Reflection Density in Photographic Color Prints : Generalizations of the Williams – Clapper Transform

above-mentioned approximate expressions with certain values in the expression taken as free parameters. 4 A third line of research has been the modification of the Williams–Clapper transformation by adding an additional empirical term to the reflectance that controls the maximum reflection density. 3–8 This term is usually attributed to light reflected off the first-surface (gelatin– air interface) back to the viewer, although it, can also be due in part to light scattered back to the viewer from within the gelatin layer (e.g., off of the dye droplets). 9 Because the addition of such a term is straightforward and is dealt with in so many other presentations, we will not explicitly include it in the discussions below except for the case of an integrating sphere where its magnitude can be calculated theoretically. A fourth line has been the generalization of Williams– Clapper to different geometries. The two generalizations of which we are aware are the unpublished work of Watt, 7 in which the ratio of the reflectance in an integrating sphere geometry (with an input angle of 8° to the normal and specular reflection excluded) to B/B' in the 45°/0° geometry was calculated, and the work of Ohta, 8 in which the reflection density was calculated for the case of diffuse (Lambertian) illumination with an arbitrary viewing angle. Finally, we note that a formal mathematical approach to the problem of light absorption and scattering in color prints based on the Kubelka–Munk model 10 has very recently been presented by Emmel and Hersch. 11 It is noted there that an extension of the approach to the situation , where the light is no longer necessarily normal to the print, 12 yields the Williams–Clapper result as a special case. In this study, we consider the transformation between transmission and reflection density for two qualitatively different geometries. One is for arbitrary incident and viewing angles; the other is for an arbitrary incident angle and an integrating sphere viewing geometry with Introduction In 1953, Williams and Clapper published a seminal paper in which they derived the transformation necessary for converting between the transmission density of the gelatin layer and the resulting reflection density for a color print. 1,2 Their calculation takes into account the multiple internal reflections that can occur in the gela-tin layer and are very important at low densities. They only presented equations for determining the reflection density for the specific case of …