Coding efficiency of CIE color spaces

Estimates were made of the efficiency with which color spaces code color information from images of natural scenes. Six spaces were tested, namely, CIE XYZ tristimulus space, and the spaces CIELUV, CIELAB, CIELAB and S-CIELAB after chromatic adaptation with CMCCAT2000, and the space CIECAM02. For each space, the information available and the information retrieved in color matching were calculated for images of 50 natural scenes under different daylight illuminants. The information available was decomposed into components associated with the individual variables of the space and the interactions between them, including redundancy and illuminant-dependence. It was found that the information retrieved was much less than the information available, and that its decomposition depended on the space. The differing efficiencies of the spaces were interpreted in relation to the effectiveness of opponent-color and chromaticadaptation transformations, and the statistics of images of natural scenes. Introduction Color spaces are used routinely for specifying or describing the color of individual samples. But they may also be incorporated within more general image descriptions, for example, of natural scenes, which often contain a complex combination of spatial and chromatic detail. Each point in the image might have its color coded by its tristimulus values (X ,Y,Z), or perhaps by the coordinates (L∗,a∗,b∗) of CIELAB space [1]. Although the choice of color space will depend on several factors, there is an advantage in choosing one that provides an efficient color code, in the sense of maximizing the amount of information about the scene provided by the description and minimizing any redundancy within it [2, 3, 4, 5]. The aim of this work was to analyze how efficiently information from a scene is coded by each of the main CIE color spaces. The images were of natural scenes under different illuminants, and information was expressed in terms of Shannon’s mutual information [6]. Information was calculated by two methods. The first was based on the information theoretically available from images of a scene. It depends only on the statistical distribution of the colorcode values in each image of the scene and how they vary with changes in illuminant. The second method was based on the information actually retrieved with a particular matching task, by which points in an image of the scene under one illuminant are matched, by color, to points in an image of the same scene under another illuminant. Estimates from the first method set an upper limit on estimates from the second. As a precursor to the analysis, the theoretical section which follows gives the definition of mutual information and an explanation of its decomposition into components. These components are associated with the individual variables of a color space and the interactions between them, including redundancy and illuminant dependence. The methods section contains details of the two kinds of information estimator, along with a brief description of the scenes and illuminants. The results for six color spaces are then summarized. Differences in the information retrieved across spaces and in redundancy and the illuminant-dependent component are considered in the discussion section. Opponent-color and chromatic-adaptation transformations were both critical in determining the efficiency of coding and the retrieval of information. Some partial results on the decomposition of the information available for different color spaces have been reported previously [7]. Theory To fix ideas, suppose that the color at each point in an image of a scene under some illuminant E is coded by its tristimulus values (X ,Y,Z), and consider, in particular, the luminance variable Y . Suppose that y is the value of Y at a particular point in an image of the scene under illuminant E and y′ is the corresponding value in the image of the scene under illuminant E ′. If the point is chosen randomly, then the values y and y′ can be thought of as samples from random variables Y and Y ′, respectively. Suppose that the probability density functions of Y , Y ′, and of the pair (Y,Y ′) are fE , fE ′ , and fEE ′ , respectively. Then the mutual information I(Y ;Y ′) between Y and Y ′ is given [6] by I(Y ;Y ′) = ∫∫ fEE ′(y,y′) log fEE ′(y,y′) fE(y) fE ′(y′) dydy′ , (1) where the integrations are taken over the spaces spanned by Y and Y ′. The logarithm is to the base 2, and mutual information expressed is in bits. Mutual information can be expressed as a combination of differential entropies [8], which are also based [6] on the probability density functions fE , fE ′ , and fEE ′ , thus h(Y ) =− ∫ fE(y) log fE(y) dy , h(Y ′) =− ∫ fE ′(y′) log fE ′(y′) dy′ , (2) h(Y,Y ′) =− ∫ fEE ′(y,y′) log fEE ′(y,y′) dydy′ . The mutual information (1) is then given by I(Y ;Y ′) = h(Y )+h(Y ′)−h(Y,Y ′) . (3) The definition of mutual information can be straightforwardly extended from the single variables Y and Y ′ to the tristimulus values (X ,Y,Z) and (X ′,Y ′,Z′), thus I(X ,Y,Z; X ′,Y ′,Z′) = h(X ,Y,Z)+h(X ′,Y ′,Z′) −h(X ,Y,Z,X ′,Y ′,Z′) , (4) 16th Color Imaging Conference Final Program and Proceedings 285 where all differential entropies are calculated as in (2) with the corresponding multivariate probability density functions. In an exactly analogous way, mutual information can be defined for the same two images of a scene for any other color space. Redundancy and illuminant dependence Again, suppose that coding is by tristimulus coordinates (X ,Y,Z). To simplify notation, let I stand for the mutual information I(X ,Y,Z; X ′,Y ′,Z′) as in (4) and let I1, I2, and I3 stand for the individual mutual-information components associated with the first, second, and third variables; that is, I1 = I(X ;X ′), I2 = I(Y ;Y ′), (5) I3 = I(Z;Z′). The difference between I and the sum I1 + I2 + I3 represents the contribution from the interactions between the variables of the color space. These interactions can be measured by the multiinformation [9, 10], which is a form of generalization of mutual information [4], and which here divides into two components. The first component is quantified by the multi-information between (X ,Y,Z) and the multi-information between (X ′,Y ′,Z′); that is, M(X ;Y ;Z) = h(X)+h(Y )+h(Z)−h(X ,Y,Z) ,