Degrees of freedom of polychromatic images

Di Francia showed that the number of degrees of freedom in two-dimensional (2-D) monochromatic or incoherent images detected by spatial-bandwidth-limited far-field receivers is approximately equal to the product of the spatial bandwidth of the imaging system and the spatial extent of the object.1 Several researchers have noted that this limit on the information content of imaging fields may be surpassed if time-domain or spectroscopic information is also analyzed.2,3 Sun and Leith described an experimental system in which one obtains improved resolution by carrying different spatial bandpass regions through a lens on different spectral components of the illuminating field.4 As noted in these papers, the number of degrees of freedom in the field scattered through a band-limited imaging system is approximately equal to the products of the space–bandwidth and time–bandwidth products of the field. This does not mean, however, that one can necessarily increase the resolution of an imaging system by increasing the temporal bandwidth of the probe field. The number of degrees of freedom independently encoded by scattering off the object and through the imaging system depends on the object. Although relationships between the spatial structure of the object and the field are well understood, less is known about the timedomain structure of fields scattered by static 2-D and onedimensional (1-D) objects. The time-domain field may be shaped by both material and form dispersion. This paper quantifies the effect of form dispersion on the field by calculating the information encoded in the space–time structure of the polychromatic fields scattered from static 1-D objects. An imaging system transforms objects and events in one region of space–time to a field in another region. By detecting the field, one hopes to reconstruct the objects or events. Reconstruction may result from direct analogy between the structure of the field and the structure of the object, as in photography, or from computational analysis of the fields, as in tomography. In either case, reconstruction is a mapping between the received field and the object. This mapping cannot be one to one because the number of demonstrably distinct fields that can be detected is finite, whereas the number of possible objects is unbounded. Reconstruction maps the detected field onto