Sample Image Restorations

To demonstrate the advances represented by our methods, we present in this section reconstructions from both real and ``mock'' data sets. We have done this for several reasons. In the mock data case, the image reconstruction conditions are essentially perfect - the noise, PSF, and true answer are known a priori with arbitrary precision. This leaves no uncertainty in how well each algorithm has performed. However, in imaging situations we rarely encounter such benign conditions. For this reason, we have also included a real data test case in which the noise and PSF characteristics are experimentally determined. Unfortunately, the true answer is also imperfectly known, making validation of the technique more difficult.

In order to make the comparisons as fair as possible, we compare our reconstructions to those performed by other professionals well versed in the competing techniques. This avoids issues of whether the competing reconstructions are the best possible. For the real data test, we have chosen IRAS (Infrared Astronomical Satellite) 60m survey scans of the interacting galaxy pair M51 (the ``Whirlpool''). This data was used for an international image reconstruction contest at the 1990 MaxEnt Workshop (see Bontekoe 1991), which was attended by leaders in the field of image reconstruction. Hence our reconstruction of M51 will be compared to the best state-of-the-art reconstructions circa 1990. From comparisons like the M51 contest, experts generally agree that ME produces results superior to GOF methods (e.g., Least-Squares and Richardson-Lucy). For this reason, we shall concentrate on comparing our reconstructions to ME reconstructions (although we shall present a Richardson-Lucy reconstruction of the M51 data set as well, allowing the reader to judge the validity of these claims). The ME code we shall make our comparisons to is MEMSYS, a powerful set of ME algorithms developed by Gull and Skilling (1991). The MEMSYS algorithms probably represent the best commercial software package available for image reconstruction. The mock data reconstruction example compares our fractal pixon methods with MEMSYS 5, the most current version of the MEMSYS algorithms (see Gull and Skilling 1991). The M51 example compares our results to those of MEMSYS 3, the current version of MEMSYS at the time.

Example 1: A Mock Data Set Reconstruction

Fig. 1 presents FPB and MEMSYS 5 reconstructions of a mock data set. The MEMSYS 5 reconstructions were performed by Nick Weir of Caltech, a recognized MEMSYS expert, and were supplemented with his multi-channel correlation method which has been shown to enhance the quality of MEMSYS reconstructions (Weir 1991, 1993a). The true, noise-free, unblurred image presented in the top row is constructed from a broad, low-level elliptical Gaussian (a two-dimensional Gaussian with different FWHMs in perpendicular directions), and two additional narrow, radially symmetric Gaussians. One of these narrow Gaussians is added as a peak on top of the low-level Gaussian. The other is subtracted to make a hole. To produce the input image, the true image was convolved with a Gaussian PSF of FWHM=6 pixels, then combined with a Gaussian noise realization. The resulting input image is displayed in the top row. The signal-to-noise ratio on the narrow Gaussian spike is roughly 30. The signal-to-noise on the peak of the low level Gaussian is about 20. The signal-to-noise at the bottom of the Gaussian hole is 12.

As can be seen, the FPB reconstruction is superior to the multi-channel MEMSYS result. The FPB reconstruction is free of the low-level spurious sources evident in the MEMSYS 5 reconstruction. These false sources are due to the presence of unconstrained degrees of freedom in the MEMSYS 5 reconstruction and are superimposed over the entire image, not just in the low signal to noise portions of the image where they are most evident. Furthermore, the FPB reconstruction's residuals show no spatially correlated structure, while the MEMSYS 5 reconstruction systematically underestimates the signal, resulting in biased photometry.

Example 2: 60 Micron IRAS Survey Scans of M51

We have also reconstructed an image from 60m IRAS survey scans of the interacting galaxy pair M51. This data was selected for several reasons. First, M51 is a well studied object at optical, IR, and radio wavelengths. Hence ``reality'' for this galaxy is relatively well known. Second, as mentioned before, this particular data set was chosen as the basis of an image reconstruction contest. Consequently, there have been a number of serious attempts at performing image reconstruction on this data set by specialists in the field. Finally, the IRAS data for this object is particularly strenuous for image reconstruction methods. This is because all the interesting structure is on ``sub-pixel scales'' (IRAS employed relatively large, discrete detectors - 1 5 by 4 75 at 60m) and the position of M51 in the sky caused all scan directions to be nearly parallel. This means that reconstructions in the cross-scan direction (i.e. the 4 75 direction along the detector length) should be significantly more difficult than in the scan direction. In addition, the point source response of the 15 IRAS 60m detectors (pixel angular response) is known only to roughly 10%accuracy, and finally, the data is irregularly sampled.

Our FPB reconstruction appears in Fig. 2 along with Richardson-Lucy and Maximum Correlation Method (MCM) reconstructions (Rice 1993) and a MEMSYS 3 reconstruction (Bontekoe et al. 1991) - see Gull and Skilling (1991) for a description of the MEMSYS algorithms. The winning entry to the MaxEnt 90 image reconstruction contest was produced by Nick Weir of Caltech and is not presented here since quantitative information concerning this solution has not been published; however, see Bontekoe (1991) for a gray-scale picture of this reconstruction. Nonetheless, Weir's solution is qualitatively similar to Bontekoe's solution (Weir 1993b). Both were made with MEMSYS 3. Weir's solution, however, used a single correlation length channel in the reconstruction. This constrained the minimum correlation length of features in the reconstruction, preventing break-up of the image on smaller size scales. This is probably what resulted in the ``winning edge'' for Weir's reconstruction in the MaxEnt 90 contest (Weir 1993b).

As can be seen from Fig. 2, our FPB-based reconstruction is superior to those produced by other methods. The Richardson-Lucy and MCM reconstructions fail to significantly reduce image spread in the cross-scan direction, i.e., the rectangular signature of the 1 5 by 4 75 detectors is still clearly evident, and fail to reconstruct even gross features such as the ``hole'' (black region) in the emission north of the nucleus; this hole is clearly evident in optical images of M51. The MEMSYS 3 reconstruction by Bontekoe is significantly better. This image clearly recovers the emission hole and resolves the north-east and south-west arms of the galaxy into discrete sources. Nonetheless, the level of detail present in the FPB reconstruction is clearly absent, e.g., the weak source centered in the emission hole (again, this feature corresponds to a known optical source).

To assess the significance of the faint sources present in our FPB reconstruction, in Fig. 3 we present our reconstruction overlaid with the 5 GHz radio contours of van der Hulst et al. (1988). The radio contours are expected to have significant, although imperfect, correlation with the far infrared emission seen by IRAS. Hence a comparison of the two maps should provide an excellent test of the reality of structures found in our reconstruction. Also identified in Fig. 3 are several prominent optical sources and H knots.

As can be seen, the reconstruction indicates excellent correlation with the radio. The central region of the main galaxy and its two brightest arms align remarkably well, and the alignment of the radio emission from the north-east companion and the IRAS emission is excellent. Furthermore, for the most part, whenever there is a source in the reconstruction which is not identifiable with a radio source, it can be identified with either optical or H knots. An excellent example is the optical source in the hole of emission to the north-east of the nucleus of the primary galaxy or the bright optical source to the north-west of the nucleus (both labeled ``Opt'' in Fig. 3). Because of the excellent correlation with the radio, optical, and H images, we are quite confident that all of the features present in our reconstruction are real.

Aside from the fact that most of the sources can be identified with emission at other wavelengths, the residual errors in our reconstruction are much smaller than in the MEMSYS 3 reconstruction. As pointed out by Bontekoe et al. (1991) the peak flux in the MEMSYS 3 reconstruction is 2650 units. The residual errors are correlated with the signal and lie between 0 and 430 units. By contrast, the peak value in the FPB reconstruction is 3290 units, the residuals are uncorrelated with the signal, and the residuals lie between and 17 units. (The contour levels for the MEMSYS 3 and FPB reconstructions of Fig. 3 are identical and are 150, 300, 600, 1200, and 2400 units.) Furthermore, the large deviation residuals in the FPB reconstruction are due to systematic errors involving incomplete scan coverage of M51. Fortunately, these errors do not lie under the significant flux emitting portions of the M51 image. The residual errors associated with emitting regions in M51 are significantly smaller ( unit) and show a roughly Gaussian distribution. Full appreciation of the sensitivity of our technique is only obtained once the reconstruction has been flux calibrated. Formally, the residual error over the majority of the image is 2.7 mJy. This compares with the 280 mJy, 90%completeness limit for the IRAS Faint Source Survey. The largest residual systematic errors associated with incomplete sampling of the M51 region correspond to 50 mJy. This is still a factor of more than five times fainter than the IRAS Faint Source Survey limit.