Harold Nyquist

If, for example, you have a CCD with 9-micron pixels and a telescope with a 1000mm focal length, the resolution will be 1.8"/pixel. This is the theoretical resolution. Why? Take an extreme example such as a CCD with tiny pixels and a telescope with a very long focal length. Try the SBIG ST-10 camera on a Celestron 14" f/11 telescope in the calculator above. The resolution of this system is 0.36"/pixel. But from a typical observing site, the seeing conditions (steadiness of the atmosphere) might never be better than 1-2". Half an arcsecond is exceptional seeing from a professional mountaintop observatory. In fact, something more like 3-4" is typical from a backyard location. Over the course of a long exposure, even 2-3" seeing tends to smear out to around 4".

Is there any advantage to having a resolution as high as 0.36"/pixel? There are some exceptions, which we will see later, but for most deep-sky imaging 0.36"/pixel is definitely overkill.

It is often stated that 2"/pixel is an ideal CCD pixel resolution. Let's see why that statement is made, why it might not be entirely accurate, and why it might not even matter at all!

Sampling

Nyquist's theorem states that the frequency of the digital sample should be twice that of the analog frequency. What is the "frequency" in the case of a telescope image? It is the resolution of the telescope system, which, as we decided above, is almost always limited by the seeing conditions. If we take the average seeing to be 4" over a long exposure, we need a "sampling rate" of 2"/pixel to satisfy the Nyquist theorem.

So why is this oft-quoted theorem not necessarily correct? Part of the problem lies in the fact that Harry did not have CCDs in mind when he developed his idea. The Nyquist theorem deals with 2-dimensional signals such as audio and electrical signals. The graph of an audio signal has two axes, intensity and time, and looks generally like a sine wave. Even this has its drawbacks, as the diagrams below demonstrate.

Above: A wave signal is sampled at certain points, spaced apart one half the frequency of the original signal. The red line in the right image shows how the signal might be reproduced.

Above: The problem becomes clearer when the sampling points do not coincide with the peaks of the original signal.

In the case of the above diagram, the signal is clipped, meaning the amplitude of the waveform is reduced. In other words, the peaks are not as high as the original signal. Take the common example of a digital music file such as an MP3. If you copy a song from a CD to a digital file, you are often given the choice of sampling rates. For MP3 files, the possible sampling rates are 32, 44.1, and 48 KHz. This means the audio signal will be sampled either 32,000, 44,100, or 48,000 times per second. If you've ever listened to a highly compressed digital sound file, you know it can sound muffled. Increasing the sampling rate better reproduces the sound -- especially the high and low frequencies -- but increases the size of the file. CD quality audio, by the way is 44.1 KHz, so 32 KHz would produce a more compressed file but with less-than-CD audio quality, whereas 48 KHz sampling would produce better sound quality than a CD. The same basic principles hold true for digitally sampling any analog signal including a telescope image.

But a telescope image adds one more complexity: it is 3-dimensional. The plot of a star image has three axes, the x-axis (which can be though of as right ascension, if you like), the y-axis (declination), and intensity. Brighter stars have higher intensities. Also, a telescope image is not quite like a sine wave. Due to some unfortunate laws of physics, an optical system forms an image of a point source (a star) as a Gaussian curve:

Note: You may have noticed that this is itself a form of sampling. The telescope optics have taken an analog signal (the star) and formed an approximate, but not exact, reproduction of it. Now the CCD must produce an accurate digital reproduction of the telescope image.

Take a look at the x-y plot of a star image on a CCD to see why Nyquist's recommendation might not hold true for CCD imaging.

Above: If a star is the same size as a pixel (say, 4"/pixel from a typical site), the star can be reproduced as a square. This is not a very accurate reproduction of the original analog signal. If the star falls on the corner of a group of pixels, the star is still square, just larger and dimmer (since the light is now split among 4 pixels).

Above: At the Nyquist sampling rate of 1/2 the resolution of the system (2"/pixel), the star is still a square if it falls on just four pixels. If the star is centered on a pixel, it is reproduced as a (somewhat) round image. This is a better approximation to the original image, but only for those stars that fall in just the right spot on the CCD.

Go back for a moment to our audio signal. What if we sampled it more often, say at three times the original frequency? A more accurate digital reproduction can be produced.

Above: Sampling at three times the frequency produces better results.

The same holds true for the CCD/telescope system.

Above: By increasing the sampling to 3 times the resolution of the system (1.33"/pixel), the star is now circular in both situations where the star image is centered on a pixel (left) and centered on an intersection of pixels (right). Also, the individual pixels can take on various intensity values to more accurately reproduce the original star image.

More pixels also gives the computer more information to work with for post-processing. The drawbacks? There are two ways to increase resolution. The first is to use a CCD chip with smaller pixels. The problem here is that you are limited by what cameras are available and by financial considerations. Also, to get a wide field of view with small pixels you need a CCD with a large number of pixels. This means more money and more data for the computer to deal with, making for more processor-intensive image manipulation.

The other option for increasing resolution is to increase the focal length of the telescope. This means making the field of view narrower and, usually, the focal ratio slower, leading to increased exposure times. A fast telescope with a long focal length is possible to obtain, but it will require a very large aperture, meaning more size, weight, and cost.

Optimal Focal Lengths for Popular Pixel Sizes (3x sampling rate)

Signal-to-Noise Ratio

Resolution also turns out to be a determinant for signal-to-noise ratio. (See the Optimum Exposures page for more details on signal-to-noise ratio.) Small details (including stars) have signal-to-noise ratios (SNR) which are a function of (amongst other factors) the resolution of the imaging system. Undersampled images can result in a reduction in SNR. To obtain the faintest possible stars (or starlike objects, such as asteroids), proper sampling is important. Higher resolution equals greater SNR.

The Solution?

Maybe the best solution is just to forget the Nyquist Theorem. You want good resolution for fine details, but this is not the most important factor in getting pretty pictures. More likely, you need enough field of view to fit large objects in the image. Also, a fast system is desirable for relatively short exposures. For a basic rule of thumb, figure you want small pixels (13-microns or less) for short-focal-length telescopes, and larger pixels (16-microns and up) for very long-focal-length scopes. But for pretty pictures, the rules can pretty easily be stretched. If your goal is achieving the best resolution of reaching the faintest possible magnitudes, sampling is more critical. For aesthetics, it matters little.

Consider that very few of the impressive images in magazines and on the web, and very few of the images on the Starizona Guide to CCD Imaging site was taken with a system producing a 2"/pixel resolution! Some were taken at 4.2"/pixel, while others are at 0.5"/pixel. Normally the most important factor is field of view. If you are imaging a large object you need a short focal length telescope. For small targets, a long focal length scope is necessary. Getting the object to fit in the field is a far more important consideration than arcseconds per pixel!