In conclusion, we have introduced a new concept, the pixon, the use of which provides large improvements in the ability to reconstruct images from blurred, noisy data. The pixon is the fundamental and indivisible unit of information required to describe the underlying signal within the accuracy allowed by the data. In this regard it is an idealized concept. However, this paper also presents a practical and capable approximation to this ideal for a broad range of problems, i.e., the fuzzy, Fractal-Pixon Basis (FPB). This basis uses the local spatial scale relevant to the underlying signal to constrain the inversion of the equations governing the measurement process. In doing so, this method provides performance superior to pure GOF (Maximum Likelihood) and ME methods. Some of the advantages of pixon-based methods are the elimination of signal-correlated residuals and the production of spurious sources typical of other methods. Practical examples from the realm of astronomical image reconstruction show that pixon-based methods can offer large improvements in resolution as well as the detection of extremely weak features in the data.
The authors would like to thank a number of people for their valuable contributions to this work. We would especially like to thank Nick Weir of Caltech for numerous fruitful discussions regarding image processing and for graciously performing the multi-channel MEMSYS 5 reconstructions presented in this paper. We would also like to thank Romke Bontekoe and Do Kester for providing the M51 IRAS test data set as well as for many helpful conversations. Finally, the authors would like to thank Steven Gull and John Skilling for a number of conversations that greatly expanded our understanding of Bayesian methods in general, and how the pixon fits into Bayesian theory in particular. This work was supported by NASA and the National Science Foundation.