Mathematical Relations between Grain, Background Noise and Characteristic Curve of Sound-Film Emulsions

Abstract

Compulations and measurements show that the background noise of film can be interpreted as the superposition of two types of noise: surface noise, and grain noise. The surface noise power decreases with the square of specular transmission; the grain noise power reaches a maximum at 50 per dent transmission. Accordingly, it is found that under conditions of variable-width recording surface noise is predominant; for variable-density recording, grain noise is the main factor. The average area of the grains or grain clusters can be calculated from the signal-to-noise ratio; their average volume from the total weight of silver per square centimeter at a given density; their average thickness from the quotient of volume and area. For equal grain sizes, surface exposure such as obtained by ultraviolet illumination is definitely noisier than penetrating exposure. — Upon the basis of random three-dimensional distribution of sensitized grains and of the quantum theoretical findings of previous investigators, the shapes of H&D curves were calculated. The assumption that a halide grain is sensitized by a single photon leads to a toe shape that is more rounded than is found in practice. The actual shape of the characteristic from toe to shoulder is accounted for by the assumption that it takes two photons to sensitize a silver halide grain. It is expressed by the equation: — <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$$D={D_{\infty}\over {\rm ln}_{\varphi}}\left[\epsilon^{-e\tau}-\epsilon^{-e}-\int^{e}_{e\tau}{1- \epsilon^{-x}\over x}dx \right]$$</tex> — in which τ represents the translucence of the unexposed emulsion to the aclinic light. — The experimental fact that the straight portions of H&D curves obtained from the same emulsion at various gammas originate from a single point which is depressed by bromide content is explainable by taking into account the fact that the emulsion contains silver halide grains of more than one size and speed.