Bayesian detection with amplitude, scale, orientation and position uncertainty

Abstract

The likelihood ratio is computed in the case where the background is known, the signal has a known shape, but the amplitude, scale, orientation and location of the signaI are unknown. The noise is assumed to be additive i.i.d. Caussian. The result is an expression involving the mutidimensional wavelet transform of the data with respect to the signal. The linear and quadratic approximations to the full likelihood statistic are examined in detail and are seen to involve operations commonly used in this context. Finally the full nonlinear ratio is formulated in germs of an amplification and modulation mechanism from the data to the prior probabilty distribution on the unknown parameters.