Maximum Entropy (ME) techniques have found widespread applicability in the reconstruction of incomplete or noisy data. These techniques have been applied in many areas of data analysis including imaging, spectroscopy, and scattering [Gull and Skilling, 1984]. The techniques have proven particularly useful in astronomy [Narayan and Nityanada, 1984]. In many of these applications the goal of the reconstruction is the detection of point objects against a noisy background. In this work we investigate the applicability of ME techniques to data sets which have strong components which are periodic in space or time. The specific interest in our laboratory is High Resolution Electron Micrographs of beam sensitive materials. However, ME techniques are of general interest for all types of data. These data mayor may not have a spatial or temporal character. Figure 1 shows an HREM image of the rigid-rod polymer poly(paraphenylene benzobisoxazole) (PBZO). The 0.55 nm spacings in the image correspond to the lateral close-packing between the extended polymer molecules. Near the center of this crystallite there is evidence for an edge dislocation. In HREM images both the frequency and position of the infonnation is important for a proper interpretation. Therefore, it is necessary to consider how image processing affects the fidelity of this information in both real and Fourier space.