Introduction

The act of measurement of physical quantities inevitably introduces artifacts. These artifacts can be associated with the statistical limits of the measurement, e.g., counting statistics, as well as characteristics of the measurement device, e.g., finite resolution. Modern approaches for deducing the underlying, uncorrupted physical quantities from the recorded data often turn to Bayesian estimation in which the measurement process is statistically modeled. For astronomical imaging, the relationship between the data, , the underlying image, , and the noise, , is given by

where is the Point Spread Function (PSF) and the integration is over the volume in -space.

The most successful methods for inverting Eq. (1) are non-linear in their approach. The simplest define a figure-of-merit, or Goodness-of-Fit (GOF) criterion, for the image and then use multi-dimensional optimization methods to maximize this function. Such methods include the familiar Least-Squares method as well as the Richardson-Lucy method (e.g., Lucy 1974). While these pure GOF techniques typically give superior inversions to Eq. (1) than do Fourier methods, they still have many undesirable properties. A common problem is over-resolution in which the algorithm attempts to fit the noise and introduces features which are unnecessary to fit the data. Lucy-Richardson reconstructions, for example, are typically stopped after an arbitrary number of iterations in an attempt to overcome this difficulty. This leaves the unpleasant (and difficult) task of determining which features are ``real'' and which are not. More sophisticated approaches such as Maximum Entropy (e.g., Skilling 1989) place additional constraints upon the solution based on prior expectations. These additional constraints greatly improve the quality of the solution by regularizing the problem and controlling over-resolution.

Recently, we have introduced a new image reconstruction method based on the pixon (Piña and Puetter 1993, Puetter and Piña 1993a). This method greatly expands upon ME methods by introducing a prior with a variable local scale. In fact, our Uniform Pixon Basis (UPB) method (Piña and Puetter 1993a) results in a ``Super-Maximum Entropy'' reconstruction in which entropy is maximized exactly. In our use of variable correlation length scales, pixon-based methods are similar in some respects to the multi-channel methods of Weir (1991, 1993a) or the Pyramidal Maximum Entropy techniques of Bontekoe, Koper, and Kester (1994). Unlike these techniques, however, pixon-based methods explicitly determine the appropriate local scale based on various criteria. Our most recent Fractal Pixon Basis (FPB) method (Puetter and Piña 1993a, 1994) selects the local correlation length based on the local structural scale of the image and represents the highest performance image reconstruction method we are aware of to date.