2D Vector field approximation using linear neighborhoods

We present a vector field approximation for two-dimensional vector fields that preserves their topology and significantly reduces the memory footprint. This approximation is based on a segmentation. The flow within each segmentation region is approximated by an affine linear function. The implementation is driven by four aims: (1) the approximation preserves the original topology; (2) the maximal approximation error is below a user-defined threshold in all regions; (3) the number of regions is as small as possible; and (4) each point has the minimal approximation error. The generation of an optimal solution is computationally infeasible. We discuss this problem and provide a greedy strategy to efficiently compute a sensible segmentation that considers the four aims. Finally, we use the region-wise affine linear approximation to compute a simplified grid for the vector field.     

非结构矢量场的特征区域可视化方法

基于特征比对的可视化方法涉及的问题主要包括,如何判断一个矢量场中是否存在另一个矢量场中的一些特定结构,如何评价两个矢量场结构间存在的相似性,如何从诸多矢量场中将这些特定的结构检测,定位并绘制出来.给出了一种基于曲线结构的矢量场特征区域可视化方法,该方法通过构造与曲线结构相关联的柱坐标系,保证了提取特征的平移、旋转和尺度不变性;采用曲线积分的Clifford Fourier变换,获取了曲线结构的频域特征并简化计算;利用多曲线结构匹配和多尺度逼近等方法,实现了区域结构的特征可视化.     

Fast combinatorial vector field topology

This paper introduces a novel approximation algorithm for the fundamental graph problem of combinatorial vector field topology (CVT). CVT is a combinatorial approach based on a sound theoretical basis given by Forman's work on a discrete Morse theory for dynamical systems. A computational framework for this mathematical model of vector field topology has been developed recently. The applicability of this framework is however severely limited by the quadratic complexity of its main computational kernel. In this work, we present an approximation algorithm for CVT with a significantly lower complexity. This new algorithm reduces the runtime by several orders of magnitude and maintains the main advantages of CVT over the continuous approach. Due to the simplicity of our algorithm it can be easily parallelized to improve the runtime further.     

Theory and Applications of Clifford Support Vector Machines

This paper introduces the Clifford Support Vector Machines (CSVM) as a generalization of the real- and complex-valued Support Vector Machines using the Clifford geometric algebra. In this framework we handle the design of kernels involving the Clifford or geometric product for linear and nonlinear classification and regression. The major advantage of our approach is that we redefine the optimization variables as multivectors. This allows us to have a multivector as output therefore we can represent multiple classes according to the dimension of the geometric algebra in which we work. We conduct comparisons between CSVM and the most used approaches to solve multi-class classification to show that our approach is more suitable for practical use on certain type of multi-class classification problems.     

Geometric neural computing

This paper shows the analysis and design of feedforward neural networks using the coordinate-free system of Clifford or geometric algebra. It is shown that real-, complex-, and quaternion-valued neural networks are simply particular cases of the geometric algebra multidimensional neural networks and that some of them can also be generated using support multivector machines (SMVMs). Particularly, the generation of radial basis function for neurocomputing in geometric algebra is easier using the SMVM, which allows one to find automatically the optimal parameters. The use of support vector machines in the geometric algebra framework expands its sphere of applicability for multidimensional learning. Interesting examples of nonlinear problems show the effect of the use of an adequate Clifford geometric algebra which alleviate the training of neural networks and that of SMVMs.     

Motor Algebra for 3D Kinematics: The Case of the Hand-Eye Calibration

In this paper we apply the Clifford geometric algebra for solving problems of visually guided robotics. In particular, using the algebra of motors we model the 3D rigid motion transformation of points, lines and planes useful for computer vision and robotics. The effectiveness of the Clifford algebra representation is illustrated by the example of the hand-eye calibration. It is shown that the problem of the hand-eye calibration is equivalent to the estimation of motion of lines. The authors developed a new linear algorithm which estimates simultaneously translation and rotation as components of rigid motion.     

结合免疫优化和LS-SVRM观测器的非线性系统自适应鲁棒控制

针对一类单输入单输出不确定非线性控制系统提出了一种自适应鲁棒控制算法. 由于最小均方支持向量回归机(LS-SVRM)的最终解可以化为一个具有线性约束的二次规划问题, 不存在局部极小, 所以该算法在不要求假设系统的状态向量是可测的条件下通过设计基于LS-SVRM的观测器来估计系统的状态向量; 同时在算法中假设LS-SVRM的最优逼近参数向量和标称参数向量之差的范数和逼近误差的界限是未知的, 因此可通过对未知界限估计的调节来提高系统的鲁棒性. 考虑到LS-SVRM本身参数对LS-SVRM性能的影响, 本文应用一种新的免疫优化算法对LS-SVRM的参数进行优化, 从而提高LS-SVRM的逼近能力. 理论研究和仿真例子证实了所提方法的可行性和有效性.     

Topology Aware Stream Surfaces

We present an algorithm that allows stream surfaces to recognize and adapt to vector field topology. Standard stream surface algorithms either refine the surface uncontrolled near critical points which slows down the computation considerably and may lead to a poor surface approximation. Alternatively, the concerned region is omitted from the stream surface by severing it into two parts thus generating an incomplete stream surface. Our algorithm utilizes topological information to provide a fast, accurate, and complete triangulation of the stream surface near critical points. The required topological information is calculated in a preprocessing step. We compare our algorithm against the standard approach both visually and in performance.     

On the curves that may be approached by trajectories of a smooth control affine system

In this paper, we give a characterization of the set of curves that may be approached by trajectories of a smooth control-affine nonlinear system, in the topology of uniform convergence. This characterization is in terms of the drift vector field and the distribution spanned by the Lie algebra generated by the control vector fields. The characterization is valid on open sets where this distribution has constant rank. These results also characterize the support of diffusion processes with smooth coefficients.     

Compression of 2D Vector Fields Under Guaranteed Topology Preservation

In this paper we introduce a new compression technique for 2D vector fields which preserves the complete topology, i.e., the critical points and the connectivity of the separatrices. As the theoretical foundation of the algorithm, we show in a theorem that for local modifications of a vector field, it is possible to decide entirely by a local analysis whether or not the global topology is preserved. This result is applied in a compression algorithm which is based on a repeated local modification of the vector field ‐ namely a repeated edge collapse of the underlying piecewise linear domain. We apply the compression technique to a number of data sets with a complex topology and obtain significantly improved compression ratios in comparison to pre‐existing topology‐preserving techniques.     

Fingerprint ridge orientation field reconstruction using the best quadratic approximation by orthogonal polynomials in two discrete variables

This paper proposes a novel algorithm for reconstructing the fingerprint orientation field (FOF). The basic idea of the algorithm is to reconstruct the ridge orientation by using the best quadratic approximation by orthogonal polynomials in two discrete variables. We first estimate the local region orientation by the linear projection analysis (LPA) based on the vector set of point gradients, and then reconstruct the ridge orientation field using the best quadratic approximation by orthogonal polynomials in two discrete variables in the sine domain. In this way, we solve the problem that is difficult to accurately extract low quality fingerprint image orientation fields. The experiments with the database of FVC 2004 show that, compared to the state-of-the-art fingerprint orientation estimation algorithms, the proposed method is more accurate and more robust against noise, and is able to better estimate the FOF of low quality fingerprint images with large areas of noise.     

多光谱图像Clifford拟微分算子及应用

采用Clifford代数理论处理多光谱图像不同于传统代数的处理方法,其可有效利用光谱层之间的联系信息.本文通过分析Clifford微分,提出了多光谱图像的Clifford拟微分算子,丰富了多光谱图像处理的Clifford代数理论体系.在边缘识别仿真实验中,与最大熵法相比较,本文提出的以Clifford拟微分算子理论为核心的边缘检测算法识别模糊边界能力更强.在医学CT影像辅助诊断边缘检测实验中,新算法能有效凸显病灶,提高了诊疗准确性和诊断效率.     

三维矢量场的流动拓扑结构抽取

提出一种基于关键点分类的三维矢量场流动拓扑结构抽取算法,可应用于三维曲线网格、结构化网格和分块网格中.在许多计算流体力学计算中,存在非滑移边界,这种边界上流体的速度为0.通过分析流场边界的表面摩擦场的拓扑,展示绕壁面流体的流动结构;使用图标定位关键点,可交互式地标记和显示涡核区域,并通过选择暗示螺旋流动的图标,沿着该关键点的实特征值对应的特征矢量方向积分流线来完成.测试结果清晰地展示了关键特征区域的流体流动特征.     

Topology preserving top-down compression of 2D vector fields using bintree and triangular quadtrees

We present a hierarchical top-down refinement algorithm for compressing 2D vector fields that preserves topology. Our approach is to reconstruct the data set using adaptive refinement that considers topology. The algorithms start with little data and subdivide regions that are most likely to reconstruct the original topology of the given data set. We use two different refinement techniques. The first technique uses bintree subdivision and linear interpolation. The second algorithm is driven by triangular quadtree subdivision with Coons patch quadratic interpolation. We employ local error metrics to measure the quality of compression and as a global metric we compute Earth Mover's Distance (EMD) to measure the deviation from the original topology. Experiments with both analytic and simulated data sets are presented. Results indicate that one can obtain significant compression with low errors without losing topological information. Advantages and disadvantages of different topology preserving compression algorithms are also discussed in the paper.     

Successive linearizations of second order multidimensional time-invariant systems

The present paper discusses a linearization method for second order multidimensional time-invariant systems. The method approximates locally the nonlinear vector field around the equilibrium, where the solution starting from a given initial state near the equilibrium is approximated by a linear solution. This linearization is usually called local trajectory-based linearization. The approximation is computed using an iterative method, which consists of successive approximations in the least square sense. Using a numerical example, it is shown that the linearized solutions exhibit good agreement with the nonlinear solutions.     

基于自适应切空间的MRI图像配准

微分同胚是一种光滑可逆的变换,在MRI图像配准中可以保证图像形变后的拓扑结构保持不变,同时避免出现不合理的物理现象。为了在空间变换中获得更合理的同胚映射,高维空间中数据的非线性结构被考虑,基于流形学习方法提出一种自适应切空间的MRI图像配准算法。首先,把MRI数据构造成对称正定（SPD）的协方差矩阵,然后形成李群;接着,利用样本点邻域的局部切空间来表示李群的几何结构的非线性;接下来,在流形上用自适应邻域选择的方法形成的线性子空间去逼近局部切空间,提高切空间的局部线性化程度,从而最大限度地保留流形的局部非线性结构,得到最优的同胚映射。仿真数据和临床数据的实验结果显示,与传统的非参数微分同胚配准算法相比,该算法在高维稠密形变场上获得更高的拓扑保持度,最终提高图像配准精度。     

Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

We present block algorithms and their implementation for the parallelization of sub-cubic Gaussian elimination on shared memory architectures. Contrarily to the classical cubic algorithms in parallel numerical linear algebra, we focus here on recursive algorithms and coarse grain parallelization. Indeed, sub-cubic matrix arithmetic can only be achieved through recursive algorithms making coarse grain block algorithms perform more efficiently than fine grain ones. This work is motivated by the design and implementation of dense linear algebra over a finite field, where fast matrix multiplication is used extensively and where costly modular reductions also advocate for coarse grain block decomposition. We incrementally build efficient kernels, for matrix multiplication first, then triangular system solving, on top of which a recursive PLUQ decomposition algorithm is built. We study the parallelization of these kernels using several algorithmic variants: either iterative or recursive and using different splitting strategies. Experiments show that recursive adaptive methods for matrix multiplication, hybrid recursive–iterative methods for triangular system solve and tile recursive versions of the PLUQ decomposition, together with various data mapping policies, provide the best performance on a 32 cores NUMA architecture. Overall, we show that the overhead of modular reductions is more than compensated by the fast linear algebra algorithms and that exact dense linear algebra matches the performance of full rank reference numerical software even in the presence of rank deficiencies.     

Smooth function approximation using neural networks

An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the approximation of smooth batch data containing the function's input, output, and possibly, gradient information. The training set is associated to the network adjustable parameters by nonlinear weight equations. The cascade structure of these equations reveals that they can be treated as sets of linear systems. Hence, the training process and the network approximation properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or approximate matching of input-output and/or gradient-based training sets. Their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques.     

线性及非线性一致性问题综述

本文综述多自主体系统线性及非线性一致性问题.首先,从自主体动态、通信拓扑和一致性协议这3个方面介绍一致性问题的基本问题.接着,从一致性问题的分析方法入手,对目前一致性问题的研究结果进行分类和讨论.然后阐释经典一致性协议是负的次梯度算法这一实质,并给出一种一致性协议的设计方法.最后列举了仍然需要解决的问题和未来的研究方向.