B-spline basis functions

Here the B-spline Bim(zf, Xj) is the ith B-spline basis function on the extended partition Xj (which contains locations of the knots in the Zj direction), and is a coefficient. We use cubic splines and sufficient numbers of uniformly spaced knots so that the estimation problem is not affected by the partition. The estimation problem now involves determining the set of B-spline coefficients that minimizes Eq. (4.1.26), subject to the state equations [Eqs. (4.1.24 and 4.1.25)], for a suitable value of the regularization parameter. At this point, the minimization problem corresponds to a nonlinear programming problem. 

The matrix M is determined by integrals of the products of the second derivatives of the B-spline basis functions used in the representation of the unknown distribution. This is a quadratic least-squares minimization problem. 

The quantitative method in Section 2.2 is used to determine the intrinsic magnetization intensity for each voxel. Cubic B-spline basis functions with a partition of 60 interior knots logarithmically spaced between 1 x 10 5 and 10 s are used to represent the relaxation distribution within each voxel. The optimal regularization parameter, A, of each voxel is found within the range between 1 x 10 5 and 5 x 10"18 s by using the UBPR9 criterion. 

If the pieces are all of equal length in abscissa, all the B-spline basis functions are just translates of the same basic function, which is typically called the basis function of the given degree. The resulting curves are then called equal interval B-splines or uniform B-splines. We shall use the shorter term B-splines here, despite the fact that it is not strictly accurate. 

The characterisation of B-splines that we shall use here is that the B-spline basis function of degree n has the following properties... 

Clearly the first derivative of the B-spline basis function must have the properties that... 

From these we see that it must be a linear combination of exactly two B-spline basis functions of degree n— 1, and in fact it is exactly the difference of two consecutive basis functions of degree n — 1. 

To evaluate the sums over virtual states in Elqs. (102) and (103), we make use of the B-spline basis functions described later in this section. 

N.(u) are the normalised B-spline basis functions defined on the knot set ... 

Stener and co-workers [59] used an alternative B-spline LCAO density functional theory (DFT) method in their PECD investigations [53, 57, 60-63]. In this approach a normal LCAO basis set is adapted for the continuum by the addition of B-spline radial functions. A large single center expansion of such... 

We employ a B-spline basis representation for the distribution function ... 

For shape design purposes it is usually far more convenient to use a basis where the basis functions sum to f. All the coefficients then transform as points, and we call them control points. For polynomials this can be achieved by using the Bezier basis, which is a special case of the B-spline basis which will be encountered shortly. 

Taking the radial wave functions and energies for states n from the B-spline basis set, we may easily carry out the double sums in (128). The partial-wave contributions to from terms in square bracket are listed in Table 2. These terms fall-off approximately as L for large L and may easily be extrapolated. We find E = —0.0373736 a.u., leading to a binding energy of -0.8990800 a.u., differing from experiment by 0.5%. 

To solve Eqs. (176-181), an angular momentum decomposition is first carried out and the equations are then reduced to coupled equations involving single-body radial wave functions only. The radial wave functions for states v, m, n, a, b, are taken from a B-spline basis set [36] and the resulting coupled radial equations are solved iteratively. The core equations (176-177) are solved first and the valence equations (179-181) are then solved for valence states of interest using the converged core amplitudes. 

The idea of using B-spline basis sets for the representation of vibrational molecular wave functions emerged rapidly. For a Morse potential and a two-dimensional Henon-Heiles potential, we have assessed the efficiency of the B-splines over the conventional DVR (discrete variable representation) with a sine or a Laguerre basis sets [50]. In addition, the discretization of the vibrational continuum of energy when using the Galerkin method allows the calculation of photodissociation cross-sections in a time-independent approach. 

A spline is a piecewise polynomial2 whose pieces meet with continuity as high as possible given the degree. The abscissa values at which consecutive pieces meet are called the knots. A B-spline is a spline expressed with respect to a particular basis, in which the basis functions are each non-zero over as small a number of consecutive pieces as possible, given the degree and the continuity, and the basis functions sum to unity. 

Higher degree B-splines can be constructed explicitly (using, for example, the Bezier basis for each span of the function) by recursion on degree, applying the same simple recipe ... 

The refined curve has a refined basis, and needs a corresponding sequence of control points. In fact it is the process of finding these new control points which is actually called knot insertion or subdivision. If we ignore end-conditions, which is a sensible way to start, there are twice as many of them. We can determine them by looking at the way in which a coarse basis function can be expressed in terms of the refined ones. Let the coarse basis be and the finer one 5"(t) where the subscripts r indicate the position in abscissa space of the central maximum4 of the particular basis function, and the superscript n the degree of the functions. Consider first B-splines of degree zero. 

For degree one B-splines, we have the relation between the new and old basis functions that... 

By looking at the extent of influence of one control point after 0,l,2,oo refinements, in the cubic B-spline scheme we can see that the refined polygons converge towards the basis function, and the last non-zero entry converges towards the end of the support region. 

In the cubic B-spline scheme each basis function has four non-zero spans, and thence each span is influenced by 4 original control points. 

However, the support is dominated by the effect of the first few refinement steps, the continuity by the last few, and so there is the possibility of using a small mask for the first few steps and a larger one later. This is elegantly exemplified by the so-called UP function32, which is the basis function of a scheme in which at the first step the coefficients used are those of the zero degree B-spline, at the second those of the linear B-spline and, in general, at the nth step those of the degree n — 1 B-spline. 

The first variant to look at is therefore to start the process slightly further up the chain of derivatives. If we start at the linear B-spline we get a basis function of support 3, if we start at the quadratic we get one of support 4, and so on. If we call the original UP function UP0, we can call the others UPi, UP2 etc. 

Just as the Holder continuity can be measured exactly at rational points with a finite computation, but only bounds can be found for the curve as a whole, the Fourier decay rate can be measured exactly at rational frequencies with a finite computation, but only bounds found for the spectrum as a whole. The two measures turn out to be identical for the B-splines, but are typically different (by a small amount) for other basis functions. Both are changed by the same amount for every additional a factor in the mask. 

In voltammetry, the spline wavelet was chosen as the major wavelet function for data de-noising. The function has been applied successfully to analyse voltammetric data since 1994 by Lu and Mo [9]. Mo and his co-workers have published more than fifteen papers on this topic in various journals. The spline wavelet is different from the Daubechies wavelet functions. Mathematically, the mth order basis spline (B-spline) wavelet, Nm, is defined recursively by convolution of the Harr wavelet function as follows [10] ... 

As stated in the previous section, most workers confine their wavelet functions in the Daubechies wavelet series only. For example, we have adopted the Daubechies wavelet function to denoise spectral data from a UV-VIS spectrophotometer [43]. In order to make use of the other available wavelet functions for chemical data analysis, Lu and Mo [44] suggested employing spline wavelets in their work for denoising UV-VIS spectra. The spline wavelet is another commonly used wavelet function in chemical studies. This function has been applied successfully in processing electrochemical signals [9,10] which will be discussed in detail in another chapter of this book. The mth order basis spline (B-spline) wavelet, Nm. is defined as follows [44] ... 

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