Imaging system resolution is a very important topic in photography. With the advent of digital camera systems the resolving power can actually exceed that of traditional film of the same size and even rival that of some medium format imaging systems. The resolving power becomes extremely important when the images are projected or printed in large formats. Very large format films, such as an 8×10 contact print, are still very difficult systems to beat when properly done, especially by an expert such as Ansel Adams.
In astrophotography circles the resolving power of the telescopes becomes even more important. The ability to resolve fine details in nebula and planets as well as the ability to “split” double stars is ultimately the end game in general amateur astronomy. I have taken on the daunting task of trying to evaluate my photographic equipment as well as one of my telescope optical tubes.
One great source for an extensive multipart primer on this subject can be found at Norman Koren’s web site: http://www.normankoren.com/Tutorials/MTF.html
The basic idea is not that complicated, take two points and determine how far apart these points need to be, at any given distance, before they can be seen as two separate points rather than appearing to be merged together. Theoretically the resolving power of any optical instrument such as a telescope or a basic camera lens is determined by the diameter of its lens for a given wavelength of light, the larger the diameter the closer the two points can be and remain visually separate or resolved. The separation distance changes with differing focus distances but one thing remains constant, the angle that is created from the lens to the two points.
The actual methods also include microscopes, diffraction gratings, as well as scanners so some rather arcane terminology has been accepted, such as line pairs per mm, line pairs per pixel, line pairs per picture height. The one measure that makes the most sense to me, from a photographic and astrophotographic perspective, is simply the angle, usually measured in arc-seconds of a degree.
The basic methods are similar; create a target with varying closely spaced lines, photograph the target and evaluate the shortest distance or angle that two lines can be separated visually. Here is where it gets complicated, there is something called a Modulation Transfer Function (MTF) that is used to quantify the evaluations scientifically with repeatable results. The MTF is very useful if the evaluations are to be accomplished automatically with software. I can attest that the evaluation can be done by hand but this is a very tedious and time-consuming task.
I used two separate targets for the images, one an ISO 12233 standard target and another downloaded from the Norman Koren web site. Here is an image of both, the larger one is the ISO 12233 and the other four are Koren’s. The other behind the far right one is a black/white/grey/RAW card used for determining exposure and white balance.
The Koren cards are designed to be photographed at a specific distance since the line spacings are printed on the target are calibrated in line pairs per mm (lp/mm). The ISO chart, on the other hand, is calibrated in hundred line pairs per picture height and can be easily photographed at any distance and can be easily converted to the other terminologies. The ISO chart, however, is not well suited for automation.
By cropping out one of the sets of targets and choosing the Region Of Interest (ROI) consisting of either the sinusoidal or square wave targets a computer program can easily evaluate the image and determine the resolution of the imaging system. Here are a couple of examples showing the cropped images:
I wrote an GNU Octave script that transforms the image into a waveform that can easily be evaluated both qualitatively and quantitatively. Here is the result of the square waveform:
By manually measuring the extrema pixel values of the waveform for each cycle the following formula is used to create the MTF graph:
(max value – min value) / (max value + min value) * 100%
This normally gives a result between zero and one hundred percent, when the MTF falls to 50% the lp/mm is determined by finding where on the target this pixel corresponds to. This is the point where aliasing and moire becomes significant, although details can be seen as low as 10% these are not reliable or accurate but do have significant effects on how the images are qualitatively seen by an observer.
The interesting thing that I noticed is when I differentiated and appropriately filtered the image waveform the envelope converges on the 50% MTF point making the evaluation very easy without the tediousness of manually measuring the pixel values and plugging the numbers into a spreadsheet or program. I tested this method by artificially changing the resolution by using the same setup at several distances and found the differentiation/filter method to be fairly accurate. I would, however, hesitate in recommending this method until further testing is performed.
Additionally the differentiated/filtered waveform also shows why this method is indeed measuring the modulation of the aliasing/moire/resolution effects. The pulses after the convergence are typical of any modulated signal waveform. These pulses are indeed showing where the moire and aliasing effects begin to overwhelm the signal.
I also tested several 50mm lenses that I had lying around and found that Nikon’s current autofocus 50mm f1.4 AF lens was far inferior to either a 1980’s era manual focus 50mm f/1.8 lens, which was also only slightly inferior to a 1960’s era 50mm f/1.4 kit lens that came with THE original SLR – the venerable Nikon F. Here is the MTF graph for these three lenses:
Another interesting thing I observed is when I switched camera bodies from a full frame FX sensor Nikon D3s with my DX sensor sized Nikon D300, the MTF function showed a pronounced improvement. However, when the distance was increased in order to obtain the same field of view the resolution was considerably poorer with the D300. The smaller pixel size did indeed increase the resolution, however, the smaller overall sensor size significantly reduced the resolution for a comparable image.
Now for measuring the Celestron 9.25 EdgeHD telescope optical tube for resolution. The theoretical resolution from a circular lens can easily be calculated by using the following approximation formula:
Theta = 1.22 (Lambda / D)
Where theta is the minimum angle measured in radians, lambda is the wavelength of light in meters, and D is the diameter of the optical tube in meters. For the Celestron this calculates to about 0.65 arc-seconds, however, my measured results show that the Celestron tube with the Nikon D3s only has a resolving power of about four arc-seconds, and this is being very generous. That is about one-sixth the theoretical resolving power, not very inspiring at all. Some of this loss is due to the camera sensor and processing so the measured results will never approach the theoretical lens minimum.
My 70mm diameter telephoto lens with the Nikon D3s, on the other hand, has a measured resolution of about
under six 12.8 arc-seconds. While the theoretical resolution power is a little over two arc-seconds. This is less than one-third one-sixth the theoretical resolving power. Tit-for-tat my Nikon 70-200mm f/2.8 is at least twice equal to the optical quality of the Celestron optical tube based only on the resolving angle. The Celestron still has a slight advantage in overall resolving power. Only a slight advantage in resolving power from a 9.25 inch lens when compared to 2.75 inch lens, I’d say the resolving quality of the Celestron optical tube was actually rather poor.
This is in perfect concert with my previous qualitative evaluations and comparisons of these two lenses with my D3s. Here is the image of the target that was approximately 200 feet (61 meters) away. The calculated distance for the Koren target would be about twice this distance and obviously would not work here.
Here is the link to the GNU Octave script along with the cropped image used for the above graphs, click on the link to download: mtf_test.zip