Generalized NEQ: Fourier analysis where you would least expect to find it

The simplest task for evaluation of image quality is detection of a known signal on a known background. For linear, shift-invariant imaging systems with stationary, Gaussian noise, performance of the ideal observer on this task is determined by the frequency-dependent noise- equivalent quanta (NEQ), defined as the ratio of the square of the system modulation transfer function (MTF) to the noise power spectrum (NPS). It is the purpose of this paper to show that a closely analogous expression applies without the assumption of shift-invariance or noise stationarity. To get this expression, we describe an object of finite support exactly by a Fourier series. The corresponding system description is the Fourier crosstalk matrix, the diagonal elements of which constitute a generalized MTF. Since this matrix is not diagonal, calculation of the ideal-observer performance requires a double integral over the frequency domain, but if we average the task performance over all possible locations of the signal, the off-diagonal elements average to zero and a single sum results. With one approximation, this expression takes the same form as in the case of shift-invariant imaging and stationary noise.