The dichromatic reflectance model introduced by Shafer [16] predicts that the colour signals of most materials fall on a plane spanned by a vector due to the material and a vector that represents the scene illuminant. Since the illuminant is in the span of all dichromatic planes, colour constancy can be achieved by finding the intersection of two or more planes. Unfortunately, this approach has proven to be hard to get to work in practice. First, segmentation needs to be carried out and second, the actual intersection computation is quite unstable: small changes in the orientation of a dichromatic plane can significantly alter the location of the intersection point. In this paper we propose to ameliorate the instability problem by regularising the intersection. Specifically we introduce a constraint on the colour of the illuminant. We show how the intersection problem in the context of convex and non-convex illuminant constraints, based on the distribution of common light sources, can be solved. This algorithm coupled with the simplest of segmentations results in good estimation results for a large set of real images. Estimation performance is significantly better than for the unconstrained algorithm.