Metamer Mismatch Volumes

A new algorithm for evaluating metamer mismatch volumes is introduced. Unlike previous methods, the proposed method places no restrictions on the set of possible object reflectance spectra. Such restrictions lead to approximate solutions for the mismatch volume. The new method precisely characterizes the volume in all circumstances. Introduction Two objects (surface reflectances) of the same colour (speaking informally) under one light can differ in colour under a second light. That is, they may be equal in CIE XYZ (for example) under the first light, but differ in CIE XYZ under the second. They are metameric matches under the first light, but fail to match, and hence are no longer metamers, under the second. If a colour under one light can become two colours under a second light, then it is natural to ask: What is the range of possible colours that the observed colour might become under the second light? For example, given an XYZ under CIE illuminant D65, what is the set of possible XYZ that could arise under CIE illuminant A? This set is commonly known as the metamer mismatch volume. In general, given the spectral power distributions of two illuminants and the colour of an object under one illuminant, the problem is to compute the metamer mismatch volume (i.e., the set of colours) that could possibly arise under the second illuminant. Metamer mismatch volumes are important because they arise in colour correction, camera design (sensors leading to the smallest metamer mismatch volumes will be colormetrically the most accurate), and lighting design (lights that lead to the smallest mismatch volumes will have the best colour rendering). Previous work on mismatch volumes has been based on finding “natural” metameric spectra or metameric blacks [1, 9] using linear programming or Monte Carlo methods [10]. There have been promising results in colour correction using the volumes computed by these methods [9, 4, 2, 3]. However, of the methods proposed thus far, none directly describes the theoretical limits of the metamer mismatch volumes. In this report we investigate the boundary of the metamer mismatch volume from the formal point of view and then provide an algorithm for computing metamer mismatch volumes for arbitrary, strictly positive illuminants without placing any restrictions on the reflectances. Metamer Mismatch Volume Theory Consider a set of colour mechanisms Φ = (φ1, ...,φn), the response of each of which to a reflecting object with spectral reflectance function x(λ ) illuminated by a light with spectral power distribution p(λ ) is given by