected by gravity. The light from a star positioned near the sun’s edge would be bent, like a mirage, to an apparent position slightly shifted from where it would normally be. But Einstein’s theory predicted the deflection to be twice that of Newton’s.


It is impossible to see such stars while looking at the sun because, well, it is daytime. Even if you block the direct view of the sun, the scattered skylight washes out all but the very brightest of stars and planets. During a total solar eclipse however, the moon’s shadow makes a hole in the sky and the stars become visible.

Eddington’s experiment then, was to photographically record the nearby stars during the eclipse and measure the apparent deflections from their true positions. He mounted an expedition to a small island off the coast of Africa for the eclipse of May 29, 1919, where he encountered the usual problems that astronomers face: bad weather. It rained in the morning, and clouds obscured the view. They had thinned somewhat by the time of the eclipse, and his team forged ahead and made sixteen exposures on various photographic emulsions on 8”x10” glass plates.

It is fascinating to read the account of obtaining these photographs (as reported in the Philosophical Transactions of the Royal Society of London). Totality for this eclipse was long: six minutes, which allowed the manipulation of the large plates to be placed in position in the telescope, the large refractor from Oxford that had been dismantled, packed and transported. It was stopped down to 8” in order to use the best part of the lens. The clock drive was driven by a weight, and a pit was dug “deep enough to allow a run of 36 minutes without rewinding.”

All but two of the plates failed to show measurable stars, mostly due to the cloud cover. But that was enough. Combined with the results from a sister expedition to Brasil, Eddington’s analysis showed that the observed deflections were best explained by Einstein’s theory. These results were reported later that year, and the world found it fascinating. Einstein, previously known and respected only in a select group of theoretical physicists, became a celebrity.

Eddington made these measurements nearly a hundred years ago. Although he suffered poor weather, he had many things going for him. A long duration eclipse, and a collection of bright stars in the near field of the sun, made it possible using early 20th century technology to make a precision measurement. He recognized this, and at the end of the report wrote: “The unusually favourable conditions of the 1919 eclipse will not recur, and it will be necessary to photograph fainter stars, and these will probably be at greater distance from the sun.”

Well, we are a hundred years on from that groundbreaking experiment. One would think that modern optics and photographic equipment would make this an easy exercise. And indeed, it has become a goal for amateur astronomers. The Astronomical League, an umbrella organization of local amateur astronomy clubs has offered a special observing award to those amateurs that can detect the deflection of starlight during the eclipse of 2017.

But even with the advantages of a century of technological advance, this is a difficult measurement to make. Yes, we don’t have to worry about how long the clock drive will run before we need to reset the weights, and our optics are better, and the photographic emulsions have been replaced by digital sensors, but the measurement still involves finding stellar deflections that are smaller than a pixel!

My exploration into this project has uncovered some interesting details, which I will share in the following blog entries.  They start with a general introduction of celestial coordinates, and an outline of my plan, but then might become somewhat tedious as I devolve into some of the image processing and mathematical details.  Much of it may be of little interest except maybe to other Eddington experimenters, but if you find it useful in some way, or wish to make contributions, feel free to add commentary.

Here are some useful entry points:

Intro to celestial coordinates 

The overall plan to measure star deflections



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