分段对称反向高斯-约当消元法及其应用

求解变系数方程的高斯消元法与高斯-约当消元法计算原理类似、问题相近,但前者计算速度高于后者.提出分段对称反向高斯-约当消元法,其中包括根据系数矩阵结构特点构成特殊增广阵,以展示和应用元素的变化规律,并分段对上下三角元素消元以大大提高计算效率.对矩阵下三角元素正向消元及对称计算可简化所有下三角元素计算,而对上三角元素反向消元可再省略所有上三角元素计算,而取倒后的对角元素作为规格化因子可大大减少除法计算.根据单位矩阵结构特点,对其规格化或对系数矩阵上下三角元素消元时均仅计算部分对角元素和下三角元素可进一步提高计算效率.所有元素均用四角规则计算而无需计算公式以简化计算和编程.新方法大大减少了高斯-约当消元法中元素的计算,且原理简单、易于编程,可快速求解各种变系数方程,还可利用元素对称性求解常系数的节点阻抗矩阵.与高斯消元法和高斯-约当消元法相比,新方法计算速度大大提高.     

扇束CT极坐标反投影重建算法的对称优化

为了提高扇束滤波反投影(FBP)算法重建图像的速度,提出一种极坐标反投影算法的优化快速重建方法。算法利用三角函数对称性对多幅预处理后的投影数据同时进行极坐标反投影运算;在反投影数据坐标转换时运用像素位置参数的对称性,以减少双线性插值的计算量。实验结果表明,在不牺牲重建图像质量前提下,与传统卷积反投影重建算法相比,优化算法的重建速度提高8倍以上。该优化方法也适应于三维锥束重建,并可推广到多层螺旋三维重建。     

改进牛顿法大规模电力系统潮流计算

电网互联导致电力系统规模不断扩大,对牛顿法进行潮流计算提出了更高的要求。探讨5种改进牛顿法应用到大规模电力系统潮流计算中。经IEEE 300、Poland多个互联的大规模电力系统共6个算例分析表明,算法1和算法2改善了初值范围,同样的迭代次数下,收敛精度较经典牛顿法高,但计算时间较经典的牛顿法并未明显提高;算法3和算法4提高潮流计算的速度和收敛精度。经UCTE 1254病态系统测试,算法3较算法5能高效地处理病态潮流问题,因而更适合于大规模电力系统潮流计算。     

关系代数中除法运算的表示

本文主要讨论了用基本运算表示除法运算的关系代数表达式.用图形法作为分析工具,给出了关系代数中除法运算的查询原理,解决了以往用抽象定义进行除法运算的难度,结合实例提出了讲授这些知识点的一些新思路和新方法,取得了较好的教学效果.     

不需模指数运算的数字签名批验证新方法

给出了一种数字签名的批验证新方法,特别是它并不需要模指数运算,而只需要除法运算,因而在计算中会节省很多工作量,解决了大的模指数运算问题。     

在FPGA中实现高精度快速除法

介绍FPGA中高精度除法运算的实现方法,给出实现高精度除法运算的VHDL源程序;实现了除数为任意八位二进制的除法,其精度可达到小数点后16位.     

用于NAND Flash的长BCH编码快速算法

为满足大容量NAND Flash的容错需求,解决传统BCH编码存在长码字编码效率低下的问题,提出一种长BCH编码的快速算法.算法利用分圆陪集和中国剩余定理,在确定生成多项式时,由每个最小多项式的根构造分圆陪集,避免了重复计算所有的根;采用等价多项式代替除法多项式,将计算的最小多项式和理想循环码的生成元加入分圆陪集,后续编码可通过查找分圆陪集得到等价余数多项式,无须每次都进行除法运算,减少了除法运算时间.实验结果表明,与传统BCH编码算法和相关算法相比,该算法在长BCH编码时具有较高的编码效率,特别是对极长BCH编码,效果更加明显.     

面向公钥密码体系的大数相除快速算法

模运算是公钥密码学的一种基本运算。做模运算前提需要做除法运算,因此除法运算也是密码学的基本运算。大整数除法的运算速度是影响公钥密码体系中效率的关键因素。针对大数相除问题,提出大数相除的快速改进算法,其基本思想是,以空间换取时间。首先,通过建立预处理表,减少试除法中大数乘法的次数,从而高效快速得出商值;然后,运用窗口滑动方法来提高大数减法的速度。实验结果表明,该算法可以提高密码学算法的运算效率。算法时间复杂度为O(n),空间复杂度为O(n)。     

放射形配网潮流计算的一种新的牛顿法

前推回推法是放射形配网潮流计算最基本的算法.通过对前推回推法求解过程的数学演化,导出一种新的牛顿类型的算法及其雅可比矩阵直接分解公式.利用比较原理,间接证明该算法是一种具有超线性收敛性的近似牛顿法.与经典牛顿法相比,该算法无须计算雅可比矩阵、无须三角因子分解等过程,直接由前代/回代或回代/前代过程就能完成;与前推回推法相比,该算法无须特定的节点和支路编号过程.文中以一个实际的中等规模配电系统为例,分析、比较前推回推法、导出的近似牛顿法、经典牛顿法等的收敛性和计算速度,证实上述研究结论.     

一种改进的SVPWM算法研究

针对二极管箝位三电平逆变器的研究,传统的基于直角坐标系下的空间矢量脉宽调制算法(SVPWM)需要大量的三角函数运算,增大了运算量,难以进行数字实现.为了提高实时性,改进算法,研究了一种基于斜角坐标系下的SVPWM改进算法,核心是通过Clarke-Park联合变换将基于三相静止坐标系下的三相电压转换为基于非正交60°斜角坐标系下的两相电压,使得电压参考矢量在进行扇区判断和开关矢量作用时间的计算上都避免了三角函数计算,只有普通的四则运算,降低了运算量.经过仿真,结果表明改进算法的有效,并且在运算时间上优于传统算法,易于实现.     

Biharmonic Coordinates

Barycentric coordinates are an established mathematical tool in computer graphics and geometry processing, providing a convenient way of interpolating scalar or vector data from the boundary of a planar domain to its interior. Many different recipes for barycentric coordinates exist, some offering the convenience of a closed‐form expression, some providing other desirable properties at the expense of longer computation times. For example, harmonic coordinates, which are solutions to the Laplace equation, provide a long list of desirable properties (making them suitable for a wide range of applications), but lack a closed‐form expression. We derive a new type of barycentric coordinates based on solutions to the biharmonic equation. These coordinates can be considered a natural generalization of harmonic coordinates, with the additional ability to interpolate boundary derivative data. We provide an efficient and accurate way to numerically compute the biharmonic coordinates and demonstrate their advantages over existing schemes. We show that biharmonic coordinates are especially appealing for (but not limited to) 2D shape and image deformation and have clear advantages over existing deformation methods.     

Maximum Likelihood Coordinates

Any point inside a d-dimensional simplex can be expressed in a unique way as a convex combination of the simplex's vertices, and the coefficients of this combination are called the barycentric coordinates of the point. The idea of barycentric coordinates extends to general polytopes with n vertices, but they are no longer unique if n > d+1. Several constructions of such generalized barycentric coordinates have been proposed, in particular for polygons and polyhedra, but most approaches cannot guarantee the non-negativity of the coordinates, which is important for applications like image warping and mesh deformation. We present a novel construction of non-negative and smooth generalized barycentric coordinates for arbitrary simple polygons, which extends to higher dimensions and can include isolated interior points. Our approach is inspired by maximum entropy coordinates, as it also uses a statistical model to define coordinates for convex polygons, but our generalization to non-convex shapes is different and based instead on the project-and-smooth idea of iterative coordinates. We show that our coordinates and their gradients can be evaluated efficiently and provide several examples that illustrate their advantages over previous constructions.     

Illustrative Parallel Coordinates

Illustrative parallel coordinates (IPC) is a suite of artistic rendering techniques for augmenting and improving parallel coordinate (PC) visualizations. IPC techniques can be used to convey a large amount of information about a multidimensional dataset in a small area of the screen through the following approaches: (a) edge‐bundling through splines; (b) visualization of “branched ” clusters to reveal the distribution of the data; (c) opacity‐based hints to show cluster density; (d) opacity and shading effects to illustrate local line density on the parallel axes; and (e) silhouettes, shadows and halos to help the eye distinguish between overlapping clusters. Thus, the primary goal of this work is to convey as much information as possible in a manner that is aesthetically pleasing and easy to understand for non‐experts.     

矩阵重心坐标

在二维重心坐标——复数重心坐标的基础上引入二维矩阵重心坐标的概念,并利用球面坐标将二维矩阵重心坐标推广到三维.三维矩阵重心坐标适用于三角控制网格、四边形控制网格甚至一般的混合控制网格.对所提出的重心坐标性质进行了研究,发现其满足大部分好的重心坐标所应具有的性质.最后对矩阵重心坐标在三维网格模型中的应用进行了细致的实验,分析了它的优缺点.     

Moving Least Squares Coordinates

We propose a new family of barycentric coordinates that have closed‐forms for arbitrary 2D polygons. These coordinates are easy to compute and have linear precision even for open polygons. Not only do these coordinates have linear precision, but we can create coordinates that reproduce polynomials of a set degree m as long as degree m polynomials are specified along the boundary of the polygon. We also show how to extend these coordinates to interpolate derivatives specified on the boundary.     

Barycentric Coordinates on Surfaces

This paper introduces a method for defining and efficiently computing barycentric coordinates with respect to polygons on general surfaces. Our construction is geared towards injective polygons (polygons that can be enclosed in a metric ball of an appropriate size) and is based on replacing the linear precision property of planar coordinates by a requirement in terms of center of mass, and generalizing this requirement to the surface setting. We show that the resulting surface barycentric coordinates can be computed using planar barycentric coordinates with respect to a polygon in the tangent plane. We prove theoretically that the surface coordinates properly generalize the planar coordinates and carry some of their useful properties such as unique reconstruction of a point given its coordinates, uniqueness for triangles, edge linearity, similarity invariance, and smoothness; in addition, these coordinates are insensitive to isometric deformations and can be used to reconstruct isometries. We show empirically that surface coordinates are shape‐aware with consistent gross behavior across different surfaces, are well‐behaved for different polygon types/locations on variety of surface forms, and that they are fast to compute. Finally, we demonstrate effectiveness of surface coordinates for interpolation, decal mapping, and correspondence refinement.     

Higher Order Barycentric Coordinates

In recent years, a wide range of generalized barycentric coordinates has been suggested. However, all of them lack control over derivatives. We show how the notion of barycentric coordinates can be extended to specify derivatives at control points. This is also known as Hermite interpolation. We introduce a method to modify existing barycentric coordinates to higher order barycentric coordinates and demonstrate, using higher order mean value coordinates, that our method, although conceptually simple and easy to implement, can be used to give easy and intuitive control at interactive frame rates over local space deformations such as rotations.     

Features in Continuous Parallel Coordinates

Continuous Parallel Coordinates (CPC) are a contemporary visualization technique in order to combine several scalar fields, given over a common domain. They facilitate a continuous view for parallel coordinates by considering a smooth scalar field instead of a finite number of straight lines. We show that there are feature curves in CPC which appear to be the dominant structures of a CPC. We present methods to extract and classify them and demonstrate their usefulness to enhance the visualization of CPCs. In particular, we show that these feature curves are related to discontinuities in Continuous Scatterplots (CSP). We show this by exploiting a curve-curve duality between parallel and Cartesian coordinates, which is a generalization of the well-known point-line duality. Furthermore, we illustrate the theoretical considerations. Concluding, we discuss relations and aspects of the CPC's/CSP's features concerning the data analysis.     

Visual Clustering in Parallel Coordinates

Parallel coordinates have been widely applied to visualize high‐dimensional and multivariate data, discerning patterns within the data through visual clustering. However, the effectiveness of this technique on large data is reduced by edge clutter. In this paper, we present a novel framework to reduce edge clutter, consequently improving the effectiveness of visual clustering. We exploit curved edges and optimize the arrangement of these curved edges by minimizing their curvature and maximizing the parallelism of adjacent edges. The overall visual clustering is improved by adjusting the shape of the edges while keeping their relative order. The experiments on several representative datasets demonstrate the effectiveness of our approach.     

Scattering Points in Parallel Coordinates

In this paper, we present a novel parallel coordinates design integrated with points (scattering points in parallel coordinates, SPPC), by taking advantage of both parallel coordinates and scatterplots. Different from most multiple views visualization frameworks involving parallel coordinates where each visualization type occupies an individual window, we convert two selected neighboring coordinate axes into a scatterplot directly. Multidimensional scaling is adopted to allow converting multiple axes into a single subplot. The transition between two visual types is designed in a seamless way. In our work, a series of interaction tools has been developed. Uniform brushing functionality is implemented to allow the user to perform data selection on both points and parallel coordinate polylines without explicitly switching tools. A GPU accelerated dimensional incremental multidimensional scaling (DIMDS) has been developed to significantly improve the system performance. Our case study shows that our scheme is more efficient than traditional multi-view methods in performing visual analysis tasks.