Reconstructions of a mock data set. The input image is shown on the far left along with a surface plot (center row). Below is the point-spread-function (PSF) and the noise added to the smoothed (PSF-convolved) data. To the right is a Pixon¹ method reconstruction and a Maximum Entropy reconstruction (the algorithms used are the MEMSYS 5 algorithms, a powerful set of commercial ME algorithms). The ME reconstructions were performed by Nick Weir, a recognized ME and MEMSYS expert. The reconstructions were supplemented by Nick's multi-correlation channel approach which has been demonstrated to greatly improve the results of ME and MEMSYS reconstructions.

The Pixon method reconstructions use what we have come to call the Fractal-Pixon Basis (FPB) approach--see references. (We have since dropped the "Fractal" nomenclature, however whenever you see FPB, this method is the "standard" Pixon method.) It can be seen that the FPB reconstruction has no signal correlated residuals and is effectively artifact (false-source) free, which are problems which are obvious in the MEMSYS reconstruction. The absence of signal correlated residuals and artifacts can be understood from underlying theory of the Pixon method--again, see references.

This example presents a Pixon method reconstruction of IRAS satellite 60 micron survey scan data of the interacting galaxy pair, M51, the "whirlpool" galaxy. The image to the left is the raw, co-added IRAS scans, and shows the rectangular signature of the 60 micron IRAS detectors. The false-color image to the right is the Pixon method reconstruction. This reconstruction has a resolution enhancement of a factor of 20 in linear dimension and because of the ability of the pixon method to reject spurious sources the reconstruction has a sensitivity which is a factor of 100 times better than the 90% completeness limit of the IRAS Faint Point Survey catalog.

The weakest feature present in false color image has a formal signal-to-noise ratio of 30. Many of these sources can easily be identified with optical field stars (some are noted in the figure, i.e. as "Opt"). Other sources are identified with radio, H alpha, and near-IR sources. (The contours are the 5 GHz radio contours and show excellent agreement with the reconstructed image--perfect agreement is not expected since the emission mechanisms are correlated, but not identical.)

In the figure above, we continue discussion of example 2. Here the results of our Pixon method reconstruction are compared to the reconstruction results produced by other techniques. The IRAS M51 data is particularly good for this purpose since this data set was used as the basis for an international image reconstruction contest held at the 1990 MaxEnt meeting.

The contour to the far right is again the raw co-added 60 micron IRAS survey scans. Shown to the left is the Pixon method reconstruction (FPB=Fractal Pixon Basis reconstruction), a MEMSYS 3 reconstruction, a Lucy-Richardson reconstruction, and a Maximum Correlation Method (MCM) reconstruction (this is the in-house method developed by IPAC, the NASA data center responsible for the analysis and distribution of IRAS imaging data).

As can be seen, the MCM and Lucy-Richardson method barely improve the co-added IRAS data. They do not even show up the absence of emission (dark region) visible in the optical (see false color image of M51 on my homepage), and in the FPB and MEMSYS 3 reconstructions. The MEMSYS 3 reconstruction was essentially the winner of the 90 MaxEnt contest. However, this reconstruction is significantly poorer than the FPB reconstruction. Formally, the resolution is 2-3 times poorer and the residuals are correlated with the emission and are almost 2 orders of magnitude larger than the errors in the FPB reconstruction. The judge officiating over the 90 MaxEnt contest has seen the FPB reconstruction, and agrees that the FPB reconstruction is a vast improvement over the MEMSYS reconstruction.

Above we display an application of Pixon method reconstruction so Yohkoh satellite hard X-ray data. The Yohkoh hard X-Ray telescope is a coded mask imager that uses 64 fixed masks to measure essentially 64 different spatial Fourier components of the object being studied.

The example above shows a time series of 3 images taken 10 seconds apart of a solar flare loop. The right-hand column shows the results of direct algebraic inversion of the problem (i.e. inverse Fourier transforms of the data). The central column shows a Maximum Entropy reconstruction, and the left column shows a Pixon method reconstruction. All of the methods are flux conserving so the total flux in each image (at a given time) is the same. This shows how poorly the direct inversion is in absolute photometry. The ME reconstruction is significantly better. However, even here we see a large number of instrumental artifacts and relative to the Pixon method reconstruction the loss of flux to false sources causes the photometry to be in error by 300%. Also note that the apparent increase in resolution in some parts of the ME reconstruction is unwarranted. Formally, the Pixon method reconstruction is a slightly better statistical fit, and the fit is done with smoother structures.

We have performed a number of mock simulations with Pixon method reconstructions with this instrument. We find that the technique consistently and reliable reproduces the input data. This gives us great confidence in the ability of the Pixon method to produce high-quality, reliable imaging results. The reconstructions of this example were performed by Tom Metcalf of the University of Hawaii (currently at Lockheed). Interested parties can visit his web page for additional information on his application of the Pixon method to Yohkoh data.

Above is another Yohkoh solar flare imaging example. In this case the false color image is the Yohkoh soft X-ray image of a flare. The contour overlays are the Pixon method reconstruction of the hard X-ray imaging data. (Yohkoh's soft X-ray imaging ability is much better than its hard X-ray imaging ability.) Also indicated is the solar limb (curved white line). As can be seen the Pixon method hard X-ray imaging results overlay the soft X-ray results very well. Compared to the ME reconstruction of the hard X-ray data the Pixon method results in this case are much better. The ME method failed entirely in this case to come up with an image. Apparently the noise was just too much for the method. Again, the reconstruction of this example was performed by Tom Metcalf.

The example above presents reconstructions of data from the OSSE instrument aboard the Compton Gamma-Ray Observatory satellite. The OSSE instrument is a set of gamma-ray collimators which each can be scanned in one direction. The images presented above includes all the data (dark region) for which there was significant exposure time in the OSSE Virgo survey (i.e. images taken in the direction of the Virgo cluster).

The above example compares two reconstructions both performed by Dave Dixon of the University of California, Riverside (interested parties should contact Dave for more information). The Non-Negative Least Square method was developed at UC Riverside and its results are displayed in the left-hand panel. As can be seen, this method has great difficulty discovering any sources which are significant relative to the reconstruction artifacts. By contrast, the Pixon method finds two strong sources, the quasar 3C 273 and the galaxy NGC 4388. (The galaxy M87 is also in the error box containing NGC 4388, but M87 is not expected to contribute significantly to the gamma-ray emission.) It also detects a mild gradient in the underlying gamma-ray background.

Above is an example of the Pixon method applied to time-series data from the Vela X-ray spacecraft. Shown in the bottom trace is the X-ray emission detected as a function of time. Above is a pixon noise filtered version of the data. (An arbitrary constant was added to the pixon-filtered data for clarity in making the plot.) In this case pixon methods where not used to enhance the temporal resolution of the Vela satellite data (although this could have been done), but merely to assess the statistical significance of any temporal structures present, and then to perform a temporally adaptive filter on the data. This procedure preserves all of the statistically significant structures and filters out the noise.

As can be seen, the Pixon method noise filter does an excellent job of describing the data. Sharp peaks retain their temporal sharpness while lower level noise features are flatted out as appropriate in a manner consistent with the information content in the data--see references.

Above is another example of using the Pixon method for X-ray mammography. The raw X-ray image appears to the left. In this example a breast phantom is used (material with X-ray absorption properties similar to the human breast). In this case a small fiber (400 micron diameter) is present in the phantom. As can be seen the signature of the fiber is rather faint in the direct X-ray image. The Pixon method reconstruction is seen to the right. Here the signature of the fiber is obvious. Such image enhancement is of clear benefit to the discovery of weak X-ray signatures. As can be seen in this case, the X-ray signature of the fiber is very close to the noise level. This is evidenced by the break-up of the continuous fiber into pieces in the Pixon method image--the Pixon method recognizes that in certain locations in the X-ray the signal present is not statistically significant. In these locations the fiber vanished in the Pixon method reconstructed image.

This example is also drawn from the field of medical imaging. Here a gamma-ray image from chemically doped tracers are used to probe cancer. Shown on the left is the raw image of a persons chest (collar bone and ribs can be seen). The raw image has poor detail and is confused by strong photon counting noise. The Pixon method processed image on the right achieves maximum possible detail and removes the photon counting noise (a minimal complexity, i.e. maximum smoothness model has been fit to the data).

Another example of the power of Pixon method reconstruction is presented in example 8. Here a NICMOS image of the "Pistol" star and its surrounding nebula taken with the Hubble Space Telescope has been processed by the Pixon method. As can be seen, the raw NICMOS data shows a dramatic diffraction pattern from the telescope. This pattern is so strong that it hides much of the emission from the much fainter nebula surrounding the Pistol star. In addition, the determination of the total flux of the surrounding stars is highly problematic due to confusion with neighboring emission. The Pixon method reconstructed image, on the other hand returns all of the emission to their points of origin. The Pistol star (and fainter stars) is thereby reduced to a single point. This allows the flux of all of the stars to be accurately measured and reveals the underlying nebular emission in unprecedented detail.

As a final example, we present in the image above, Keck K-band (2.3 micron) imaging of the IRAS source FSC 10214+4724. This object was once thought to be perhaps the most luminous object in the universe. However, it is now known to be a gravitationally lensed system with large magnification (approx. 100 times).

In the left-hand panel is presented the raw, flat-fielded Keck K-band image of Graham and Liu (1995, ApJ Letters, 449, L29). To the right is the Pixon method reconstruction. Unlike other reconstructions (e.g. Maximum Entropy--see the Graham and Liu paper) and imaging of FSC 10214+4724, the Pixon method reconstruction reveals the entire Einstein ring of the gravitational lens. This has strong implications for the size of the emitting region (at least twice as large as previously suspected) and clearly indicates (through image warping) that other objects in the field significantly contribute to the gravitational lensing potential.

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