Splines and multiresolution are two independent concepts, which – considered together – yield a vast variety of bases for image processing and image analysis. The idea of a multiresolution analysis is to construct a ladder of nested spaces that operate as some sort of mathematical looking glass. It allows to separate coarse parts in a signal or in an image from the details of various sizes. Spl...

In order to define the full set of B-splines required, eight additional knots 1; 2; 3; 4 and nþ4; nþ5; nþ6; nþ7 are inserted automatically by the routine. The first four of these are set equal to the smallest xr and the last four to the largest xr. The representation of sðxÞ in terms of B-splines is the most compact form possible in that only nþ 3 coefficients, in addition to the nþ 7 knots, fu...

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two–scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spli...

This paper approximates simulation models by B-splines with a penalty on high-order finite differences of the coefficients of adjacent B-splines. The penalty prevents overfitting. The simulation output is assumed to be nonnegative. The nonnegative spline simulation metamodel is casted as a second-order cone programming problem, which can be solved efficiently by modern optimization techniques. ...

The static polarizability of the 1s, 2p, and 3d states of the hydrogen atom has been calculated using the B-splines basis set method, and the portion of the polarizability without poles has been calculated using the B-splines Galerkin method. Excellent agreement with exact values has been obtained. The current algorithm improves the efficiency and accuracy over the finite perturbation method. I...

This paper approximates simulation models by B-splines with a penalty on highorder finite differences of the coefficients of adjacent B-splines. The penalty prevents overfitting. The simulation output is assumed to be nonnegative. The nonnegative spline simulation metamodel is casted as a second-order cone programming model, which can be solved efficiently by modern optimization techniques. The...

This paper presents a short, simple, and general proof showing that the control polygons generated by subdivision and degree elevation converge to the underlying splines, box-splines, or multivariate B~zier polynomials, respectively. The proof is based only on a Taylor expansion. Then the results are carried over to rational curves and surfaces. Finally, an even shorter but as simple proof is p...

In this paper we develop a novel image processing technique to register two dimensional temporal mammograms for effective diagnosis and therapy. Our registration framework is founded upon triangular B-spline finite element method (TBFEM). In contrast to tensor-product B-splines, which is widely used in medical imaging, triangular B-splines are much more powerful, associated with many desirable ...