Projective Holder-Minkowski Colors: A Generalized Set of Commutative & Associative Operations with Inverse Elements for Representing and Manipulating Colors

TL;DR

The paper tackles the lack of a unified algebraic framework for color operations in rendering by introducing Hölder-Minkowski Colors, a generalized Minkowski/Holder operation defined in an extended projective space that accommodates negative and complex values. By recasting colors as complex-time functions in projective space and defining an associative, commutative binary operation with inverses, the authors achieve a cohesive algebra for point and vector color elements, enabling stable composition, inverse problem solving, and wave-based color representations. They establish mathematical foundations (associativity, commutativity, scale-invariance, inverses) and present concrete illustrative models, extensions to complex-valued colors, and potential applications in rendering, compositing, and filtering. The framework promises flexible, invertible, and phase-aware color manipulation, with implications for HDR workflows, wave-based color physics, and future shader/format developments that support complex color values.

Abstract

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.

Figures (11)

• Figure 1: A sphere illuminated by four point-lights located in the vertices of a tetrahedron. The images show the impact of different parameters that are based on different weighted averages of the direct illumination term $max(\cos \theta,0)$, which is a discontinuous derivative. The case $p=1$ corresponds to the classical weighted average (or addition), visual artifacts caused by the derivative discontinuity of the function $max(\cos \theta,0)$. Parameter values greater than $p=1$ create smoother results by removing these visual artifacts caused by the discontinuity of the derivative at zero. Very high values $p$ approach the maximum operator and create another visual artifact. The $p$ values smaller than $1$ are not meaningful; therefore, we did not include them.
• Figure 2: A sphere illuminated by six point-lights located in the vertices of an octahedron. The images show the impact of different parameters that are based on different weighted averages of the direct illumination term $max(\cos \theta,0)$, which is a discontinuous derivative. The case $p=1$ corresponds to the classical weighted average (or addition), visual artifacts caused by the derivative discontinuity of the function $max(\cos \theta,0)$. Parameter values greater than $p=1$ create smoother results by removing these visual artifacts caused by the discontinuity of the derivative at zero. Very high values $p$ approach the maximum operator and create another visual artifact. The $p$ values smaller than $1$ are not meaningful; therefore, we did not include them.
• Figure 3: An example that demonstrates how to create complex representations using multiple images. This example also demonstrates the impact of the phase term. Note that each image consists of three channels, red, green, and blue. We assume that in phase images the color value $c$ corresponds to $2c \pi$. For example, the image of the white phase corresponds to $(2\pi, 2\pi, 2\pi )$.
• Figure 4: An example of the effect of the value of $p$ on the aliased boundaries. Bottom images are details. In this case, we use the $15\times15$ box filter by changing the values to see the impact of the $p$ value on aliasing.
• Figure 5: Following up the Figure \ref{['fig_filterscircle']} demonstrating more examples on the effect of the value of $p$ on the aliased boundaries. We observe that $p=1$ followed by $p-1$ gives the best results, which is similar to erosion followed by dilation.

Theorems & Definitions (11)

• Theorem 9.1
• Theorem 9.2
• Theorem 9.3
• Theorem 9.4
• Lemma 9.5
• Theorem 9.6
• Theorem 10.1
• Theorem 10.2
• Theorem A.1
• Theorem A.2