Deconvolution of images

The search for high spatial resolution

The images obtained with ground-based telescopes are blurred through the action of two main effects: an INSTRUMENTAL EFFECT and ATMOSPHERIC EFFECT

The instrumental blurring, which is present in all images obtained through any optical device, is the result of the finite aperture of the instrument (the size of the primary mirror in a telescope) as well as of possible optical aberrations.

The atmospheric blurring (the so-called "seeing") is caused by turbulence in the earth's atmosphere.

The spatial resolution of the images (i.e. the size of the smallest details that can be distinguished) is thus limited by these two effects. In the best sites on Earth, the resolution is of the order of half an arcsecond to an arcsecond, i.e. the angular size of a man seen from a distance of 400 km.

One of the major challenges of modern astrophysics is to obtain images with a spatial resolution as high as possible. The recent developments take place along 4 main lines:

 

  • The interferometric techniques, which use several telescopes and which are quite complex to set up in the optical domain;

  • The use of space telescopes, which allow to get rid of atmospheric effects but are extremely expensive;

  • Adaptive optics, which aim at correcting the atmospheric perturbations in real time by modifying the shape of one of the mirrors;

  • The deconvolution techniques, which consist in a numerical processing of the image, posterior to its acquisition at the telescope.

 

decon001Airy rings: image of a point source seen through a circular aperture. The central spot and the rings are larger for smaller apertures (smaller primary telescope mirror).

speckleImage of a star caught through the earth's atmosphere, with a short exposure time. For longer exposure times, the moving individual images (speckles) average to form a seeing spot.

This image has been obtained from the Applied Optics group at Imperial College London

deconv002Centre of a large cluster of galaxies observed with different spatial resolutions. When spatial resolution improves, finer details can be distinguished (as, here, the arcs formed by gravitational lensing).

 

Traditional deconvolution techniques

Among these different methods, deconvolution is by far the cheapest, and its development has followed the one of computers. Nevertheless, until recently, the results were rather disappointing: the gain in spatial resolution was hampered by the appearance of spurious structures called artefacts (e.g., rings around point sources). Moreover, it was impossible to perform astrometric or photometric measurements on the deconvolved images. It was thus nearly impossible to derive any quantitative information from these images.

 

From an analysis of these problems, we have concluded that, for a large part, their origin is to be found in the fact that these traditional methods attempt to correct the totality of the blurring, thus to obtain a perfect spatial resolution. However, it is clear that, if the image is in the form of pixels, it is impossible to represent details smaller than the pixel size. Such an attempt is a violation of a theorem well known in signal processing, namely, the "sampling theorem", established by Shannon in 1949.

 

deconv004

Example of deconvolution with a traditional method, illustrating some of the common defects.
Left: exact light distribution, which would be obtained with a perfect deconvolution technique.
Middle: simulated observation, with lower resolution and measurement errors.
Right: deconvolution of the centre image with the maximum entropy method. Rings can be seen around the stars superposed to a diffuse background, as well as "bridges" connecting nearby stars. The magnitudes and positions of the latter are not accurately recovered.

MCS deconvolution method

The solution to that problem lies in an algorithm which does not attempt to correct the totality of the blurring, but to keep some of it, so that the finest details in the resulting image are compatible with the size of the pixels chosen to represent it. One thus seeks the shape of the light distribution which would be obtained, not with a perfect instrument, but with a better instrument (e.g. a 10 m space telescope).

 

The size of the pixels and the spatial resolution of the deconvolved image are chosen in order to conform to the sampling theorem. A further advantage of this method is that the observer knows a priori the shape of all point sources in the deconvolved image, since he chooses this shape. The light distribution can thus be written as the sum of a number of point sources (with positions and intensities to be determined), plus a diffuse background which must itself conform to the sampling theorem. One thus introduces a smoothing constraint in order to avoid details smaller than the pixel size allows.

 

The algorithm searches for the positions and intensities of the point sources, as well as the shape of the background, which best reproduce the observed image, within the error bars. This is a rather complex mathematical problem, since one must find the minimum of a function in a space with (N+3M) dimensions, N being the number of pixels and M the number of point sources. As an example, the deconvolution of a 256 x 256 image with 10 point sources requires to work in a 65566 dimensions space.

 

Unlike traditional methods, the MCS algorithm provides a deconvolved image in which the positions and intensities of the different sources are respected. This allows to perform astrometric and photometric measurements with high accuracy.

 

deconv006Deconvolution of a simulated image of a star cluster partly superimposed on a background galaxy.
Top left: true light distribution with 2 pixels FWHM resolution;
bottom left: observed image with 6 pixels FWHM and noise;
top middle: Wiener filter deconvolution of the observed image;
bottom middle: 50 iterations of the accelerated Richardson-Lucy algorithm;
top right: maximum entropy deconvolution;
bottom right: deconvolution with our new algorithm.

 

deconv008Infrared image of the gravitational lens PKS 1830-211 similar to the one obtained with the 10 m Keck I telescope. The animation shows the MCS deconvolution.