The relationship between second order stochastic processes and coherent power spectra are explored for developed grain size frequency distributions of silver halide emulsion/developer combinations. Although Wiener spectra represent incoherent power spectra, the coherent case has been unresolved due to the difficulty of filtering the optical system effect. The emulsion plates were exposed with a uniform irradiance distribution from an incoherent tungsten source to create a series of uniform density samples. Conventional processing chemistry is utilized with substitution of different developer types, from predominantly physical development to direct chemical development. The exposed and processed density samples were liquid gated with refractive index matching fluid and inserted at the front focal plane of a conventional on-axis Fourier transform system. The power spectra are ensemble and frequency domain averaged at the Fourier transform plane with a special annular ring detector. The coherent nature of the optical analyzer necessitates consideration of the interaction or interference of the power spectra for the optical system, and the spectra of the input transmittance sample. An algorithm was derived to estimate the power spectra of just the input grain distributions, which was applied to all of the measured spectra of the emulsion/developer combinations. The calculated spectra demonstrate significant power shifts from the low spatial frequency region with increasing power levels as a function of increasing developed grain size. The bandwidths of the power distributions are inversely proportional to developed grain size with the larger grains the predominant factor. The midpoints of the power spectra also shifted toward increasing spatial frequencies for decreasing values of developed grain size. The peak power levels for each emulsion/developer family were proportional to those predicted by the well known random checkerboard granularity model. Theoretical power spectra models are demonstrated for both second order and first order auto-regressive stochastic models. The second order process reveals a significant shift in the power distribution away from the low frequency region analogous to the calculated coherent power spectra. The power functions for the first order process are centered at zero spatial frequency, similar to conventional Wiener spectra. It is shown that a first order complex transmittance model transforms to the power spectra proportional to second-order stochastic processes when the real and imaginary parts are correlated. Although the filtered and measured coherent spectra are strongly dependent upon the developed grain size frequency distributions, the significant departure in shape and location of the spectra is related to the functional change from first order to second-order stochastic process characteristics
