Spatial reconstruction

Fig. 3 shows the theoretical descriptors S2(r) and p(r) and their recalculations from construct B depicted in Fig. 1. RES A shows high angularity, and the deviation of the computed and the theoretical p(r) at low r reflects the difficulty of constructing the shape of contacting and sUghtly penetrating spheres precisely. However, the principal character of the morphology is recovered. A complete agreement between target and spatial reconstruction was observed at no time [29,30,34,35]. The construction was fairly successful with respect to the geometrical parameters, as the comparison of the data in the third row in Table 1 with the theoretical data in the first row shows.

Several decades ago it was realized that chemistry of the shells of benthic foraminifera (carbon isotope and Cd/Ca ratios) carried an imprint of the nutrient content of deep-water masses (Shackleton, 1977 Broecker, 1982 Boyle, 1981). This led rapidly to the recognition that the water masses in the Atlantic Ocean were arrayed differently during the last glacial maximum than they are today, and the hypothesis that the glacial arrangement reflected a diminished contribution of low-nutrient North Atlantic deep water (NADW) (Curry and Lohmann, 1982 Boyle and Keigwin, 1982). More detailed spatial reconstructions indicated a shallow nutrient-depleted water... 

In this section, we introduce KBFVM for the moment-transport equations and show how they can be formulated to guarantee that the updated moment set is realizable. A more detailed presentation can be found in Appendix B. For clarity, we begin with the PBF and then discuss the additional complications that arise with the KF and, finally, the GPBF. Because the advection term is the principal cause of unrealizable moment sets, we limit our discussion to this term and focus on the high-order spatial reconstruction needed in order to reduce numerical diffusion. The material in this section follows roughly the work of Vikas et al. (2011a, 2012), and the reader is referred there and to Appendix B for more details. 

The reader should notice that this realizability condition on the time step At applies even when a high-order spatial reconstruction is used for the velocity. It is also worth noting that Eq. (8.26) applies even when the velocity is discontinuous (e.g. at a shock). 

If a first-order spatial reconstruction is used for the abscissas, then the interface values of the abscissas will be the same as the volume-average values = 1/2= i- j2a,r-On putting this into Eq. (8.30), the numerical NDE becomes... 

When high-order spatial reconstruction is used for the weights, the updated moment... 

However, for the special reconstruction for which t a(mAt,x) = is constant in each grid cell, it is correct to refer to the weights and abscissas as the volume-average values. Nevertheless, the nonlinear dependence between the moments and the abscissas makes it challenging to use high-order spatial reconstructions for the abscissas, while preserving the values of the volume-average moments. 

Following Vikas et al. (2011a), in order to ensure that the spatial reconstruction of the NDF is realizable, we will use first-order spatial reconstmction for the quadrature abscissas and high-order spatial reconstruction for the quadrature weights. Formally, this can be written as... 

For each order of the spatial reconstruction, it is then necessary to define the spatial gradient terms appearing in Eq. (B.59). 

For second-order spatial reconstruction, the weights at the interfaces are defined by... 

In fact, this expression is the same as the second-order expression found when working directly with the diffusion term written using the transported moment set. In general, the second-order spatial reconstruction on unstructured grids will also be realizable. Perhaps for this reason, the diffusion term is not usually identified as a potential source of unrealizable moment sets. 

Eor higher-order spatial reconstruction, the NDF will be realizable only if the abscissas in neighboring cells are employed to construct the scheme. In other words, we must continue to use Eqs. (B.61) and (B.63). We can then approximate the spatial derivatives of the weights using a higher-order scheme. For example, with a fourth-order scheme, we will use... 

Note that if the abscissas are constant in all cells, then the overall scheme will have the same order as the spatial reconstruction of the weights. On the other hand, if the weights are constant, then the overall scheme for the abscissas is of second order. 

Allard P., Duhaime M., Labelle H., et al. 1987. Spatial reconstruction technique and kinematic modeling of the ankle. IEEE Eng. Med. Biol. 6 31. 

The visible light produced by positron decay and neutron capture is recorded using 1300 photomultiplier tubes. Events are selected according to the magnitude of the energy depositions of the positron and neutron, and the spatial and time coincidence between the positron and neutron. The spatial reconstruction resolution is an estimated 10 cm for a 1 MeV event, while the time resolution of the phototubes is less than 10 ns. 

Figure 8. Comparison of the different modalities for generating and evaluating image series. Upper part images taken at different times give insight into 2-D anatomy and function. With a comparable number of pictures taken at different angles (lower part) multiple cross sections of a static 3-D object can be obtained. It requires the same number of pictures and processed data for each image if a dynamic spatial reconstruction (DSR) of the beating heart should be achieved.

Spatial reconstruction of former soils geographical patterns is possible in GIS through a back analysis of soil patterns based on reclassification of current soil types and soil boundaries. 

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