Scaled convex hull

A drawback of the gamut-constraint method is that it may fail to find an estimate of the illuminant. This may happen if the resulting intersected convex hull A4n is the empty set. Therefore, care must be taken not to produce an empty intersection. There are several ways to address this problem. One possibility would be to iteratively compute the intersection by considering all of the vertices of the observed gamut in turn. If, as a result of the intersection, the intersected hull should become empty, the vertex is discarded and we continue with the last nonempty hull. Another possibility would be to increase the size of the two convex hulls that are about to be intersected. If the intersection should become empty, the size of both hulls is increased such that the intersection is nonempty. A simple implementation would be to scale each of the two convex hulls by a certain amount. If the intersection is still empty, we again increase the size of both hulls by a small amount. 

Since the set of illuminants Hs is a convex hull, this set will also be a convex hull. The set of maps is simply scaled by the inverse of the observed color of the standard patch viewed... 

The three-dimensional gamut-constraint method assumes that a canonical illuminant exists. The method first computes the convex hull TLC of the canonical illuminant. The points of the convex hull are then scaled using the set of image pixels. Here, the convex hull would be rescaled by the inverse of the two pixel colors cp and Cbg. The resulting hulls are then intersected and a vertex with the largest trace is selected from the hull. The following result would be obtained for the intersection of the maps Mn-... 

We merely have to scale up the convex hull by a factor of Z. To be more precise, each band corresponding to a face orientation needs to be scaled up about its centroid by a factor of Z. 

---