基于拟插值理论的三维体数据场高精度重建算法

三维体数据场重建是体数据建模与分析的一个非常基础和重要的步骤,重建算法的精度直接决定了数据重采样和后续分析的准确性.文中发展了基于拟插值方法的三维体数据场重建理论,首次提出一种充分利用体数据自身信息与核函数优化相结合的预处理算法.在合理假设的前提下得到重建误差的表述式;然后充分利用三维体数据自身的信息全局优化该误差,使得重建误差在L2意义下达到最小.最后通过丰富的重建实例,显示了文中算法能有效地提高重建精度,并能够更好地保持三维体数据的高频信息.     

Applying multiquadric quasi-interpolation for boundary detection

In this paper, we propose a novel scheme for simulating geometric active contours (geometric flow) of one kind, applying multiquadric (MQ) quasi-interpolation. We first represent the geometric flow in its parametric form. Then we obtain the numerical scheme by using the derivatives of the quasi-interpolation to approximate the spatial derivative of each dependent variable and a forward difference to approximate the temporal derivative of each dependent variable. The resulting scheme is simple, efficient and easy to implement. Also images with complex boundaries can be more easily proposed on the basis of the good properties of the MQ quasi-interpolation. Several biomedical and astronomical examples of applications are shown in the paper. Comparisons with other methods are included to illustrate the validity of the method.     

多层积分值三次样条拟插值

目的 在实际问题中,某些插值问题结点处的函数值往往是未知的,而仅仅知道一些连续等距区间上的积分值。为此提出了一种基于未知函数在连续等距区间上的积分值和多层样条拟插值技术来解决函数重构。该方法称之为多层积分值三次样条拟插值方法。方法 首先,利用积分值的线性组合来逼近结点处的函数值;然后,利用传统的三次B-样条拟插值和相应的误差函数来实现多层三次样条拟插值;最后,给出两层积分值三次样条拟插值算子的多项式再生性和误差估计。结果 选取无穷次可微函数对多层积分值三次样条拟插值方法和已有的积分值三次样条拟插值方法进行对比分析。数值实验印证了本文方法在逼近误差和数值收敛阶均稍占优。结论本文多层三次样条拟插值函数能够在整体上很好的逼近原始函数,一阶和二阶导函数。本文方法较之于已有的积分值三次样条拟插值方法具有更好的逼近误差和数值收敛阶。该方法对连续等距区间上积分值的函数重构具有普适性。     

An adaptive IgA-BEM with hierarchical B-splines based on quasi-interpolation quadrature schemes

The isogeometric formulation of the boundary element method (IgA-BEM) is investigated within the adaptivity framework. Suitable weighted quadrature rules to evaluate integrals appearing in the Galerkin BEM formulation of 2D Laplace model problems are introduced. The proposed quadrature schemes are based on a spline quasi-interpolation (QI) operator and properly framed in the hierarchical setting. The local nature of the QI perfectly fits with hierarchical spline constructions and leads to an efficient and accurate numerical scheme. An automatic adaptive refinement strategy is driven by a residual-based error estimator. Numerical examples show that the optimal convergence rate of the Galerkin solution is recovered by the proposed adaptive method.     

一种新的Multiquadric拟插值

Multiquadric（MQ）是由Hardy提出来的一种径向基函数,至今它已在大地测量学、微分方程数值解等许多方面得到了应用。目前已知的multiquadric拟插值有4种,即,Beatson和Powell的LA、LB和LC以及Wu和Schaback的LD,其中,LB是常数再生的,LC和LD是线性再生的。该文首先给出形如LD的拟插值线性再生时其基函数应具有的性质,然后构造了一种具有线性再生性和保单调性的multiquadric拟插值,并对其逼近误差进行了理论分析。最后,通过两个实例进行数值实验,从算例的结果来看,该拟插值具有良好的逼近精度。     

Multivariate quasi-interpolation in L p (ℝ d ) with radial basis functions for scattered data

In this paper, quasi-interpolation for scattered data was studied. On the basis of generalized quasi-interpolation for scattered data proposed in [Z.M. Wu and J.P. Liu, Generalized strang-fix condition for scattered data quasi-interpolation, Adv. Comput. Math. 23 (2005), pp. 201–214.], we have developed a new method to construct the kernel in the scheme by the linear combination of the scales, instead of the gridded shifts of the radial basis function. Compared with the kernel proposed in [Z.M. Wu and J.P. Liu, Generalized strang-fix condition for scattered data quasi-interpolation, Adv. Comput. Math. 23 (2005), pp. 201–214.], the new kernel, which is still a radial function, possesses the feature of polynomial reproducing property. This opens a possibility for us to propose a different technique by obtaining a higher approximation order of the convergence.     

某型拟插值的多项式再生性

研究某种特定形式的拟插值满足多项式再生性的充分必要条件。给出了几个定理,分别描述了常数再生、线性再生、二次多项式再生乃至一般的k次多项式再生的充分必要条件,并通过这些定理推出：L_A不是常数再生的;L_B是常数再生的但不是线性再生的;L_D是线性再生的但不是二次多项式再生的。     

一种新的Multiquadric拟插值

Multiquadric(MQ)是由Hardy提出来的一种径向基函数,至今它已在大地测量学、微分方程数值解等许多方面得到了应用。目前已知的multiquadric拟插值有4种,即,Beatson和Powell的LA、LB和LC以及Wu和Schaback的LD,其中,LB是常数再生的, LC和LD是线性再生的。该文首先给出形如LD的拟插值线性再生时其基函数应具有的性质,然后构造了一种具有线性再生性和保单调性的multiquadric拟插值,并对其逼近误差进行了理论分析。最后,通过两个实例进行数值实验,从算例的结果来看,该拟插值具有良好的逼近精度。