On Curvature of Color Spaces and its Implications

In this paper we discuss the role of curvature in the context of color spaces. Curvature is a differential geometric property of color spaces that has attracted less attention than other properties like the metric or geodesics. In this paper we argue that the curvature of a color space is important since curvature properties are essential in the construction of color coordinate systems. Only color spaces with negative or zero curvature everywhere allow the construction of Munsell-like coordinates with geodesics, shortest paths between two colors, that never intersect. In differential geometry such coordinate systems are known as Riemann coordinates and they are generalizations of the well-known polar coordinates. We investigate the properties of two measurement sets of just-noticeable-difference (jnd) ellipses and color coordinate systems constructed from them. We illustrate the role of curvature by investigating Riemann normal coordinates in CIELUV and CIELAB spaces. An algorithsm is also shown to build multipatch Riemann coordinates for spaces with the positive curvature. Introduction The fact that a color space is a Riemann space, or a curved space, rather than a Euclidean space was first understood by Helmholtz, whose well known line element is the first effort in vision research to define a Riemann metric tensor for the color space characterizing human color vision. An historical account of related studies of color spaces and their foundations in Riemann geometry can be found in [14] and [13]. A major difference between a curved space and a Euclidean space is that cartesian coordinates are only meaningful in local neighborhoods of points. Therefore it is usually very hard to characterize quantitative properties and relationships among whole distributions of color stimuli. An example is the difficulty in investigating large color differences. Riemannian geometry provides powerful tools with which one can use to construct coordinate systems in a color space that are similar to a coordinate system in a Euclidean space. In particular, the color-difference between two color stimuli can be measured by the geodesic distance between them. As an analogy to the Munsell system, the surfaces of constant brightness correspond to the surfaces with constant geodesic distance from the origin, the lines of constant hues are geodesics starting from the achromatic origin on the constant brightness surface and the lines of constant chroma are the closed curves with constant geodesic distance from the origin on the constant brightness surface (for more information see [14]). This Munsell-like coordinate system in a color space is known as the Riemann normal coordinate system which is a generalization of the polar system in a Euclidean space. This coordinate system has many favorable properties and plays an important role in various applications. Examples are the construction of isometry or color-difference-preserving maps for uniform color spaces, color-weak correction and color reproduction (see [4], [5] and [10] for some examples). It would be of great advantage if one could construct such a Riemann normal coordinate system in all color spaces, but this is unfortunately not always possible. In fact, the existence of such a coordinate system depends on one of the most important properties of a color space as a Riemann manifold, the Riemann curvature. The Riemann curvature tensor describes the bending of the space, that is how much and in what way the Riemann space deviates from a flat space. It is known as an invariant of a Riemann space under isometries or distance preserving maps. Therefore the curvature of a color space is very important for both theoretical and practical reasons, especially when one transforms one color space into another. A simple first test to check color-difference preservation is to see if the curvature is preserved. e.g. since a uniform color space is isometrc to the Euclidean space, the curvature in a uniform color space should be zero everywhere. However, although the metric is well studied in colorimetry, the curvature issue seems to have been regarded as a purely theoretical subject and has not attracted sufficient attention until now. An interesting illustration of the importance of curvature is the conclusion that a chromaticity plane cannot be embedded into 2D Euclidean space because of its nonzero curvature (see [12] and [9]). Curvature is also used in [14] to show that the Stiles line element model is not compatible with MacAdam’s ellipses since they have different signs in the xy chromaticity diagram. In fact, the discrepancy between the Stiles line element and the MacAdam ellipses has much more serious implications than expected. In particular, the curvature plays such an important role in a color space that it is vital for the existence of Riemann normal coordinates. Radiating straight lines emerging from the origin in a Euclidean space will never intersect each other but in a general Riemann space geodesics may intersect each other. A well-known example is the sphere where the great-circles are the geodesics and infinitely many of them intersect at the two poles of the sphere. In that case we may not be able to obtain well defined coordinates and lines of constant hue or constant chroma from geodesics. Thus the possibility to draw geodesics through the whole chromaticity diagram means that the later can be covered by a single Riemann normal coordinate neighborhood, which requires that the whole diagram has negative or zero curvature. Therefore it is not always possible to have a global Munselllike or Riemann normal coordinate system for a color space. In CGIV 2010 Final Program and Proceedings 393 other words, usually such a Riemann normal coordinate system will only exists locally, i.e., at certain neighborhood of a point in the space. Consequently, one may be able to uniformize, or to build a color-difference-preserving map, around a neighborhood of every point in such a color space using the local Riemann normal coordinates, but usually this can not be extended to a global uniformization or color-difference preservation. In this paper we will show how curvature effects the properties of the color spaces. The curvatures of two sets of measurements of threshold ellipses will be calculated for the CIELUV and CIELAB spaces. We will present the intersection problem of geodesics and show that the problem originates in the positive curvature. In such a case, the Riemann normal coordinate system does not exists. We show that by applying smoothing one can reduce fluctuation of the curvature, but it also changes the shapes of jnd ellipses, and therefore we need a better interpolation strategy. Finally, we also show that in a case where we have a positive curvature that we can not circumvent, we will use the comparison theory of Riemann manifolds to find the injective radius or the minimal distance before the geodesics intersect. Using this information, one is able to build a multi-patch geodesics coordinate system for the whole color space. Moreover, a combination of smoothing with the multi-patch strategy is also discussed in order to reduce integral error and for fast implementation using parallel processing. Curvature of color spaces We know that the squared length of a vector in a Euclidean space is measured by the self-inner product of the vector. An n-D Riemann space is a space in which the distance can be only measured locally, or the length of a very small vector (dx1, ...,dxn)′ around a point x= (x1, ...,xn) can be measured using an extended inner product as follows or a quadratic form defined by an n×n matrix G(x) = (gi j(x)) called a metric tensor.[2]