Hull, convex

A convex hull is a molecular surface that is determined by running a planar probe over a molecule. This gives the smallest convex region containing the molecule. It also serves as the maximum volume a molecule can be expected to reach. 

Fig. 3.62 An example of overlapping convex-hulls in the 4-site (center-site excluded) von Neumann neighborhood voting rule with threshold = 2 see equation 3.77.

Convex hull formulations of MILPs and MINLPs lead to relaxed problems that have much tighter lower bounds. This leads to the examination of far fewer nodes in the branch and bound tree. See Grossmann and Lee, Comput. Optim. Applic. 26 83 (2003) for more details. 

Classification of Protein Structures Based on Convex Hull Representation by Integrated Neural Network. 

Figure 2.44 The Ti-Zn system. The calculated enthalpy of formation of the intermetallics and the resulting convex hull are shown according to Ghosh et al. (2006). Experimental values of about 22-19 kJ/mol of atoms were measured in the composition range 50-75 at.% Zn. The calculations yield a ground state convex hull defined by the following structures (1) Ti3Zn-Ll2 (2)...

Fig. 1. Geometry concepts tor analyzing protein cavity. The protein bulk is represented in black, probes as little spheres. The convex Hull ot the protein is represented in dash line. The plain vectors emerging trom the probe in grey are pointing towards the bulk solvent, whereas the dash vectors will encounter protein atoms within a radius of 8 A (radius of the influence circle m dot line).Jhe probe in front ot the clett defines the degree of precision in represenfing the molecular surface.

The convex hull of a set of points is the volume surrounding these points such that any segment between any two of these points stay inside the volume. For proteins, we might relax this definition to include inside the protein volume, the closed protein cavities that are not open to bulk solvent. A more adequate... 

Remark 3 Any point x in the convex hull of a set S in 5ftn can be written as a convex combination of at most n + 1 points in S as demonstrated by the following theorem. 

In this chapter, the basic elements of convex analysis are introduced. Section 2.1 presents the definitions and properties of convex sets, the definitions of convex combination and convex hull along with the important theorem of Caratheodory, and key results on the separation and support of convex sets. Further reading on the subject of convex sets is in the excellent books of Avriel (1976), Bazaraa et al. (1993), Mangasarian (1969), and Rockefellar (1970). 

Figure 6.13 An IT8 target (made by Wolf Faust www.coloraid.de) viewed under sunlight (a). The image was taken with a Canon 10D. Two different views of the three-dimensional convex hull of the colors are shown in (b) and (c).

Let v e H0 be a vertex of the convex hull computed from the input image. The set of feasible maps M. (v) for the given vertex v is... 

For each vertex of the convex hull of the observed colors, we compute the feasible maps. We then intersect all these maps, as the actual illuminant must lie somewhere inside the intersection of these sets. Therefore, each vertex of the convex hull of the observed gamut gives us additional constraints to reduce the set of possible illuminants that may have produced the observed image. Let Ain be the computed intersection. 

The intersection will also be a convex hull, as the intersection of two convex hulls is again a convex hull. Therefore, the basic operations of gamut-constraint algorithms involve computing the convex hull of a set of points and then intersecting these hulls. 

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 set of illuminants allowed is constructed as follows. First we choose a standard surface. Let s be the color of the standard surface viewed under the canonical illuminant. The same surface is then illuminated with a large number of illuminants. Let Hs be the convex hull of the observed colors of the surface. The convex hull contains all possible colors that could be observed when the illuminants are added in different amounts. If we choose a white patch as the standard surface, then the vertices of the convex hull will be just the chromaticities of the illuminants. The standard surface does not necessarily have to be white. Therefore, one computes the set of maps that take the observed color of the standard surface when viewed under the canonical illuminant to the color of the same patch when viewed using a different illuminant. This set of maps A4 is given by... 

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... 

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