Processing measurements of the directional content of Fourier spectra

Directional analysis and filtering make use of the most straightforward properties of optical Fourier transforms. A quantitative study of the detectability of angular maximums in Fourier spectra and its relationships with the shape and distribution of elements in the object is proposed. The problem is modeled by using a simple object which consists of the superimposition of two random distributions of rectangular grains, one being rotated with respect to the other. Computing the Fourier spectrum of this object allows the expression of the amount of light integrated through a wedge which scans the spectrum. An index of angular separability of the two distributions is made up from such measurements. It is shown to depend on the rotation angle as well as the grain shape. Using an additional spatial filter can improve this index. The influence of its radius is studied. Experimental results obtained either in real time with optical fibers or on photographic records with a microdensitometer are compared with the theoretical ones. They show the necessity of the implementation of optimal estimation schemes to reduce the influence of the noise. Two linear least squares filters are used: an angular Wiener filter and an autoregressive one. In the case of a high signal-to-noise ratio, the Wiener filter reduces to a Laplacian filter which does not depend on the shape of the grains.