Deconvolution of images and spectra
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.
Airy 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).
Image 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.
Centre 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.
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.
Deconvolution 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.
Infrared 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.
Deconvolution of spectra
Following the same fundamental principles as for our image deconvolution algorithm, a technique for spatial deconvolution of spectra has been developed. The spectrum of a stellar-like object can be used to spatially resolve the spectra of very blended objects. As in the image deconvolution algorithm, the spectrum is decomposed into a sum of point sources and diffuse numerical background, so that the spectrum of extended sources blended with point sources may be obtained.
The images below show examples of applications to simulated and real spectra. The algorithm takes into account seeing variations as a function of wavelength and atmospheric refraction, which are both included in the present simulations.
Deconvolution of two very blended spectra
From left to right:
1. A simulated two-dimentional spectrum of 2 blended point sources. The seeing is 4 pixels FWHM and the separation between the two objects is only 2 pixels. The two blended input spectra consist of a QSO (continuum + emission lines) about 1.5 magnitude brighter than a star (continuum only). Note the curvature of the spectrum, simulating the effect of strong atmospheric refraction.
2. The deconvolved spectra, where the 2 objects are now visible.
3. Residual map, i.e., data minus deconvolved model (reconvolved by the PSF), in units of the photon noise. The residual map is flat and equal to 1 almost everywhere in the field, indicating the result of the deconvolution is good.
Spatial profiles of the spectrum before and after deconvolution.
1-D deconvolved spectra compared with input spectra.
One dimensional deconvolved spectra of the QSO (top) and the star (bottom). For both objects, the inset show the division of the deconvolved spectrum by the input spectrum, and demonstrates how well the deconvolution procedure recover the spectra, in spite of their severe degree of blending.
Deconvolution of the spectrum of a lensed QSO and its lensing galaxy
From left to right:
1. Simulated spectrum of a doubly imaged QSO. The lensing galaxy is 4.5 magnitudes fainter than the QSO images and is situated at only 2 pixels away from the QSO images on the left. The seeing is 4 pixels FWHM.
2. Deconvolution of the simulated spectrum
3. Deconvolved spectrum of the lensing galaxy
4. Input spectrum of the lensing galaxy
5. Residual map, as in the first example.
1-D deconvolved spectra of the QSO images (top). The bottom panels show the result of the division of the deconvolved spectra by the input spectra.
1-D deconvolved spectrum of the faint lensing galaxy with simulated 3000 Å break and OII emission line (top). As for the QSO images, the bottom panel show the result of the division of the deconvolved spectrum by the input spectrum. Note the very good agreement between the recovered and input spectrum. Only the very right part of the spectrum is slightly (2σ) overestimated by the deconvolution. However, (i) this is visible on the upper part of the residual map of figure 1 (2σ residuals in the upper part of the residual map), (ii) the position of the emission line is recovered with an accuracy of 0.1 pixels and allows one to measure for example the redshift of the lensing galaxy. (iii) Such results can not be obtained with any other existing method.
VLT/FORS1 spectrum of the quasar HE 1503+0228 at a redshift of 0.13. The figure shows the total spectrum (quasar + host galaxy). Note the very high signal-to-noise ratio attained despite the relatively high dispersion, R=700. This spectrum is not flux-calibrated.
After deconvolution and subtraction of the quasar spectrum, the spectrum of the host galaxy is obtained. Note that the spectrum shows no trace of contamination by the central quasar.
Zooms on two emission lines in the spectrum of the host galaxy. In each figure, the raw 2-D spectra are shown on the left. The right part shows the spectrum of the galaxy alone, with a spatial resolution of 0.2" after deconvolution. This will allow to study the variation of the spectral properties with distance from the central quasar. The emission lines visible on these spectra are Hβ and OIII (left figure), and OII (right figure). The latter also shows the H and K lines of Ca II in absorption, as well as the 4000 Å break. Note the tilt of the lines as a consequence of rotation. The total field is 10".