Object-space coordinates

Simultaneous estimation of object space coordinates with measures for algorithmic quality control. 

Abdel-Aziz, Y. I., and Karara, H. M. (1971). Direct linear transformation from comparator coordinates into object-space coordinates, Close-range photogrammetry. American Society of Photogrammetry, Falls Church, Va. 

Our main motivation to develop the specific transient technique of wavefront analysis, presented in detail in (21, 22, 5), was to make feasible the direct separation and direct measurements of individual relaxation steps. As we will show this objective is feasible, because the elements of this technique correspond to integral (therefore amplified) effects of the initial rate, the initial acceleration and the differential accumulative effect. Unfortunately the implication of the space coordinate makes the general mathematical analysis of the transient responses cumbersome, particularly if one has to take into account the axial dispersion effects. But we will show that the mathematical analysis of the fastest wavefront which only will be considered here, is straight forward, because it is limited to ordinary differential equations dispersion effects are important only for large residence times of wavefronts in the system, i.e. for slow waves. We naturally recognize that this technique requires an additional experimental and theoretical effort, but we believe that it is an effective technique for the study of catalysis under technical operating conditions, where the micro- as well as the macrorelaxations above mentioned are equally important. 

Space-coordinate density transformations have been used by a number of authors in various contexts related to density functional theory [26,27, 53-64, 85-87]. As the free-electron gas wavefunction is expressed in terms of plane waves associated with a constant density, these transformations were introduced by Macke in 1955 for the purpose of producing modified plane waves that incorporate the density as a variable. In this manner, the density could be then be regarded as the variational object [53, 54]. Thus, explicitly a set of plane waves (defined in the volume V in and having uniform density po = N/V) ... 

The objective of any heat-transfer analysis is usually to predict heat flow or the temperature which results from a certain heat flow. The solution to Eq. (3-1) will give the temperature in a two-dimensional body as a function of the two independent space coordinates x and y. Then the heat flow in the x and y directions may be calculated from the Fourier equations... 

The vacuum interface is the source of all quantum effects. Interaction with the interface causes particles to make excursions into time and bounce back with time-reversal and randomly perturbed space coordinates. Different from classical particles, quantum objects can suffer displacement in space without time advance. They can appear to be in more than one place at the same time, as in a two-slit experiment. 

From a geometrical point of view, we can consider a v-dimensional space, in which each dimension is associated to one of the variables. In this space each sample (object) has coordinates corresponding to the values of the variables describing it. 

Each molecule is described in an elaborative manner by transforming the coordinates of its nuclei to their respective strategical position very much well within the molecules s object space . Thus, alterations with respect to the relative position of atoms e.g, rotation of atoms about a covalent bond i.e., single bond) are duly achieved well within the object space . 

Here, we have dubbed /y as the atomic scattering factor of atom j, and its atomic position in the unit cell is given by the real-space coordinates Xj, yj, and Zf, the sum runs over all atoms in the unit cell. Thus, the structure factor F i l (in reciprocal space because it depends on the reciprocal Miller indices) results as the Fourier-transform of a real-space infinite object (the entire crystal), schematically sketched in Figure 2.7. 

Camera Calibration. Each camera must be calibrated before it can contribute to locating a target in object-space. Camera calibration defines a mapping from three-dimensional object-space into the two-dimensional u, v coordinates of the camera. This mapping is expressed in Eq. (5.2) in terms of homogeneous coordinates ... 

FIGURE 5.6 The X, Y, Z coordinates of the control points (i.e., reflective targets) are known relative to the origin of the object-space. Once the cameras have been calibrated, the hanging strings with the control points are removed from the field of view of the cameras. The black rectangle flush with the floor is a force platform (see Sec. 5.4.5). 

FIGURE 5.7 Cameras 1 and 2 each have a unique perspective of the tracking target in object-space (i.e., silver circle). The X, Y, Z coordinates of the target can be calculated by using the u, v coordinates and the betas determined during calibration. 

While only two cameras are necessary to reconstmct the three-dimensional coordinates of a tracking target in object-space, more than two cameras are recommended to help ensure that a minimum of two cameras see the target every point in time. The cameras should be positioned so that each has a unique perspective of the wortepace. Ideally the cameras should be placed at an angle of 90 with respect to one another. In practice this may not be possible, and every effort should be taken to maintain a minimum separation angle of at least 60 . 

Force platforms are used to resolve the load a subject applies to the ground. These forces and moments are measured about X, Y, and Z axes specific to the force platform. In general, the orientation of the force platform axes will differ from the orientation of the reference axes of the object-space. This is illustrated schematically in Fig. 5.14. Thus, it is necessary that the ground reaction forces be transformed into the appropriate reference system before they are used in subsequent calculations. For example, the ground reaction forces acting on the foot should be transformed into the foot coordinate system, if ankle joint forces and moments are expressed in an anatomically meaningful reference system (i.e., about axes of the ACSfooi). 

For the case of two objectives. Fig. 1 shows an objective space with a concave part, the CHIM (which is the discontinuous line joining the individual minima of objectives) and several important points. We have to define the Nadir point, pNadin which is the vector of upper bounds of each objective in the entire Pareto-optimal set (in practice, its coordinates can be estimated from the coordinates of the shadow minimum [5-6]). The line that joins the shadow minimum F and the Nadir point represents the quasi-normal direction used by NBI. This line intersects the CHIM in a point given by the vector p ax = [0-5,0.5]. Thus, the upper bound is defined by the distance from this point to the shadow minimum. Similarly, the lower bound is given by the distance to the Nadir point. It is clear that tj = -In. t is positive if the normal is pointing towards the shadow minimum, and negative in the opposite sense). It should be noted that these bounds are set for a two-objective problem. 

As the original data matrix can be represented in the variable and object space, the results of the PCA can be displayed in a corresponding score and loading plot. The score plot is the realization of the principal components and represents the position of the objects in the new coordinate system. In this study, score plots were employed to identify similar objects (i.e., columns). Similarly, the loading plots depict the influence of the original variables on a principal component. Usually, it is assumed that principal components which describe a large part of the variance also represent important physical or chemical factors (e.g., in this case, the special characteristics of the different columns). Smaller factors can be interpreted as noise. 

In a score plot each point corre.sponds to an object the coordinates are given by the scores. This plot is the result of projecting the multivariate space. The distances between objects in the score plot are approximations of the distances in the multivariate feature space therefore groups (clusters) of similar objects can be detected visually. 

Under circumstances that this condition holds an ADT matrix, A exists such that the adiabatic electronic set can be transformed to a diabatic one. Working with this diabatic set, at least in some part of the nuclear coordinate space, was the objective aimed at in [72]. 

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