Applications of fuzzy sets in curve fitting

This paper presents several methods for functional approximation with variable-knot variable-degree splines, with variable-knot first order splines, which are relatively easy to find, as the intermediate input. By means of a fuzzy characteristic function determining how “sharp” the angle at each knot is (the higher the polynomial degree, the “sharper” its first order spline approximation bends), we can decide whether we can group certain adjacent segments together to be approximated by a single higher order polynomial segment. Some simulation experiments have also been done.     

Explicitly given pairs of dual frames with compactly supported generators and applications to irregular B-splines

We consider systems of functions appearing by letting a class of modulations act on a countable collection of functions. These systems correspond to shift-invariant systems, considered on the Fourier side. We provide sufficient conditions for the system to be a frame, as well as an explicit construction of a class of frames and associated duals. We use the result to construct frames based on B-splines with knot sequences satisfying a natural condition, as well as explicitly given duals.     

Convergence of local variational spline interpolation

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local variational spline increases. By piecing together these locally defined splines, one can obtain a very good C⁰ approximation of the global variational spline. Finally we generalize this idea to approximate global tensor product B-spline interpolatory surfaces.     

Alignment using genetic programming with causal trees for identification of protein functions

A hybrid evolutionary model is used to propose a hierarchical homology of protein sequences to identify protein functions systematically. The proposed model offers considerable potentials, considering the inconsistency of existing methods for predicting novel proteins. Because some novel proteins might align without meaningful conserved domains, maximizing the score of sequence alignment is not the best criterion for predicting protein functions. This work presents a decision model that can minimize the cost of making a decision for predicting protein functions using the hierarchical homologies. Particularly, the model has three characteristics: (i) it is a hybrid evolutionary model with multiple fitness functions that uses genetic programming to predict protein functions on a distantly related protein family, (ii) it incorporates modified robust point matching to accurately compare all feature points using the moment invariant and thin-plate spline theorems, and (iii) the hierarchical homologies holding up a novel protein sequence in the form of a causal tree can effectively demonstrate the relationship between proteins. This work describes the comparisons of nucleocapsid proteins from the putative polyprotein SARS virus and other coronaviruses in other hosts using the model.     

Interpolating Cubic Spline Wavelet Packet on Arbitrary Partitions

In many applications, the splines on an arbitrary partition are very useful. In this paper, a spline wavelet structure is created in the way that it provides a multiresolution approximation of the spline subspaces with arbitrary partition in the space of continuous functions on a finite interval. Based on the wavelet basis and the wavelet packet in this structure, a multi-level interpolation method is developed for decomposing a function into wavelet series and reconstructing it from its wavelet representation.     

A method of adding–removing knots for solving smoothing problems with obstacles

We study a method of adding–removing knots that has been proposed in the literature for solving the smoothing problem with obstacles. The method uses the coefficients of natural splines in the expansion by radial basis functions. We present examples of cycling and counterexamples to possible use of some ideas. We also give some sufficient conditions for finiteness of the method.     

Regularity and well-posedness of a dual program for convex best <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-spline interpolation

An efficient approach to computing the convex best C ¹-spline interpolant to a given set of data is to solve an associated dual program by standard numerical methods (e.g., Newton’s method). We study regularity and well-posedness of the dual program: two important issues that have been not yet well-addressed in the literature. Our regularity results characterize the case when the generalized Hessian of the objective function is positive definite. We also give sufficient conditions for the coerciveness of the objective function. These results together specify conditions when the dual program is well-posed and hence justify why Newton’s method is likely to be successful in practice. Examples are given to illustrate the obtained results. The work was supported by EPSRC grant EP/D502535/1 for the first author and by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 5141/01E) for the second author.     

实分片代数曲线的拓扑结构

The piecewise algebraic curve is a kind generalization of the classical algebraic curve.By analyzing the topology of real algebraic curves on the triangles,a practi-caUy algrithm for analyzing the topology of piecewise algebraic curves is given.The algrithm produces a planar graph which is topologically equivalent to the piecewise algebraic curve.     

Spline Smoothing on Surfaces

This article presents a method for estimating functions on topologically and/or geometrically complex surfaces from possibly noisy observations. Our approach is an extension of spline smoothing, using a finite element method. The article has a substantial tutorial component: we start by reviewing smoothness measures for functions defined on surfaces, simplicial surfaces and differentiable structures on such surfaces, subdivison functions, and subdivision surfaces. After describing our method, we show results of an experiment comparing finite element approximations to exact smoothing splines on the sphere, and we give examples suggesting that generalized cross-validation is an effective way of determining the optimal degree of smoothing for function estimation on surfaces.