Abstract
In this paper, as data, ellipsoids in a color coordinate called the Commission Internationale de l’Eclairage (CIE)-Lab system are given as data for 19 colors. Each ellipsoid is a region where all points are visually recognized as the same color at the center of the coordinate system. Our aim here is to predict the shape of an ellipsoid whose center is given by a new color. We proposed two prediction methods of positive definite matrices determining ellipsoids. The first one is a nonparametric method with Gaussian kernel. The prediction is provided as a weighted sum of positive definite matrices corresponding to 19 ellipsoids in the training data. The second one is to use a matrix-valued regression model applied to a logarithm of positive definite matrices where explanatory variables are three elements of color centers. The best result was obtained by the nonparametric methods with three bandwidth parameters. The log normal regression had a weaker performance, but even so the model estimation was easily carried out.
Highlights
When car motor companies develop a new color or texture, it is very important to ensure color matching between the various products or parts in order to provide a sophisticated appearance and design
We developed a nonparametric method of Nadaraya– Watson type estimation based on Gaussian kernel with several bandwidth parameters
We propose a matrix-valued regression to logarithm of positive denite matrices as response variables and three elements of color centers as explanatory variables
Summary
When car motor companies develop a new color or texture, it is very important to ensure color matching between the various products or parts in order to provide a sophisticated appearance and design. One example of a color coordinate system is the Commission Internationale de l'Eclairage (CIE) This is an Open Access article published by World Scientic Publishing Company. Predicting a dead zone ellipsoid corresponding to a new color is required in the Lab space in order to know the acceptable zone. This is known as the color matching problem. The problem considered here is how to predict an ellipsoid or a positive denite matrix for a new color center given as u0 1⁄4 ða0; b0; L0ÞT 2 1⁄2À90; 902Â 1⁄20; 100. G11ðu0Þ g12ðu0Þ g13ðu0Þ Gðu0Þ 1⁄4 @ g12ðu0Þ g22ðu0Þ g23ðu0Þ A; ð2Þ g13ðu0Þ g23ðu0Þ g33ðu0Þ is a positive denite matrix corresponding to the color with center u0. Both of the proposed methods can greatly reduce the cost of color matching compared with the conventional cut-andtry method, which would make the development of new colors more e±cient
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