Riemann Geometry for Color Characterization and Mapping

In this paper, we use Riemann geometry to develop a general framework for the characterization of and mapping between color spaces. Within this framework we show how to construct maps, so-called isometries, between two color spaces that preserve color differences. We illustrate applications of this framework by constructing a uniform color space and developing algorithms for color reproduction on different printers and correction of color-vision for color-weak observers. Introduction Many traditional color science methods are based on statistical averages. The most well-known is the standard observer describing the basic color vision properties of an observer with average color vision. This is useful in applications like the evaluation of color reproduction properties in the printing industry where comparison of methods or products on average are required. It is less useful where applications require more exact descriptions of individual properties or where the target is another device like a printer. Obviously methods using the standard observer fail completely for problems involving color blindness. For these applications more precise, tailored procedures are necessary. Similar problems are handled in the framework of color management systems. But methods used there are of limited use when human observers are the ”devices” to be handled since it is impossible to ”calibrate” a human observer by measuring input/output value pairs. Thus, in the widest sense of color reproduction or color management, one has to consider mappings between various devices and individual human observers. In particular, one of the most natural, and therefore useful maps, is a map which preserves subjective color-differences. In the language of Riemann geometry, such a map is called an isometry. In general it is non-trivial to construct an isometry. In fact, a map which preserves small color-differences around one point can be easily obtained by linear algebra, since locally the map can be represented by its piece-wise linear approximation at the point, which can be found using the local isometry condition, or simply matching the threshold ellipses/ellipsoides at the preimage point and the image point. However, a map preserving color-differences at every point (an isometry) between two spaces and a global isometry or a map which preserves large color-differences are usually more difficult to construct since the above piece-wise linear approximation should take a form of an affine transformation in global coordinates. The affine or parallel shift information after the mapping is not easy to come by. Another, and more serious problem, is that the map between color spaces are usually assumed to be monotonic for simplicity such as in the ICC profiles. When the map is not monotonic the inverse map is not injective or one-to-many and thus hard to find. In this paper, we introduce Riemann geometry as a tool to compute geometrical descriptions of color spaces from measurement data and mapping between color spaces. The input data are descriptors of the local properties of the color space under investigation. The best known examples are the MacAdam ellipses. (see [1], [2], [3] and [4]). Other examples are measurements of the color coordinates of a printed test chart or descriptors computed from other color systems. We first show how to compute an isometry between a color space and the Euclidean space. From a geometrical point of view, a coordinate system is the most straightforward characteristic of a color space. Another characteristic object of a geometrical space are its straight lines, or geodesics in Riemann spaces. From local threshold measurements the geodesics can be calculated by solving differential equations known as the geodesic equations. We will show how to obtain a ”polar coordinate system” in a color space using the geodesics. The correspondence of this system with the polar coordinate system in the Euclidean space then provides us an isometry between the two spaces. As an example of this method we show the construction of uniform space for CIELUV. Then we show how to build an isometry between two color spaces based on the above method. In particular, once the geodesic polar coordinate systems are obtained for these two color spaces, one can easily read out the isometry between these two color spaces. As the first application of the framework, we show a color reproduction on two printers, MP790 by Epson and PM970C by Canon. Here the isometries between the working color spaces of different devices provide us a way to handle device -independent color reproduction. Apart from investigations involving color blindness it is often ignored that there is also a variation in the color vision properties between observers with full color vision. Problems involving color weakness have received little attention in color technology. As the second applications, we will therefore illustrate how to characterize the color spaces of color normal observers and a color weak observer. An exact correction method is obtained using an isometry between this two color spaces called a color-weak map. The framework described here has also applications in other fields. In [5], for example, the related model of a fibre bundle of the color spaces was used in image processing. Riemann Geometry for Color Spaces Intuitively a Riemann space is a generalization of Euclidean flat space where the distance between two points is measured in the same way everywhere. An intuitive way to understand how this can be generalized to a Riemann space is to consider a surface S like the unit sphere. Locally a neighborhood of a point x on S looks like a plane and can be approximated by the tangent plane TxS. On this tangent plane we have a positive definite matrix G(x) defining the scalar product between tangent vectors u,v ∈ TxS as uT G(x)v. This scalar product defines an arcCGIV 2008 and MCS’08 Final Program and Proceedings 277 length on the Riemann space in the same way as in Euclidean geometry. A formal definition of a Riemann space is a space S with a positive definite matrix G(x) smoothly defined on every point x such that the infinitesimal distance near x is measured by (dx,dx)G = dxT G(x)dx. Let x = (x1, ...,xn)T ,G(x) = [gi j], then (dx,dx)G = gi jdxdx j (where the Einstein convention is used to sum over indices that appear as both subscripts and superscripts as abi = ∑i abi). Such a Riemann space is usually denoted as S = (Rn,G). The global distance of any two vectors x,x′ in S is measured as follows. If a spatial curve x(t) = (x1, · · · ,xn)T in a Riemann space (Rn,G) is smooth for a ≤ t ≤ b,x(a) = x,x(b) = x′, i.e., if ẋi = dx j dt exist and are continuous, then the integral