A decision surface-based taxonomy of detection statistics

摘要

Current and past literature on the topic of detection statistics - in particular those used in hyperspectral target detection - can be intimidating for newcomers, especially given the huge number of detection tests described in the literature. Detection tests for hyperspectral measurements, such as those generated by dispersive or Fourier-transform spectrometers used in remote sensing of atmospheric contaminants, are of paramount importance if any level of analysis automation is to be achieved. The detection statistics used in hyperspectral target detection are generally borrowed and adapted from other fields such as radar signal processing or acoustics. Consequently, although remarkable efforts have been made to clarify and categorize the vast number of available detection tests, understanding their differences, similarities, limits and other intricacies is still an exacting journey. Reasons for this state of affairs include heterogeneous nomenclature and mathematical notation, probably due to the multiple origins of hyperspectral target detection formalisms. Attempts at sorting out detection statistics using ambiguously defined properties may also cause more harm than good. Ultimately, a detection statistic is entirely characterized by its decision boundary. Thus, we propose to catalogue detection statistics according to the shape of their decision surfaces, which greatly simplifies this taxonomy exercise. We make a distinction between the topology resulting from the mathematical formulation of the statistic and mere parameters that adjust the boundary's precise shape, position and orientation. Using this simple approach, similarities between various common detection statistics are found, limit cases are reduced to simpler statistics, and a general understanding of the available detection tests and their properties becomes much easier to achieve.