White-point preservation enforces positivity

Abstract

It is commonplace to use a 3 × 3 linear transform to map device RGBs to XYZs. Two particular types of transforms have been developed based on the assumptions that we either maximally ignorant or maximally prescient about the world. Under the maximum ignorance assumption, it is assumed that nothing is known about the spectral statistics of the world and so the best correction transform is the one that maps device spectral sensitivities so they are as close to observer sensitivities as possible. Under maximum prescience, we know the spectral statistics that we will observe and so the maximally prescient transform maps, with minimum error, the RGBs (that we know we will see) onto corresponding XYZs. In general the two assumptions lead to quite different color corrections. In previous work we have argued against total ignorance or prescience and have instead developed compromise transforms. Our work is based on two observations. First, one is never completely ignorant about the world - color signal spectral power distributions are everywhere all positive. Second, it is accepted that it is much more important to correct some colors than other. In particular, white is central to color vision and color imaging, so it is imperative that white should always look right. However, to date these two compromise solutions have been studied in isolation. Surely, it would be advantageous to combine the constraints of whiteness and positivity? In fact we show that this is not the case: by preserving white we enforce positivity. This is an important result. Not only does it add to our understanding of color correction, but it helps explain color correction results published in the literature (the assumptions of positivity and white-preservation lead to very similar results). Moreover, it helps us to derive a new measure for assessing the goodness (color correctability) of camera sensors that is strictly less pessimistic (and more accurate) than the existing Vora Value.