We present two generalizations of the Williams–Clapper transform for converting between transmission and reflection densities of a color print. The first generalization allows for arbitrary incident angle, viewing angle, and index of refraction. The second generalization is to the geometry of an integrating sphere with the specular reflection either included or excluded. Our derivation also clarifies a potential source of confusion: Williams and Clapper had noted that, because of the cancellation of two factors, a reflection print with a perfectly transmitting gelatin layer (and a base of perfect reflectance) has the same brightness as it would have in the absence of the gelatin layer. We find that this cancellation is in fact only approximate. The approximate nature of the cancellation was not readily apparent because of the fortuitous closeness of the cancellation for the particular geometry and indices of refraction considered by Williams and Clapper. We also discuss efficient numerical implementation of the transform and outline an example of an application.
Joel D. Shore, John P. Spoonhower, "Reflection Density in Photographic Color Prints: Generalizations of the Williams–Clapper Transform" in Journal of Imaging Science and Technology, 2001, pp 484 - 488, https://doi.org/10.2352/J.ImagingSci.Technol.2001.45.5.art00010