For multiprimary displays that have four or more primaries, a color may be reproduced using multiple alternative control vectors. We provide a complete characterization of the Metameric Control Set (MCS), i.e., the set of control vectors that reproduce a given color on the display. Specifically, we show that MCS is a convex polytope whose vertices are control vectors obtained from (parallelepiped) tilings of the gamut, i.e., the range of colors that the display can produce. The mathematical framework that we develop: (a) characterizes gamut tilings in terms of fundamental building blocks called facet spans, (b) establishes that the vertices of the MCS are fully characterized by the tilings of the gamut, and (c) introduces a methodology for the efficient enumeration of gamut tilings. The framework reveals the fundamental inter-relations between the geometry of the MCS and the geometry of the gamut developed in a companion Part I paper, and provides insight into alternative strategies for color control. Our characterization of tilings and the strategy for their enumeration also advance knowledge in geometry, providing new approaches and computational results for the enumeration of tilings for a broad class of zonotopes in R3.

Displays that render colors using combinations of more than three lights are referred to as multiprimary displays. For multiprimary displays, the gamut, i.e., the range of colors that can be rendered using additive combinations of an arbitrary number of light sources (primaries) with modulated intensities, is known to be a zonotope, which is a specific type of convex polytope. Under the specific three-dimensional setting relevant for color representation and the constraint of physically meaningful nonnegative primaries, we develop a complete, cohesive, and directly usable mathematical characterization of the geometry of the multiprimary gamut zonotope that immediately identifies the surface facets, edges, and vertices and provides a parallelepiped tiling of the gamut. We relate the parallelepiped tilings of the gamut, that arise naturally in our characterization, to the flexibility in color control afforded by displays with more than four primaries, a relation that is further analyzed and completed in a Part II companion paper. We demonstrate several applications of the geometric representations we develop and highlight how the paper advances theory required for multiprimary display modeling, design, and color management and provides an integrated view of past work on on these topics. Additionally, we highlight how our work on gamut representations connects with and furthers the study of three-dimensional zonotopes in geometry.