Error-Less Colour Correction Graham D. Finlayson and Peter Moroviˇ c School of Computing Sciences University of East Anglia NR4 7TJ Norwich, England {graham, peter}@cmp.uea.ac.uk Abstract Colour correction is the problem of mapping device de- pendent RGBs to standard CIE XYZs. Traditionally it is solved for by an error minimising one-to-one linear trans- form. However this problem is ill-posed. There exist multi- ple reflectances, known as metamers, which induce the same RGB but different XYZs (and vice versa). In this paper we propose that this ill-posedness might be viewed positively. Indeed, that it leads to an error-less transform for colour correction. We propose that a map- ping is error-less if it takes an RGB to an XYZ such that there exists a real reflectance spectrum which integrates to this RGB-XYZ pair. We show how we can solve for a map- ping which satisfies this error-less criterion. As in previous studies, we seek a linear transform that is error-less. We show that we can solve for such a transform by quadratic programming. Experiments demonstrate 3 important results. First, that a linear least squares transform is not error-less. Specif- ically, saturated RGB-XYZ pairs do not correspond to a plausible reflectance. Second, there exists a linear trans- form that is error-less. Finally, that the best error-less transform performs almost as well as least-squares, but sub- stantially better for saturated colours. It is possible to map RGB to XYZ with zero error. 1 Introduction Typically, colour input devices such as scanners and cameras coarsely sample the colour signal through three fil- ters, resulting in three values known as the RGB. These val- ues express the interaction of light, a surface and the spec- tral characteristics of the device’s filters. In general, each device has a different set of filters, which in turn are differ- ent from those of a human observer. As a consequence, all devices need to transform the raw, device dependent RGBs into a standard colour space. This is usually the CIE XYZ system, that represents the colour matching functions of hu- man observers[13]. This process of transforming RGBs into XYZs is known as colour correction. Most methods in lit- erature model this transform using a simple 3 × 3 error- minimising linear transform[10]. However, unless the de- vice’s filters are an exact linear transform of the CIE XYZ curves (the Luther condition is satisfied), colour correction is prone to error[2]. Since colour image formation maps reflectances, fun- ctions of wavelength, to three values, an RGB, it becomes apparent that the linear transform is under-determined. That is, from the RGB, we cannot uniquely identify the corre- sponding surface reflectance. Indeed it is well known in theory[12] as well as in practice[3], that there are many re- flectances, called metamers, that induce the same RGB. One consequence of metamerism is that reflectances that induce the same RGB might induce a set of XYZs. On the one hand this implies that colour correction is ill-posed. If an RGB potentially maps to many XYZs, how do we choose the cor- rect answer? We can not. However, we argue here that this uncertainty is a help and not a hinderance. We propose that colour correction is error-less if the pair of RGB and XYZ put in correspondence through the mapping is such that there exists a physically realisable surface reflectance that induces this RGB-XYZ pair. In the next section we introduce briefly the framework of metamerism. In section 3 we show how the metamer set is used to formulate the error-less constraint of a linear transform. In section 4 we present results comparing the error-less colour correction to standard linear least squares and discuss our findings. Finally, in section 5 we summarise and conclude our study. 2 The Metamer Set It has been shown before that given an RGB, the spec- tral sensitivities of the device, the illuminant spectral power distribution and a linear model of surface reflectances of ar- bitrary dimension (≥ 3), it is possible to solve for the con- 0-7695-2128-2/04 $20.00 (C) 2004 IEEE