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.

Tensor fields (TF) are a special kind of multidimensional data, in which a tensor is given for each point in space. Often, it is a 3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\times 3$$\end{document} array in each voxel. To detect the patterns of interest in the field, special matching methods must be developed. We propose a method for the description and matching of TF patterns under an unknown affine transformation of the field. Transformations of TFs act not only in the spatial coordinates but also on the field values, which makes the detection more challenging. To measure the similarity between the template and the field patch, we propose original invariants with respect to affine transformations designed from moments. Their performance is demonstrated by experiments on real data from diffusion tensor imaging.

Classical vector field topology has proven to be a useful visualization technique for steady flow, but its straightforward application to time-dependent flows lacks physical meaning. Necessary requirements for physical meaningfulness include the results to be objective, i.e., independent of the frame of reference of the observer, and Lagrangian, i.e., that the generalized critical points are trajectories. We analyze whether the theoretical concept of distinguished hyperbolic trajectories provides a physically meaningful generalization to classical critical points and if the existing extraction algorithms correctly compute what has been defined mathematically. We show that both theory and algorithms constitute a significant improvement over previous methods.We further present a method to visualize a time-dependent flow field in the reference frames of distinguished trajectories. The result is easy to interpret because it makes these trajectories look like classical critical points for each instance in time, but it is meaningful because it is Lagrangian and objective.

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.