Convex hull method

Geometric methods where most definitions rest on defining the smallest convex area that covers the training set compounds in descriptor space. This method is also known as the convex hull method (Figure 14.7). 

Figure 14.7 Schematic representation of the convex hull method for a two-parameter description.

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. 

This process is repeated until the intersection is nonempty. It would also be possible to compute the two closest points between the two convex hulls and choose the point lying half way between the two convex hulls or we could increase the size of all convex hulls by a certain percentage before computing the intersected hull in order to avoid an empty intersection. This could also be done iteratively. First we see if the intersection is indeed empty. If the intersection is empty, we increase all convex hulls by a certain percentage. If it is still empty, we increase it even further. This process continues until the intersection is nonempty. For our experiments we have used the latter method, as this method produced the best results. 

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-... 

Around the same time, Glasser et al. (17) retrieved and extended the insightful methods of Horn (18) and presented graphical procedures known as the attainable region (AR) method. Their approach requires the graphical construction of the convex hull of the problem and helps to exemplify the need for a systematic and general methodology. In principle, the reactor network with maximum performance in terms of yield, selectivity, or conversion can be located on the boundary of the AR in the form of DSR and CSTR cascades with... 

One technique which is applicable for surfaces that are not everywhere differentiable is also suitable for the shape characterization of dot representations of molecular surfaces such as the Connolly surfaces [87], which are not only nondifferentiable, but are not even continuous. The method of 1-hulls [351] is based on a generalization of the concept of convex hull. The convex hull of a set A is the smallest convex set that contains A. Consider a three-dimensional body T. The T-hull of a point set A is the intersection of all rotated and translated versions of T which contain A. The T-hull method is suitable for shape comparisons with a common reference shape, chosen as that of the body T. Alternatively, when the shapes of two molecules, T and A are compared, one molecular body can be chosen as T and the T-hull of the other molecular body A provides a direct shape comparison [351]. 

Figure 8.11 The RCC method apphed to the three-dimensional Van de Vusse. The method may be summarized into four broad construction phases (a) Initiahzation (PFR from the feed point), (b) growth (initial DSR trajectories from the feed), (c) iteration (constant a DSR trajectories from the extreme points), and (d) pohsh (PFR trajectories from the convex hull points).

Geometric Methods Convex Hull AD. AD is defined as a convex hull of points in the multidimensional descriptor space (Fechner et al. 2008). 

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