TRANSFORM NOISE STATISTICS AND FOURIER COMPONENT ESTIMATION

As a first step towards more absolute quantitative procedures for evaluating images, new and better ways are presented of estimating the spacings and complex amplitudes present in the image of a crystalline specimen by examining the peaks in its calculated transform. So that the expected performance of different estimators can be compared, the statistical properties of the noise in calculated Fourier transforms are established in some detail, and related to those of the noise in the image itself: the variance is found at each pixel, the covariance between pixels, and the actual distribution of the transform noise. The role of image windowing in minimising systematic errors due to interference between different Fourier components is made clear, and the properties of three different windows evaluated; the half-cosine window is recommended as a useful compromise between the (trivial) unit window and the von Hann window recommended previously. An alternative approach involving image resampling is shown to have excellent properties for low frequency components, and the degradation of high frequencies arising on re-interpolation is characterised quantitatively. Key Words: Transform noise, transform statistics, windowing, spectral estimation, lattice spacin