The Two-Dimensional Clifford-Fourier Transform

Recently several generalizations to higher dimension of the Fourier transform using Clifford algebra have been introduced, including the Clifford-Fourier transform by the authors, defined as an operator exponential with a Clifford algebra-valued kernel. In this paper an overview is given of all these generalizations and an in depth study of the two-dimensional Clifford-Fourier transform of the authors is presented. In this special two-dimensional case a closed form for the integral kernel may be obtained, leading to further properties, both in the L ₁ and in the L ₂ context. Furthermore, based on this Clifford-Fourier transform Clifford-Gabor filters are introduced. AMS subject classification numbers: 42B10, 30G35 Fred Brackx received a diploma degree in mathematics from Ghent University, Belgium, in 1970 and a Ph.D. degree in mathematics from the same university in 1973. Since 1984 he is professor for mathematical analysis at Ghent University and currently he is leading the Clifford Research Group. His main interests are function theory and functional analysis for functions with values in quaternion and Clifford algebras. The research covers Clifford distributions, generalized Fourier, Radon and Hilbert transforms, orthogonal polynomials and multi-dimensional wavelets. Nele De Schepper received a diploma degree in mathematics from Ghent University, Belgium, in 2001. Since then she holds an assistantship at the Department of Mathematical Analysis of Ghent University and is a member of the Clifford Research Group. Her main interests are function theory and functional analysis for functions with values in Clifford algebras. The research covers generalized Fourier transforms, orthogonal polynomials and multi-dimensional wavelets. Frank Sommen received a diploma degree in mathematics from Ghent University, Belgium, in 1978, a Ph.D. degree in mathematics from the same university in 1980, and a habilitation degree in mathematical analysis in 1984. From 1978 until 1999 he was at the National Fund for Scientific Research (Flanders). Since 2000 he holds a Research professorship at Ghent University. His main interests are function theory and functional analysis for functions with values in quaternion and Clifford algebras. The research covers Clifford distributions, generalized Fourier, Radon and Hilbert transforms, orthogonal polynomials and multi-dimensional wavelets, algebraic analysis, hyperfunctions and radial algebra.     

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.     

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.     

广义左Clifford半群的半直积

在不含单位元的情况下,通过半群S和T的子半群T~e={t~e|t∈T},给出了广义左Clifford半群的半直积的刻画,得到了2个半群的半直积为广义左Clifford半群的充要条件。     

奇数次复Clifford代数的不可约表示

在数学和数学物理的研究领域中 ,Clifford代数起着重要的作用。有关复 Clifford代数的许多重要应用来源于复 Clifford代数的不可约表示。然而 ,周期性同构现象是复 Clifford代数中的一个重要特性 ,利用有关复 Clifford代数周期性定理的结果 ,本文作者得到了奇数次复 Clifford代数的不可约表示     

Almost rotation-minimizing rational parametrization of canal surfaces

Almost rotation minimizing parametrization of the canal surface is given. The basic building block of our approach is the curve approximation scheme that enables us to construct a curve, called the parameter curve, on the canal surface that produces, when projected and rescaled, an almost parallel normal vector field on the spine curve. Its construction relies on our earlier methods and results on relating the geometry of the canal surface to the Lorentzian geometry of the Minkowski geometry via the Clifford algebra formalism. We then iteratively construct patches out of the parameter curves via suitable interpolation procedure. Furthermore, its rotational deviation, i.e., the angle deviation from the parallel (no rotation) frame along the spine curve, can be controlled with the use of the rotation deviation estimate of the parameter curves. Our numerical experiment shows that the rotation deviation is minuscule. When compared with other earlier results including our earlier one, our result fares extremely favorably. To facilitate the implement, a practitioners' summary is given in the appendix.     

矢量场数据演示的快速Clifford傅立叶变换

对于结构化的矢量场数据,一般是先由均匀网格取样后,再由Clifford卷积来分析.通过证明快速Clifford傅立叶变换的可行性和有效性,介绍了应用快速Clifford傅立叶变换对矢量场数据进行分析的新方法.快速Clifford傅立叶变换把Clifford卷积由空间域转化到频域运算,加速了卷积运算,并且可以很容易、准确地模拟矢量场图像,对图像处理的特征提取和纹理分割也将起着重要的作用.     

电磁场方程微分形式的另一种导出方式以及它的几何代数形式

文中叙述电磁场方程的两种表达形式。一是基于Ｇｒａｓｓｍａｎｎ代数的微分形式。一是基于Ｃｌｉｆｆｏｒｄ代数的几何代数形式。只做到导出Ｍａｘｗｅｌｌ方程。它的推论和应用不在文中讨论。     

THE PHASE WEB PARADIGM

We describe an information-oriented, computational physical theory based on expressing knowledge of the world via patterns of synchronization. The systems this theory describes are self-organizing, display emergent phenomena, and can be expressed in terms both of Clifford algebras and algebraic topology, and practical running programs.     

Hyperbolic Hopfield neural networks with four‐state neurons

In recent years, applications of neural networks with Clifford algebra have become widespread. Clifford algebra is also referred to as geometric algebra and is useful in dealing with geometric objects. Hopfield neural networks with Clifford algebra, such as complex numbers and quaternions, have been proposed. However, it has been difficult to construct Hopfield neural networks by Clifford algebra with positive part of the signature, such as hyperbolic numbers. Hyperbolic numbers are useful algebra to deal with hyperbolic geometry. Kuroe proposed hyperbolic Hopfield neural networks and provided their continuous activation functions and stability conditions. However, the learning algorithm has not been provided. In this paper, we provide two quantized activation functions and the primitive learning algorithm satisfying the stability condition. We also perform computer simulations and compare the activation functions. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.