This is a topic that is often debated in the amateur literature and in amateur internet forums. We present some material a little later on aimed at matching a CCDs resolution element size to your telescope. Here, we will talk a bit about the philosophy and aesthetics of pixel sizes as well as some practical points of interest and a brief mention of the gains to be had by those seeking to use the CCD for some real science. We hope that this will help both novice and advanced users to see all or most of the variables that fold into this decision point.
If we had two CCDs with equal QEs (Quantum Efficiency) but one has 9um pixels and the other has 18um pixels, (i.e. the pixels on CCD#2 are twice the linear size of those on CCD #1) and we put both of these CCDs into cameras that operate identically, then the image taken with CCD#1 will require 4X the exposure of the image taken with CCD#2. This seeming discrepancy is due in its entirety to the area of the pixels in the two CCDs and could be compared to the effectiveness of rain gathering gauges with different rain collection areas: A rain gauge with a 2-inch diameter throat will collect 4X as much rain water as a rain gauge with a 1-inch diameter throat. In various amateur forums is often argued that the only thing accomplished by having CCDs with smaller pixels is to increase the exposure time required to obtain a decent image. While it is possible to have a major mismatch between pixel size and equipment capabilities, we generally feel that the 'get the shortest exposure' clan is quite mistaken in its view of CCDs with smaller pixels. Now, if one really IS limited by the light available to the CCD by the nature of the program (For example a supernova survey where imaging a large number of galaxies each night is the primary goal or in a program where narrow band filters are being used to search for ultra faint traces of sodium near IO or H II regions in the galaxy) then there is certainly a case for using larger pixel CCDs (or of using shorter focal length systems) for such a program. Even when detection is the primary imaging goal some care needs to be taken to match the pixel size to the size of the object(s) that are being studied or searched for. However, if image quality and resolution are major goals then it is hard to find justification for the large pixel CCD cameras unless one is using an imaging system with a very long EFL where larger pixel CCDs are a good match for the imaging system. When image quality, from either the aesthetic point of view or the scientific point of view, are important then it is very important to carefully match the pixel size of the CCD to the capabilities of the imaging system and the expected best possible observing conditions and we will discuss this in detail in short order.
The counterside of this smaller pixel argument is that an image taken with a CCD that has 9um pixels contains 4X as much information as does an image taken with a CCD that has 18um pixels. This means that each image will occupy 4X the disk space and it may also mean that image display requires a more capable computer display card and a higher quality display monitor. This may not seem like too serious a problem until one counts up all the bias images and dark frames and flat field images that re required to fully process the images produced on any good quality night and then begins to think about how to save all of those megabytes of image data for future use and display. Since most of the night's ancillary images (flats, bias and dark frames) can be disposed of after the images have been partially processed this problem is somewhat solved, but how does one go about saving the images taken in 1995? Removable hard disks offer one temporary solution, but hard disks CRASH and wear out. Tapes have a useful lifetime of 4-5 years and so offer a slightly better solution. Writeable CDROMs are rapidly appearing and they have a probable lifetime of 10-20 years which is quite reasonable but are still semi-expensive at this point in time. Finally there is good ole film which has a storage lifetime of over 100 years and which is relatively easy and inexpensive to use but which has the disadvantage of not preserving the digital form of the image. In any case, the downside of smaller pixels is largely an economic one requiring more capable computers, displays, and data archiving systems.
We have discussed a lot about the role of pixel size and
image
aesthetics but there is also the science end of the scale for
CCD
users who may be interested in applications beyond just pretty
pictures: astrometric or photometric uses for example. In
astrometry, the finer pixels and the oversampling of the image
makes for more accurate position determinations and thus
enhances the usefulness of the CCD images. In photometric
applications, having a stellar image that is poorly sampled
*(i.e.
occupies 2-5 pixels) makes for noisy data and poor photometric
reductions. While having a stellar image that is well sampled
or
oversampled (i.e sampled by 16-25+ pixels) makes for a much
less noisy data set and thus to better photometric
determinations.
(* The REAL ISSUE here is not over BIG or SMALL pixels, but over IMAGE SAMPLING. With a 24-inch f/15 instrument, 25 um pixels will provide a very well sampled image but this same 25 um pixel camera on an 8-inch f/6 scope provides for very poor image sampling)
Finally before leaving this subject behind there is a final area in which smaller pixels provide an added plus for the user: Focusing. If a star image is close to the size of the CCD pixel then its image may easily be shifted about during the focusing task and move rapidly from pixel to pixel as well as move from falling on a single pixel to a position where its light is spread over 3-4 pixels. These rapid movements create random intensity fluctuations and make focusing into a very difficult task whether one is watching the maximum intensity of a star or the shape of the star during the focusing task. However, if the star's image is well sampled, the focusing task becomes much simplified. As the focus is approached the star's image will shrink in a linear manner with variations only caused by seeing conditions and the star's peak intensity will also behave in a much less random manner as well.
In selecting a camera system for your scope the best system resolution will be obtained by matching the focal length of the telescope to the pixel size of the detector (the CCD camera in this case). The strategy here is to try to select the pixel size so as to provide adequate sampling of the smallest resolution element that the imaging system is capable of producing. In fact, it is far better to oversample the image than it is to undersample the image (as we have discussed at length above). In many systems, the image is undersampled and the most obvious result of this undersampling is the appearance of rectangular or square star images. The compromise here is between an optimally sampled image and the size of the total field of view. For planetary work this is of little import and in general the planetary image will need to be greatly oversampled.
In order to match the system's resolution to the pixel size it is first of all necessary to determine what the optimum resolution of the system actually is. In AMOACP we presented a detailed discussion based upon actual measurements from photographs as to what we had found to be the determining factors behind the limits of photographic resolution on long exposure astronomical photographs. Those limits are a combination of the structure of the diffraction pattern produced by the system and atmospheric turbulence which blurs that pattern during long exposures. Our arguments were empirically derived from measurements of star images (FWFM) in high resolution images from various imaging systems. The bottom line equation that resulted from that study was that the very best resolution of the any earthly imaging system operating at prime focus was about six times larger than the Raleigh criteria for visual resolution.
*** This expression is a great simplification of the discussion and expression derived in AMOACP. In reality, one will be limited by the worst factor that plays into the image size equation. The factors that must be considered as primary limiting factors include: APERTURE, SEEING, GUIDING, OPTICAL QUALITY, and FOCUS. In practical applications, whichever factor is the worst will be the primary determinant of image size. In general, seeing is the primary factor in this system. This was derived from numerical data for systems that were hand guided in excellent seeing and which used no active optical compensation of fast guiding for image quality enhancement.
We also transformed the above formula into a form which
allows
one to determine the smallest star images (FWFM) that can be
expected for any photographic imaging system :
Rpl (microns) = 3.062 *
f
where:
Rpl.....is the linear size of the smallest star
images (FWFM) in microns
f.......is the f/ratio of the optical system
In order to not under sample this resolution element, one should get at least TWO PIXELS across the size of Rp. Another approach would be to set the resolution element to the very best seeing conditions that one could normally expect to experience (i.e. 1 arc second) or perhaps one-half of that value. For planetary work, one would pick a very small value for the pixel element in order to insure oversampling of the image (also, the resolution criteria for planetary (continuous) details is far more relaxed than is the Raleigh criteria).
Thus, one could select one of several criteria to determine the optimum resolution element:
(where N represents the degree of oversampling sought and where N should quite probably be 3 or 4 for achieving results that approach the potential of the system. N MUST BE SET TO AT LEAST 2 FOR ADEQUATE SAMPLING.
Once this resolution element is set it is then possible to
calculate
the optimal focal length of the system to match the CCD camera's
capabilities by matching the size of the resolution element to
the
pixel size of the CCD. To do this we simply use an adaptation
of
the image scale formula that was presented earlier in this work.
F (inches) = 8.12 * {Pixel Size
(microns)} / S
or
Pixel Size (microns) = { F (inches) *
S } / 8.12
It is quite important to oversample the imageto some degree. Not only is it a matter of aesthetics to not end up with rectangular star images, but it also makes for more accurate astrometric positional determinations and makes for more accurate photometry as well. A star that is ideally only sampled by a single pixel can take on a number of rectangular and/or irregular shapes as the star image is moved about the sampling array (Images shown in FIGURE 1). While well sampled stars retain a consistant shape as one moves about the image (Image shown in FIGURE 2). A star which is so badly undersampled can not be well measured for accurate centroiding for astrometry and the totally irregular shape of the star image becomes a source of noise in photometric applications. In addition, if such undersampled images are to be used in tri-color work one will find that all stars in the field show color fringes of Red, Green, or Blue depending upon how the image was sampled during the exposures. Finally, if your are a user or lover of maximum entropy (ME) algorithms and have an ME processor which can be tuned you will find that oversampled images are far less likely to produce the tell-tale black-ring effect around stars that are in undersampled images which have rectangular star images. As a side effect of oversampling, you will discover that focusing will become a bit easier since the energy of a single star image is now spread over a few pixels instead of all falling on a single pixel. (See CCD Focusing section for more details on this aspect of pixel sizing).
FIGURE 3 shows (greatly zoomed) the effect of undersampling on the shape of a star image. In this example, the same star has been slightly shifted between exposures resulting in totally different final image shapes as the CCD's sampling array moves about the star image and departs from the ideally centered sampling geometry ( upper row, center). The CCD pixels were approximately the same size as the expected resolution element (i.e. N=1). The lower row shows how the effective resolution of a system can be improved by using a SHIFT AND ADD imaging technique developed by Brad Wallis. (all images by Benoit Schillings)
I would like to offer some explanations and analogies on the matter of big and little pixels and how they relate to image quality. An analogy between big and little pixels can be drawn by comparing images taken with the old grainy 103a- spectroscopic emulsions and the current popular use of ultra-fine grained Technical Pan. Images with 103a- films were coarse and grainy looking yet the grain sizes with this emulsion were considerably smaller than 9um pixels of today's KODAK CCDs. When 103a- and Tech Pan images are laid side-by-side there is no question that the Tech Pan image is not only much more aesthetically pleasing but that it contains much more information and detail than does the 103a- image (See AMOACP, p.87,88 for illustrations of such a comparison).
When this comparison is made though it is also clear that while the image detail is greatly increased using Tech Pan that the star images on the Tech Pan image are only a very small amount smaller than the images on the 103a- image. Reasons for this seeming discrepancy include the fact that the often quoted Raleigh Limit was never meant to be applied to details in continuous portions of the image (planets , nebulae, galaxies, etc. ) and such areas of continuous detail the size of the star image or the Airy disk are not good guides as to the resolution capabilities of the system. In visual work the Sparrow criteria is best applied to these areas and it says that a given system can resolve details that are slightly finer than the Raleigh criteria.
The image of M51
demonstrates the degradation of an
excellent image (taken by Benoit Schillings) by the use of
larger
and larger pixel sizes. This set of images was created using
IDL to
perform the image binning to synthesize the effects of larger
pixels.
UPPER LEFT: 9 um pixels UPPER RIGHT: 18 um pixels
LOWER LEFT: 27 um pixels LOWER RIGHT: 36 um pixels
ADDENDUM: Comments on THE FOSTER SERIES text.
If you have seen discussions of PIXEL SIZE out on various
internet forums you will probably notice that our conclusions
and recommendations are contrary to those cited from a work
called "The Foster Series". There is a very basic
reason for
this discrepancy as those who have actually read the Foster
book will recognize
The FOSTER SERIES was written as a VERY basic and
introductory
text on CCD imaging. It basically assumes that one is using
POOR OPTICS, in POOR SEEING, and that the instrument will NOT
BE PROPERLY FOCUSED. It assumes that the CCD user will take
NO STEPS to improve any of the above conditions and thus will
only be seeking images of mediocre quality.
WE ASSUME that we are speaking to an audience that IS interested in achieving results that are near optimal.