A myriad of design rules for what constitutes a "good" colormap can be found in the literature. Some common rules include order, uniformity, and high discriminative power. However, the meaning of many of these terms is often ambiguous or open to interpretation. At times, different authors may use the same term to describe different concepts or the same rule is described by varying nomenclature. These ambiguities stand in the way of collaborative work, the design of experiments to assess the characteristics of colormaps, and automated colormap generation. In this paper, we review current and historical guidelines for colormap design. We propose a specified taxonomy and provide unambiguous mathematical definitions for the most common design rules.

The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features like linearity or a shift theorem. As a result, we provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context. Furthermore, the standard theorems do not need to be shown in a slightly different form every time a new geometric Fourier transform is developed since they are proved here once and for all.Mathematics Subject Classification (2010). Primary 99Z99; Secondary 00A00.

Pseudocoloring is one of the most common techniques used in scientific visualization. To apply pseudocoloring to a scalar field, the field value at each point is represented using one of a sequence of colors (called a colormap). One of the principles applied in generating colormaps is uniformity and previously the main method for determining uniformity has been the application of uniform color spaces. Here we present a new method for evaluating the feature discrimination threshold function across a colormap. The method is used in crowdsourced studies for the direct evaluation of nine colormaps for three feature sizes. The results are used to test the hypothesis that a uniform color space (CIELAB) gives too much weight to chromatic differences compared to luminance differences because of the way it was constructed. The hypothesis that feature discrimination can be predicted solely on the basis of luminance is also tested. The results reject both hypotheses and we demonstrate how reduced weights on the green-red and blue-yellow terms of the CIELAB color space creates a more accurate model when the task is the detection of smaller features in colormapped data. Both the method itself and modified CIELAB can be used in colormap design and evaluation.

Abstract. The large variety of Fourier transforms in geometric algebras inspired the straight forward definition of "A General Geometric Fourier Transform" in Bujack et al., Proc. of ICCA9, covering most versions in the literature. We showed which constraints are additionally necessary to obtain certain features like linearity, a scaling, or a shift theorem. In this paper we extend the former results by a convolution theorem.

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.

Hypercomplex Fourier transforms are increasingly used in signal processing for the analysis of higher-dimensional signals such as color images. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. The present paper develops and studies two conceptually new ways to define convolution products for such transforms. As a by-product, convolution theorems are obtained that will enable the development and fast implementation of new filters for quaternionic signals and systems, as well as for their higher dimensional counterparts. MSC 2010 : 30G35; 42B10; 44A35; 94A12

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