Richard Puetter (Center for Astrophysics and Space Sciences, UCSD)
The problem of image restoration is to find the best unblurred, underlying model of the image which is consistent with the blurred, noisy data. The more general problem of image reconstruction has the same goals when the measured data is in a more complex form, e.g., an interferogram. Standard techniques normally model an image on a rectangular grid of pixels selected to provide the resolution obtainable from the data. In the case in which the data is also pixelated, the image grid is usually taken to be the same as the data grid. The problem with this approach is that often in astronomical imaging situations, the data is of insufficient quality to constrain the full set of image pixel values. This leads to the classical problems associated with image reconstruction, i.e., over fitting of the noise, signal correlated residuals, and the generation of spurious sources.
There have been many attempts to address these problems. The maximum entropy method (MEM), for example, imposes an image smoothness penalty on the image. This penalty makes it harder to over fit the data. However since it is a global penalty function, it over-smooths in some regions and under-smooths in others. The Pixon method offers a more general approach. It asks: ``What is the simplest image model that can be constructed to fit the data?'' In our most successful incarnation of the Pixon method, we use a multiresolution language to build an image model. Specifically, we ask how much can the image be smoothed locally and yet still fit the data. This changes the global smoothness idea of MEM into a local condition in which local maximum smoothness of the image is imposed. From an information science point of view, one is selecting a model with the minimum Algorithmic Information Content from the family of models constructible within the multiresolution language and which statistically fit the data. Such a model has maximum a priori probability within this set. Furthermore, because this image model has the smallest number of parameters possible while still fitting the data a number of important benefits arise. These include: (1) Since the model has minimum complexity, spurious sources cannot arise. (2) Each parameter is determined using a larger fraction of the data. Hence it is determined more accurately. This results in superior photometric and positional accuracy. (3) Since the minimum number of parameters are used, the data cannot be over fitted. (4) Better fitting of the data means that signal correlated residuals are eliminated. Practical tests by scientists at a variety of institutions have shown that relative to competing techniques, the Pixon method typically provides factor of a few improvements in linear spatial resolution and order-of-magnitude (or more) improvement in sensitivity to faint sources.