  One of the key problems in dealing with color in rendering, shading,
compositing, or image manipulation is that we do not have algebraic structures
that support operations over colors. In this paper, we present an
all-encompassing framework that can support a set of algebraic structures with
associativity, commutativity, and inverse properties. To provide these three
properties, we build our algebraic structures on an extension of projective
space by allowing for negative and complex numbers. These properties are
important for (1) manipulating colors as periodic functions, (2) solving
inverse problems dealing with colors, and (3) being consistent with the wave
representation of the color. Allowance of negative and complex numbers is not a
problem for practical applications, since we can always convert the results
into desired range for display purposes as we do in High Dynamic Range imaging.
This set of algebraic structures can be considered as a generalization of the
Minkowski norm Lp in projective space. These structures also provide a new
version of the generalized Holder average with associativity property. Our
structures provide inverses of any operation by allowing for negative and
complex numbers. These structures provide all properties of the generalized
Holder average by providing a continuous bridge between the classical weighted
average, harmonic mean, maximum, and minimum operations using a single
parameter p.
