Super-resolution is found an ambigous term in the disciplines of informatics, physics, optics, astronomical imaging and spectroscopy (Bauer, 2011). It is ambigous in the sense, that super-resolution has at least two different meanings: (1) Resolution beyond the optical diffraction-limit, and (2) resolution below the barrier of the dimension of a pixel of the imaging sensor, either in spectroscopy or in imaging. In other words, super-resolution tries to find information in and between pixels, and possibly breaks the optical limit of resolution (i.e. photo camera optics, microscope or an astronomical telescope). To make things more confusing, one shall distinguish between the term super-resolution as a measure of resolution found and the method used to obtain that resolution.

In principal, there is little difference between the different meanings of super-resolution in terms of resolution that is achieved. More than that: Resolution is not limited in any way, because any limit appears arbitrary (Den Dekker & Van den Bos, 1997). Depending on the scale of the image and dimension of the optics, one can distinguish and mean both at the same time. To give an example: A well sampled image taken without atmospheric blur means, the optical diffraction pattern or abberration is well sampled or slightly undersampled (the reader might think of it as a "sharp image"). This is the typical case with any photo optics, well adapted to the imaging sensor of the digital camera. Achieving super-resolution may mean to break the optical limit of diffraction. This has been shown to be possible under certain circumstances. With any ground-based telescope and aperture of more than 10 cm, resolution is not only limited by the optics, but also by atmospheric blur (also called seeing). In this case super-resolution may mean to first overcome the limit of the atmospheric blur, then possibly break the diffraction limit of the telescope at a finer pixel grid.

Both situations of having found super-resolution, the diffraction limited case and the case of different blur (eg. from seeing) demonstrate any resolution beyond the principal design and theoretical capabilities of the optical system and detector under certain circumstances. However, not only the different meanings may cause confusion. Super-resolution will be achieved by no physical process, but only digital image processing (although there also exist physical super-resolution techniques, like interference contrast). However, we cannot expect an omnipotent machine, "Deus ex machina", or a perpetual motion machine creating information. Not everything is possible to be achieved from a certain algorithm. In fact, every algorithm underlies physical limits, especially in this case of super-resolution. That does not mean any of the limits of resolution mentioned above. Instead, the problem lies in the understanding of the nature of the limits of the method. No wonder, super-resolution was discussed and denied by some people even in the 90ies of the last century. Some claimed it impossible to achieve super-resolved images. Puschmann & Kneer (2005) still presented remarks of having found no gain in resolution from techniques expected to provide super-resolved images. However, the reasons may have to be interpreted in a different way. It is true: we cannot produce information or energy, which is not already contained in the data. Super-resolution is possible due to certain statistical processes, which are needed to achieve Super-Resolution. Therefore, without noise in the digital images we wouldn't be able to obtain super-resloved images. Statistical image motion is the essential requirement for the task of super-resolution to work.

Fig: The technique of super-resolution applied to a set of images of Messier 57. Click to enlarge.

The above image of Messier 57, the famous ring nebula, demonstrates well the method of super-resolution. The original raw image (left) is a sample of 15 images taken for the super-resolved image processed with an appropriate method (right). The basic ideas of the method are partially described by Bauer (2011). The super-resolution output presents a more detailed image of Messier 57 at a finer pixel grid and better resolution. 

Discussion 

The low resolution input images have been taken with an amateur telescope of 20 cm aperture (f/6.5) at a scale of only 1.2 arcsec per pixel. A photographer might think of the telescope as a 1200 mm teleoptics. Surprisingly, this is not a very large telescope. The camera was a digital single-lens reflex camera (DSLR). The image scale represents a very small angle of view per pixel and well adapted to the typical seeing disc diameter. However, and in this case the pixel dimension appears rough-textured compared to the dimension of the faint nebula. As a result, the small nebula looks blocky at higher magnification of the low resolution input image. Stars appear larger than the dimension of a pixel in the low resolution images and are distorted by the atmospheric turbulence. Notice the double star below the nebula. While the binary star appears blocky and smeared, separated only 3 pix in the original image (left), it is well separated in the super-resolved image (right). The fine details reconstructed from the noisy images of the nebula are seen as real features, when compared to an image of the Hubble Space Telescope! Therefore, this method of super-resolution provides an excellent image fidelity and works for both, single point sources and extended light sources (which is not the case with every method discussed in the literature). The super-resolved image (right) demonstrates well to have broken the pixel barrier by a factor of four and reflects an image scale of only 0.3 arcsec per pixel. This is well below the diffraction limit of the telescope of about 0.65 arcsec. Due to the small amount of images, it is not as sharp, as this would be theoretically possible. The image processing is a very complex algorithm including automatic estimate of image motion, image deconvolution and a sophisticated noise estimate and removal. The famous Drizzle technique also yields a finer pixel grid at an arbitrary scale. However, the Drizzle method is restricted on the image registration. Therefore, the output of Drizzle provides no better or even a worse image resolution, compared to the low-res input images.

Conclusion and Remarks 

Super-resolution is a technique to overcome certain resolution limits of an image, like atmospheric blur and limited pixel dimension. Therefore, an amateur with a small telescope will be able to reproduce some resolution of a larger telescope, while larger telescopes can be improved in resolution, too. However, there is a certain amount of effort. In general, super-resolution will help to remove atmospheric blur from ground-based imaging and may provide better spectral resolution with a spectrograph. But things are more complicated with spectroscopic devices, because some statistical processes intentionally are supressed with a spectrograph design, like statistical image motion along the wavelength scale, as this would disturb the measurement of precise radial velocities. However, it will be possible to introduce these effects artificially, which again will enable super-resolution. This can be thought of a statistical micro-scanning spectroscopic device. Such a device can be created by an engineer or amateur astronomer. There are already similar microscope designs built by several optical companies. However, this will be outside the scope of this article about Super-Resolution Imaging in Astronomy.

Observational data

References

Bauer, T., 2011. Super-Resolution Imaging: The Use Case of Optical Astronomy. Proceedings of the IADIS International Conference Computer Graphics, Visualization, Computer Vision and Image Processing, CGVCVIP 2011, p. 49-59, Rome, Italy, 2011, ISBN 978-972-8939-48-9

Den Dekker, AJ and Van den Bos, A., 1997. Resolution: a survey. Journal of the Optical Society of America A, vol. 14, no.3, p.547-557. 

Puschmann, K. G. and Kneer, F. 2005. On super-resolution in astronomical imaging. Astronomy & Astrophysics, Vol. 436, p. 373–378