Roxana Barbara Bujack

Although scientists agree that a perceptual color space is not Euclidean and color difference measures, such as CIELAB's \Delta E_{2000} ΔE2000 , model these aspects of color perception, colormaps are still mostly evaluated through piecewise linear interpolation in a Euclidean color space. In a non-Euclidean setting, the piecewise linear interpolation of a colormap through control points translates to finding shortest paths. Alternatively, a smooth interpolation can be generalized to finding the straightest path. Both approaches are difficult to solve and are compute intensive. We compare the 11 most promising optimization algorithms for the computation of a geodesic either as the shortest or as the straightest path to find the most efficient one to use for colormap interpolation in real-world applications. For two control points, the zero curvature algorithms excelled, especially the 2D relaxation method. For multiple control points, only the mimimal curvature algorithms can produce smooth curves, amongst which the 1D relaxation method performed best.

Significance For over 100 y, the scientific community has adhered to a paradigm, introduced by Riemann and furthered by Helmholtz and Schrodinger, where perceptual color space is a three-dimensional Riemannian space. This implies that the distance between two colors is the length of the shortest path that connects them. We show that a Riemannian metric overestimates the perception of large color differences because large color differences are perceived as less than the sum of small differences. This effect, called diminishing returns, cannot exist in a Riemannian geometry. Consequently, we need to adapt how we model color differences, as the current standard, Δ E , recognized by the International Commission for Weights and Measures, does not account for diminishing returns in color difference perception. The scientific community generally agrees on the theory, introduced by Riemann and furthered by Helmholtz and Schrödinger, that perceived color space is not Euclidean but rather, a three-dimensional Riemannian space. We show that the principle of diminishing returns applies to human color perception. This means that large color differences cannot be derived by adding a series of small steps, and therefore, perceptual color space cannot be described by a Riemannian geometry. This finding is inconsistent with the current approaches to modeling perceptual color space. Therefore, the assumed shape of color space requires a paradigm shift. Consequences of this apply to color metrics that are currently used in image and video processing, color mapping, and the paint and textile industries. These metrics are valid only for small differences. Rethinking them outside of a Riemannian setting could provide a path to extending them to large differences. This finding further hints at the existence of a second-order Weber–Fechner law describing perceived differences.

Colormapping is one of the simplest and most widely used data visualization methods within and outside the visualization community. Uniformity, order, discriminative power, and smoothness of continuous colormaps are the most important criteria for evaluating and potentially improving colormaps. We present a local and a global automatic optimization algorithm in Euclidean color spaces for each of these design rules in this work. As a foundation for our optimization algorithms, we used the CCC‐Tool colormap specification (CMS); each algorithm has been implemented in this tool. In addition to synthetic examples that demonstrate each method's effect, we show the outcome of some of the methods applied to a typhoon simulation.

Continuous colormaps are integral parts of many visualization techniques, such as heat-maps, surface plots, and flow visualization. Despite that the critiques of rainbow colormaps have been around and well-acknowledged for three decades, rainbow colormaps are still widely used today. One reason behind the resilience of rainbow colormaps is the lack of tools for users to create a continuous colormap that encodes semantics specific to the application concerned. In this paper, we present a web-based software system, CCC-Tool (short for Charting Continuous Colormaps ) under the URL https://ccctool.com , for creating, editing, and analyzing such application-specific colormaps. We introduce the notion of "colormap specification (CMS)" that maintains the essential semantics required for defining a color mapping scheme. We provide users with a set of advanced utilities for constructing CMS's with various levels of complexity, examining their quality attributes using different plots, and exporting them to external application software. We present two case studies, demonstrating that the CCC-Tool can help domain scientists as well as visualization experts in designing semantically-rich colormaps.

We present a state-of-the-art report on time-dependent flow topology. We survey representative papers in visualization and provide a taxonomy of existing approaches that generalize flow topology from time-independent to time-dependent settings. The approaches are classified based upon four categories: tracking of steady topology, reference frame adaption, pathline classification or clustering, and generalization of critical points. Our unique contributions include introducing a set of desirable mathematical properties to interpret physical meaningfulness for time-dependent flow visualization, inferring mathematical properties associated with selective research papers, and utilizing such properties for classification. The five most important properties identified in the existing literature include coincidence with the steady case, induction of a partition within the domain, Lagrangian invariance, objectivity, and Galilean invariance.