Predicting Cone Quantum Catches under Illuminant Change

Given LMS cone quantum catches from a surface under a first illuminant what is the best method of predicting what the corresponding quantum catches will be for the same surface under a second illuminant given only the quantum catches of a white surface under both illuminants? The von Kries rule is one well known method. In this paper, two new prediction methods along with a variation on an existing third method are introduced and then compared experimentally. In contrast to the von Kries rule which is equivalent to a diagonal transformation, all three methods estimate a full 3-by-3 linear transformation mapping LMS values between illuminants. All the new methods perform better than the von Kries rule. Introduction When there is a change in illumination, the cones’ quantum catches change. We address the problem of predicting how they change. In particular, we are concerned with predicting the LMS cone signals under a second illuminant given the LMS cone signals under a first illuminant along with the LMS cone signals of a white surface under each of the illuminants. Although the problem we address here of predicting cone quantum catches may be relevant to chromatic adaptation, it is not the same as chromatic adaptation. Models of chromatic adaptation [7,8] aim to predict which colors have corresponding appearances to a human subject under a change in illuminant. Accurate prediction of cone quantum catches will not necessarily lead to accurate appearance matches; however, it might be useful as a subcomponent of a full color appearance model. One common method of predicting LMS under a second illuminant is the von Kries rule[6], which involves a diagonal model of illuminant change. Given the LMS quantum catch a x observed for a surface under illuminant a the diagonal model predicts the corresponding LMS quantum catch of the same surface under illuminant b as: a b Dx x = where D is a 3-by-3 diagonal matrix. The diagonal model is limited to 3-parameters, We would prefer to use the more general full 9-parameter, 3x3 linear model of the form: a b Mx x = (1) where M is 3-by-3. One reason the full 3x3 linear model is not used is that usually there is not enough information available to determine the 9 coefficients directly, especially if all we know about the two illuminants are the LMS values of a white surface under each illuminant. The question we address here is: Are there other nondiagonal models that would perform better than the diagonal model? We answer the question with 3 methods we will call the Lighting-Matrix Estimation Method (LME), the Palette Method (PM) and the Characteristic Vector Method (CVM). PM is entirely new. LME is closely related to a derivation by Maloney [10]. CVM is a modification of an earlier PCA-based method [2]. LME employs the assumptions that illuminant spectra and reflectance spectra are approximated well by 3dimensional linear models along with the machinery of Maloney and Wandell’s “lighting matrix”[9]. As will be shown below, the restriction to 3 dimensions means that LMSs from the 2 whites provide enough information to solve for the 3-by-3 lighting matrix mapping from one 3D illuminant to the other. The method starts from the fact that given the LMS of a known reflectance (white in this case), a 3D model of the illuminant spectrum can be calculated. Then, based on a 3D model of illumination, the transform mapping one lighting matrix to another is calculated. PM involves an analysis of the space of possible palettes of LMS signals that occur under different illuminants. We define an illuminant’s palette as the set of all LMS signals obtained from a training set of surface reflectances under that illuminant. We use the term ‘palette’ rather than ‘gamut’ to avoid confusion with the Appears in: CIC’11 Proc. Eleventh Color Imaging Conference, Scottsdale, Nov. 2003. Page 162 Copyright: Society for Imaging Science and Technology, 2003 similar, but different, us of the term ‘gamut’ in the context of gamut mapping algorithms. Characteristic Vector Analysis is used to extract a 3-dimensional linear model approximating the set of possible palettes. A change in illumination causes a change in observed palette. CVM considers the 9-dimensional space of 3x3 transformations that model illuminant change and then finds the 3-dimensional subspace that best approximates these matrices. The required illuminant transformation matrix is then built up from the 3 basis matrices based on the LMS of white. LME: Lighting-Matrix Estimation Method Suppose finitely sampled illuminant spectra and surface reflectances are modeled by 3-dimensional linear models: ) ( ) ( 3 n i j n E E λ λ ≈ ) ( ) ( 3 n j j n S S λ λ ≈ Let the cone sensitivity functions be . 3 1 ) ( = k R n k λ Following Wandell[14] and Maloney[10], we can construct a lighting matrix Λ mapping surface reflectance weightsσ to LMSs: σ Λ = l where the kj entry of Λ is