Uniform B-splines

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. 

Uniform B-splines are much less sensitive to poor spacing of the control points, because the first derivative is given by a lower degree B-spline with control points the first differences of the control points. If the control points have first differences all more or less in the same direction, the curve cannot kink back on itself. 

The first derivatives can, of course, vary fairly wildly in magnitude if the first differences do, and there is a perfectly good theory of knot-insertion into non-uniform B-splines and this can indeed be expressed in terms of choosing the coefficients in a subdivision implementation. 

Strictly speaking these flow simply and naturally from a multiple knot at the end of the domain, and are understood best in terms of unequal interval (non-uniform) B-spline theory, but we can equally well approach them in an ad-hoc fashion, modifying the top left hand corner of the matrix, adding columns as well as rows. For schemes other than box-splines some element of ad-hoc design will be necessary. 

T. J.Cashman, N.A.Dodgson and M.A.Sabin Non-uniform B-spline subdivision using rehne and smooth. ppl21-137 in Mathematics of Surfaces XII, (eds Martin, Sabin and Winkler), Springer LNCS4647, 2007 ISBN 3-540-73842-8... 

The only requirement for the knot sequence is that it must be a non-decreasing sequence of numbers. When ti = ti+i it indicates a multiple knot and the segment Qj is reduced to a point. This is one of the great advantages with non-uniform B-splines since it offers great flexibility in the representation ol functions. For example [0, 0, 1, 1, 1, 1, 2, 3, 4, 4] is a valid sequence of knots. The knot value 0 has multiplicity of 2, knot value 1 has multiplicity of 4 and so on. The multiplicity is used to control the continuity of a point. The higher the multiplicity, the less smooth the spline function at this point becomes. A curve segment Qj in cubic B-splines is defined by four control points... 

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 space of binary uniform stationary schemes is therefore understandable in terms of two components one contains all primal schemes, with an odd number of entries in the mask, which is a linear combination of odd degree B-splines the other all dual schemes, with an even number of entries in the mask, which is a linear combination of even degree B-splines. In principle each is only a countably infinite-dimensional space, and in practice each is only a finite dimensional space because we shall not wish to include B-splines above some maximum degree in order to keep the support limited. 

Handling non-uniformity is a known research topic at the time of writing. In the case of B-splines, the end conditions can elegantly be treated as a specific case of non-uniformity. Can this be extended neatly to other subdivision... 

S.Schaefer and R.Goldman Non-uniform subdivision for B-splines of arbitrary degree. CAGD 26(1), pp75-81, 2009... 

To control the shape of the curve, control points or spline coefficients are used. For a uniform cubic parametric B-spline over a region i we have... 

The word uniform means that the distance between the individual knots is the same. For the B-splines to become a more versatile tool, it must relax its assumption about constant distance between the knots. Now the blending functions for the different segments will be different. 

When parameter intervals along the knot vector are equal, the B-spline curve is uniform-. 

A sculptured surface is obtained by interpolation with two-dimensional B-spline-functions. A B-spline surface is considered as a collection of surface patches and the whole surface is a mosaic of these patches linked together with proper continuity (Figure 9). Due to its computational efficiency a uniform bicubic B-spline surface has been implemented. The bicubic B-spline patch can be written in matrix form as... 

F. Mazzia, A. Sestini, D. Trigiante, The continuous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes, Applied Numerical Mathematics 59(3-4)-Special Issue, 723-738(2009). 

The result of polygon mesh optimization stage is a fully closed model (fig. 6) ready to generate NURBS (Non-uniform Rational B-Spline) curves or surfaces network. 

We will use uniform translates of the tensor product of cubic B-splines that are dilated according to a resolution parameter N. These translates constitute a basis for a finite dimensional subspace V of S. Since these basis functions have finite support the Gramian matrix B is sparse. Moreover, due to fundamental properties of cubic B-splines, V2 is included in V. 

The schema provides for regular conic curves (e.g. circle, ellipse, etc.), and quadric surfaces (e.g. cylinder, cone, etc.). Each of these has a specified parameterisation which must be adhered to when providing parameter values for other entities (e.g. trimmed curves). This is an attempt to exclude the idea of default parameterisation, the interpretation of which may differ from one implementation to another. In addition, parametric curves and surfaces are provided for by B-spline entities. These are specified by their control points, and may be rational or non-rational, uniform or non-uniform. Bezier representation of curves and surfaces is provided as a... 

B SPLINE CURVE rational uniform bezier degree... 

Fig. 11. Wavelet decomposition (a) dyadic sampling using Daubechies-6 wavelet (b) uniform sampling using cubic spline wavelet.

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