In theory the resolution limit of a telescope is a function of the size of the main mirror, due to the effects of Fraunhofer diffraction. This results in images of distant objects being spread out to a small spot known as the Airy disk. A group of objects spread out over a distance smaller than this limit look like a single object. Thus larger telescopes can not only image dimmer objects because they collect more light on the larger mirror, but are also able to image smaller objects as well.
This theory breaks down due to the practical limits imposed by the atmosphere, who's random nature disrupts the single spot of the Airy disk into a pattern of light and dark spots about 30 times larger. A similar effect can be seen by looking at the bottom of a swimming pool on a sunlight day, the sunlight is spread out into a pattern of light and dark patches. Due to the nature of the airmass, the practical resolution limits are at mirror sizes well within existing mechanical limits, at about 4m in diameter. For many years telescope growth was limited by this effect, until the introduction of speckle interferometry and adaptive optics provided paths to remove this limitation.
Speckle interferometry re-creates the original image through image processing techniques. Key to the technique is to take very fast images, in which the atmosphere is effectively "frozen" in place. For infra-red images this is on the order of 100ms, but for the visible region it is as small as 10ms. In images of this time scale, or smaller, the movement of the atmosphere is too small to have an effect; the speckles recorded in the image are a snapshot of the airmass at that instant.
Speckle interferometry uses this effect to regain information otherwise lost. By taking a series of images at high speed, and then overlapping them by lining up the brightest speckle, a single image in regained. However this averaged image still contains more information than would be captured on a single longer-time image. By measuring the "average bispectrum", which is a combination of both the spectrum and the phase. This information can then be used to calculate the Fourier spectrum, which is then inverted to obtain an image.
The interesting part of this process is that the image can often contain a far higher resolution than would otherwise be possible. This is because the technique is effectively using the large bubbles in the atmosphere as a series of large lenses, creating a huge optical interferometer, one with better resolving power than the telescope would otherwise have.
Of course there is a downside, taking images of this sort of time frame is difficult, if the object is too dim, not enough light will be captured to make the analisys possible. Early uses of the technique in the early 1970s were made on a limited scale using photographic techniques, but since photographic plates capture only about 7% of the incoming light, only the brightest of objects could be processed in this way. The introduction of the CCD into astronomy, which capture about 70% of the light, lowered the bar on practical applications enormously, and today the technique is widely used on almost all objects.
Another limitation of the technique is that it requires extensive computer processing of the image, which was also hard to come by when it was first applied. Although the almost-universal Data General Nova served well in this role, it was slow enough to limit the application to only "important" targets. Again, this limitation has largely disappeared over the years, desktop computers have more than enough power to make such processing a trivial task.
More recently another use of the technique has developed for industrial applications. By shining a laser, who's smooth wavefront is an excellent simulation of the light from a distant star, on a surface the resulting speckle pattern can be processed to regain detailed images about flaws in the material.
See also: holographic interferometry optical interferometry Knox-Thompson interferometry