Contour threshold

The entire 3D property function P(r) can be represented by an infinite family of IPCOs, by taking one such surface G a) for each value of the contour parameter a, throughout the entire range a in a< ainiax- The minimum and maximum values Amin and flmax of the contour threshold a depend on the property P and the actual molecular system studied. 

As an example, one may consider an isodensity contour Gi = Gi(<2i) and a MEP contour Gz = G2(threshold values ai and az-If interpenetration occurs, then it defines one or several closed loops on both surfaces. The points along these loops belong to both contour surfaces. Hence on each surface the loops define the boundaries of subsets characterized by the function value of the other physical property the value of the property is either greater than the threshold for all points within a subset, or it is lower than the threshold for all points in the subset. In fact, the interpenetration pattern can be generated simultaneously for a series of threshold values. For example, by considering k different MEP threshold values, az, az, <22, . .. the pattern of interpenetration may be used to generate ranges of the electrostatic potential... 

The availablility of an estimate of the conditional cdf F CzjCN)) at each nodal joint x allows an assessment of the risks a(x) or p(x) of making wrong decisions. Consider the contour map of a particular estimate p (x), x (see Figure 3a). Suppose that the threshold value 500 has been selected to declare any sub-area of A hazardous. The contour line 500 on Figure 3 delineates the zones which are candidates for cleaning. Within these zones, the probability that the concentration is actually under 500, i.e. the risk a(x) of cleaning unduly, can also be mapped ... 

The importance of this case is that OS-CFAR can be analytically analysed without any approximations. Furthermore the resulting scaling factor Tos is completely independent of p. Figure 14 shows the typical behaviour of OS-CFAR in clutter edge and multiple target situations. The threshold follows the clutter contour with a certain safety distance. In two target situations the threshold is more or less unchanged compared with a pure noise situation. 

Figure 17.7 (a) Cylindrical flame sheet area as a function of gray scale threshold (6) corresponding flame image and (c) image contours at threshold values of 0.34, 0.54, and 0.69... 

Figure 17.6 shows the results for a typical composite image of the entire flame in which the image intensity, or gray scale value, is plotted on the vertical axis against radial and axial distance coordinates. To find the flame area, a threshold intensity is first picked and a binary image is created with all pixels with intensities above the threshold set to 1 (white) and the rest set to 0 (black). The number of white pixels is then proportional to the surface area of the flame at that intensity contour. This process is then repeated to build up a plot of flame area versus a normalized threshold number (intensity value/256) such as that shown in Fig. 17.7a. 

Fig. 7 Plane wave focusing by a NA = 1.35 objective lens, calculated using vectorial Debye theory, a The normalized 3D intensity distribution with the cutoff threshold at 1% intensity. The lateral cross-sections are plotted on a log-scale at the axial positions z = 0 (b) and z = 7-/2 (c), respectively. Contour lines are plotted at 0.5 (inner) and 1/e (outer) levels, respectively. Polarization of the plane wave was horizontal (along x)...

Fig. 8.9. Contour plot of the potential energy surface of H2O in the BlA state as a function of the H-OH dissociation bond Rh-oh and the HOH bending angle a the other O-H bond is frozen at the equilibrium value in the ground electronic state. The energy normalization is such that E = 0 corresponds to H(2S ) + OH(2E, re). This potential is based on the ab initio calculations of Theodorakopulos, Petsalakis, and Buenker (1985). The structures at short H-OH distances are artifacts of the fitting procedure. The cross marks the equilibrium in the ground state and the ellipse indicates the breadth of the ground-state wavefunction. The heavy arrow illustrates the main dissociation path and the dashed line represents an unstable periodic orbit with a total energy of 0.5 eV above the dissociation threshold.

A contour plot hides many evils. Because we can set the threshold as high as we want, we can eliminate impurities and artifacts without being accused of fraud To be honest (and... 

One approach to the approximate representation of molecular bodies is based on molecular isodensity contours, MIDCOs, defined with respect to some fixed nuclear configuration K and some electron density threshold a. A MIDCO G(a,K) is defined (in the fixed nuclear configuration approximation) as the collection of all those points r of the three-dimensional space where the electronic density is equal to the threshold a ... 

In general, a locally convex domain D2(b),j(a> Fj) of a functional group F, relative to a reference curvature b, shows local shape complementarity with a locally concave domain Do( b),j(a, F2) of a complementing functional group F2, relative to a reference curvature of -b. The threshold values a and a are also likely to complement each other the shape complementarity between the higher electron density contours of one functional group and the lower electron density contours of the other functional group is relevant. 

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