Observations of the mono-static (single frequency) field radiated by a source provide far too little data to uniquely identify the source. Indeed, there are many non-radiating sources, which do not radiate any field at one or more frequencies. Nevertheless, the principal investigator has developed tools to draw useful conclusions about the size and shape of the source or sources that radiate a particular field from just such limited and noisy data. In particular, the far field radiated by a source can be used to compute a convex set which must be contained within the convex hull of true source (which may be bigger but cannot be smaller). The PI continues to develop algorithms to compute these properties and develop other more detailed descriptions of sources or scatterers from data that is easily acquired. The PI's methods utilize the natural phenomenon of evanescence to estimate the size and location of an source.
Many areas of science and technology, especially those that take place in the laboratory or the medical center, are inundated with data. Remote sensing does not share that luxury. Data typically consists of measurements made by a single radar antenna attached to an airplane or a satellite, dragged behind a snowmobile, or supported between two skiers. The data contains valuable information for locating a land mine, predicting an avalanche, or inferring past variations in climate from the properties of snow and ice. However, in virtually all cases, the data is too limited to compute a complete description of the medium it is probing. The PI's program is to describe precisely what properties one can reliably infer, and to develop algorithms to compute them, even in the presence of substantial noise.
The project outcomes include the development and analysis of fundamental mathematical techniques for imaging and remote sensing, with special attention paid to incorporating physical and engineering insights, and formulating mathematical results in a physically meaningful way. Specific results make precise statements about the applicability of the Born, or linear scattering approximation, which widen our understanding of the circumstances where the complicated effects of multiple scattering can safely be ignored without causing artifacts or otherwise damaging the image. In particular, the methods here are consistent with the well-known phenomenon that the farther apart two objects are, the less the scattering of each interferes with that of the other. We have also developed quantitative statements relating the size of a source of radiation to the amount of energy that can be concentrated in a narrow beam. The qualitative version of this statement is a well known; our contribution was a to calculated the dependence on angular width, and the diameter of the source. We have shown for the first time with rigorous mathematics that the (convex) corner of an object will be visible at all frequencies and under all illumination patterns. While a spherical object can appear invisible to a spherical illumination pattern at the correct frequency, it is believed, and very likely, that this is only possible for the special geometry of a sphere. Work in this project exhibited for the first time, an object where this phenomenon cannot occur. All of these results are part of an ongoing attempt to produce mathematics that explains and quantifies physical expectations, and to find physical interpretations of all mathematical insights.
