Reconstruction defined

Before the data can be visualised, ie displayed in a two or three-dimensional representation, the ultrasonic responses from the interior of the test-piece must be reconstructed from the raw ultrasonic data. The reconstruction process projects ultrasonic indications into 3D space. As well as reconstructing the entire ultrasonic data set within an acquisition file, it is possible to define an arbitrary sub-volume of the test object over which reconstruction will take place. The image resolution may also be defined by the user. Clearly, larger volumes or greater resolution will increase the computation time for both the reconstruction and visualisation processes. 

The three-dimensional synnnetry that is present in the bulk of a crystalline solid is abruptly lost at the surface. In order to minimize the surface energy, the themiodynamically stable surface atomic structures of many materials differ considerably from the structure of the bulk. These materials are still crystalline at the surface, in that one can define a two-dimensional surface unit cell parallel to the surface, but the atomic positions in the unit cell differ from those of the bulk structure. Such a change in the local structure at the surface is called a reconstruction. 

The main problem in the analysis of E vs. 0 plots is that the two quantities are usually measured independently on different samples. It may happen that the surface structure differs somewhat so that for the sample on which E is measured is different from that of the sample used in UHV experiments. This is especially the case with polycrystalline surfaces, whose structural reproducibility is occasional, but it is also the case with well-defined crystal faces if reconstruction phenomena are possible.60 The problem persists also in the absence ofreconstruction since the concentration and/or distribution of surface defects may be differ-... 

Although it sounds reasonable to use the maximum likelihood to define our esfimafe of the displacement, there are two questions that remain. Firstly, what is the variance of the error associated with this estimate This defines N which was used in Eq. 22 fo defermine fhe error in fhe wavefront reconstruction. Secondly, is it possible to do better than the centroid In other words is it optimal ... 

The task of reconstructing the diet can be defined as that of finding D and F, by first evaluating or estimating the matrix com-ponents dij and f,j(), and applying these values to measurements of B. 

We can now define the rank of the column-pattern as the number of linearly independent columns or rank of X. If all 50 points are coplanar, then we can reconstruct each of the 50 columns, by means of linear combinations of two independent ones. For example, if x, and Xj 2 linearly independent then we must have 48 linear dependences among the 50 columns of X ... 

Algebraically, the reconstruction of the values of X has been defined by the matrix product of the scores S with the transpose of the loadings L (eq. (31.22)). Geometrically, one reconstructs the value Xy by perpendicular projection of the point represented by upon the axis represented by s, as shown in Fig. 31.3c ... 

A bipolar axis is defined by two rows S and s,- in Fig. 31.3e. If we project a column upon this bipolar axis, we reconstruct the difference between two elements in X. This follows readily from eq. (31.22) ... 

A close analogy exists between PCoA and PCA, the difference lying in the source of the data. In the former they appear as a square distance table, while in the latter they are defined as a rectangular measurement table. The result of PCoA also serves as a starting point for multidimensional scaling (MDS) which attempts to reproduce distances as closely as possible in a low-dimensional space. In this context PCoA is also referred to as classical metric scaling. In MDS, one minimizes the stress between observed and reconstructed distances, while in PCA one maximizes the variance reproduced by successive factors. 

Modern NMR software covers all facets of MR applications and assists the laboratory staff and the research groups not only in the standard procedures of scan preparation, data acquisition, reconstruction and analysis, but also offers an appropriate development environment for user defined measurement methods and data analysis algorithms and provides easy-to-use tools for data management, documentation, export and archiving. The software allows the user to run complex NMR machines in a routine manner and to integrate the spectrometer into the laboratory infrastructure [7]. 

By its very definition, the MaxEnt method is optimally suited to flexibly reconstruct distributions whose main features are well represented in the available data, that is either in the observations or in the prior structural knowledge. When this is the case, the missing structure can be reasonably approximated by a collection of randomly and independently distributed constituents (by missing structure here we mean all those structural details which are not completely defined by the prior knowledge). 

Preliminary studies into a third variable, the partial pressure of water vapor in the system, are discussed in Part 3 of the Results and Calculations section. Each calorimetric sample ( 1 g, 13.47 mass % bitumen) came from a large sample of "reconstructed" oil sand consisting of Athabasca bitumen loaded onto a chemically inert solid support material (60/80 mesh acid washed Chromosorb W) of well-defined particle size. 

This generalization is therefore quite straightforward. One difficulty is that in higher dimensions, reconstructing A( ) from its derivatives is not straightforward. For example, let us choose a reference point o and decide that, 4Alil/(o) = 0. Say we define AABF( ) by... 

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