Color Correction by Considering the Distribution of Metamers within the Mismatch Gamut

Color correction describes the transformation process between device RGB values and CIEXYZ or CIELAB values. This mapping is in general not unique, because the spectral sensitivities of most of the devices do not satisfy the Luther condition and the acquisition and viewing light sources have a different power spectrum. Therefore, there exists a set of colors with different reflectance spectra which result in the same device RGB response (device metamerism), but leads to different tristimuli for an observer under the viewing light source. To determine an optimal mapping between a given device RGB and a CIELAB color, the distribution of metamers in a metamer mismatch gamut has to be characterized in the viewing CIELAB space. We present a novel method by estimating the distribution of metamers within the mismatch gamut using a Monte Carlo method. The main idea is the construction of a basic collection of metameric blacks (used by the Monte Carlo method) that is calculated by using a representative set of reflectance spectra and performing principle component analysis (PCA) within the black space of the device. The transformation of the sum of a fundamental metamer for the sensor response and the basic collection in the CIELAB color space leads to a point cloud with a centroid approximating the center of gravity of the mismatch gamut. This point is the optimal color correction in the sense of the smallest mean error. Introduction Color correction describes the transformation process between device RGB values and an device independent color space of an observer (e.g. CIEXYZ, CIELAB). In each metamer color reproduction system this is the first color transformation after image acquisition. The mapping is in general not unique, because the spectral sensitivities of most of the devices do not satisfy the Luther-Ives condition [1][2] and the acquisition and viewing light sources have a different power spectrum. Therefore, there exists a set of colors with different reflectance spectra which result in the same device RGB response (device metamerism), but have different color appearances for an observer under the viewing light source (metamer mismatch gamut). For this reason color correction is a classical one-to-many transformation problem. In recent years various color correction methods have been proposed. Besides common target based methods like linear transformation into the CIEXYZ color space using a simple 3x3 matrix or multi-order polynomial regression into the CIEXYZ color space, Hardeberg [3] and Koenig [4] proposed multi-order polynomial regression into the CIELAB color space after initially performing different transformation steps to consider the nonlinear relationship of human color vision to the intensity of the RGB values. They achieved a distinct improvement of error rates. Vrhel et al. [5] used neural networks, Koenig [4] proposed a matrix method that was robust to noise; Hung [6] used LUT-interpolation and extrapolation; and Finlayson [7] proposed a constrained least square-regression to preserve the white point. This list of target based methods is not exhaustive. Other methods use the model of a linear image acquisition system for color correction. Finlayson, et al. [8] characterized the metamer mismatch gamut by an enclosing cube in the CIEXYZ color space using linear programming and chose the center of gravity of this cube for color correction. The results outperform the standard target based methods. Urban, et al. [9] used also a linear programming technique to calculate a metamer boundary descriptor matrix that characterizes the metamer mismatch gamut within the CIELAB color space. The center of gravity of the matrix entries had been chosen for color correction. Other approaches used the linear image acquisition model to calculate a matrix based transformation function [10][11][12]. The aim of this paper is to determine an optimal mapping between a given device RGB value and a device independent color in the sense of the minimal mean color difference. Our approach is to estimate the density distribution of metamers inside the metamer mismatch gamut (see [13][14][15]) within an observer’s perceptual color space (i.e. the device independent color space) using a Monte Carlo method and choose the center of gravity of the metamers considering this density distribution for color correction. The main idea is the construction of a basic collection of device metameric blacks which is calculated by using a representative set of reflectance spectra and a PCA within the black space of the device. For a RGB sensor response we calculate a fundamental metameric spectrum and add the spectra of the basic collection according to the well-known metameric black method. Each of the resulting spectra leads by construction to the given RGB value. Transforming the whole set into the perceptual color space of the observer yields to a point cloud which density distribution is assumed as a good approximation of the density distribution of metamers within the metamer mismatch gamut. This method improves the performance of target and regression based methods especially in the area of saturated colors. 222 Copyright 2007 Society for Imaging Science and Technology The Metamer Mismatch Gamut The following text is a short introduction to metamer mismatch gamuts. A detailed description can be found in [10][11][12]. If all spectra are sampled at N equi-spaced wavelength a linear acquisition system can be described algebraically as c = SLar+ = Ωar + ε (1) where c = (R,G,B) is the sensor response, S is a 3×N matrix which contains the channel sensitivities as row vectors, La is a N × N diagonal matrix with the radiant spectrum of the acquisition illuminant along the diagonal, r is a N × 1 vector of the reflectance sample and is additive noise. The matrix product of the sensor response matrix S and the illuminant matrix La is the acquisition lighting matrix Ωa. By means of the lightning matrix Ωa the device black space is defined as follows Kernel(Ωa) := {r | Ωar = 0} (2) The device black space contains all spectra which sensor responses are zero, i.e. black. The set of all device metameric spectra which lead to the sensor response c can be derived as follows Rc = {r | r = fc +Kernel(Ωa)} ∩ Rall (3) where fc is the so called fundamental metameric spectrum of the sensor response c and Rall is the space of all natural reflectance spectra. Each reflectance spectrum with sensor response c can be used for fc. To calculate such a spectrum from the sensor response c, various methods can be used, e.g. pseudoinverse, Wienerinverse or principle eigenvector method. The intersection with Rall is to ensure physically useful spectra which are positive, bounded and smooth. In an analogous manner to the sensor response the observer’s tristimulus value o ∈ CIEXYZ can be described