Symmetrized regression for hyperspectral background estimation

We can improve the detection of targets and anomalies in a cluttered background by more effectively estimating that background. With a good estimate of what the target-free radiance or reflectance ought to be at a pixel, we have a point of comparison with what the measured value of that pixel actually happens to be. It is common to make this estimate using the mean of pixels in an annulus around the pixel of interest. But there is more information in the annulus than this mean value, and one can derive more general estimators than just the mean. The derivation pursued here is based on multivariate regression of the central pixel against the pixels in the surrounding annulus. This can be done on a band-by-band basis, or with multiple bands simultaneously. For overhead remote sensing imagery with square pixels, there is a natural eight-fold symmetry in the surrounding annulus, corresponding to reflection and right angle rotation. We can use this symmetry to impose constraints on the estimator function, and we can use these constraints to reduce the number or regressor variables in the problem. This paper investigates the utility of regression generally -- and a variety of different symmetric regression schemes particularly -- for hyperspectral background estimation in the context of generic target detection.