Global basis functions

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

Ab initio calculations reproduce the experimentally observed geometrical changes. Ethylene is planar (absolute true minimum global minimum) at all theoretical levels. The computed geometry of disilene depends strongly on the basis functions and on electron... 

Set up a global basis xi)X2,— of basis functions and divide the global space into three subspaces (i) functions of a core space (ii) iVvai functions of a valence sp>ace and Na,m functions of a complementary subspace. 

Global basis functions. Common global basis functions, where the interpolation functions for multi-dimensional domains can be obtained, come from expansions of Pascal s triangle. In 2D, Pascal s triangle is defined by,... 

Either MTO or ACO functions are valid as basis functions for expanding a global wave function b in all atomic cells. By construction, they are regular in r, smooth at ex, and bounded outside. When the matrix C is nonsingular, modified canonical basis functions can be defined such that... 

For an exact solution, the external function ijrv would be identical to the global matching function on each interface. An alternative algorithm can be based on fitting a linear combination of basis functions in each cell to %, which is uniquely determined. The VCM equations for an orbital basis at fixed energy reduce to [282]... 

If the total number of basis functions f l for all cells is N,p, then for each global solution index X there are 2N,p equations for the 2N,p elements of the column vectors oj1"- and Thus the variational equations derived from Ea provide exactly the number of inhomogeneous linear equations needed to determine the two coefficient matrices, oj and /I These equations have not yet been implemented, but they promise to provide an internally consistent energy-linearized full-potential MST. 

Figure 4 Example of a spectral simulation result for NAA from a PRESS TE = 30 ms simulation reconstituted for use as a basis function. The ideal area, frequency and phase values of all transition lines in the simulation have global lineshape, phase and B0 shift applied and are summed to create the representation of a whole metabolite.

Cardinal data led us to the basis function polynomial data led to global approximation properties of the limit curve. 

If a FEM is to be used instead, we need to replace the global basis functions with element basis functions. In this case, we need to discretize the domain. In general we may discretize z = [0, L] into K + I elements with K internal node points in addition to the boundaries. 

The expansion of four-component one-electron functions into a set of global basis functions can be done in several ways independent of the particular choice of the type of the basis functions. For instance, four independent expansions may be used for the four components. However, we might also relate the expansion coefficients of the four components to each other. In contrast to these expansions, the molecular spinors can also be expressed in terms of 2-spinor expansions. This latter ansatz appears to be quite common and will be described in greater detail now. Obviously, analogous thoughts also apply for the first two possibilities (for a detailed discussion compare Dyall etal. 1991a). 

A A th single-particle wavefunction in which the global phase, 6, on the two basis functions is the same. 

Another method of refinement under consideration is the inclusion of a global square top-hat basis function in the FDDI basis set. The size, location, and orientation of the square basis function in the measurement domain are extremely important. Work with phantom data has illustrated the sensitivity of the FDDI reconstruction to changes in the relative positions and orientations of the scalar field being reconstructed and the basis function (Fig. 2.4). 

Figure 2.5 (a) Adaptive FDDI reconstruction of phantom square jet data (6) adaptive FDDI reconstruction utilizing the new global basis function in (c), notice that while the structure on top of the peak has become more complex, the sides more closely approximate that of a square top-hat and (c) normalized globbasis function constructed from smoothing the adaptive FDDI reconstruction in (a). 

A third refinement method includes the development of a global basis function specific to each reconstruction that could be added to the current basis function set. This global basis function can be developed based on preliminary results of an adaptive FDDI reconstruction of the scalar field and included in a subsequent implementation of adaptive FDDI (Fig. 2.5). 

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