Clifford Fourier transform on vector fields

Image processing and computer vision have robust methods for feature extraction and the computation of derivatives of scalar fields. Furthermore, interpolation and the effects of applying a filter can be analyzed in detail and can be advantages when applying these methods to vector fields to obtain a solid theoretical basis for feature extraction. We recently introduced the Clifford convolution, which is an extension of the classical convolution on scalar fields and provides a unified notation for the convolution of scalar and vector fields. It has attractive geometric properties that allow pattern matching on vector fields. In image processing, the convolution and the Fourier transform operators are closely related by the convolution theorem and, in this paper, we extend the Fourier transform to include general elements of Clifford Algebra, called multivectors, including scalars and vectors. The resulting convolution and derivative theorems are extensions of those for convolution and the Fourier transform on scalar fields. The Clifford Fourier transform allows a frequency analysis of vector fields and the behavior of vector-valued filters. In frequency space, vectors are transformed into general multivectors of the Clifford Algebra. Many basic vector-valued patterns, such as source, sink, saddle points, and potential vortices, can be described by a few multivectors in frequency space.     

基于Adaboost与Clifford代数的人脸检测

Adaboost算法在光照不均、背景复杂的条件下进行人脸检测时误检率较高。为解决该问题,提出一种基于Adaboost算法与Clifford代数矢量积性质的人脸检测方法。利用Adaboost算法初步定位人脸可能存在的区域,对该区域进行基于知识的校验,如果校验失败,根据Clifford矢量积性质,寻找与待验证区域相似度较高的人脸,当相似度大于阈值时,判断其为人脸。实验结果表明,与Viola-Jones方法相比,该方法在保持较高检测率的同时,降低了误检率,且鲁棒性较好。     

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.     

基于Clifford代数矢量积的掌纹提取方法

基于Clifford代数矢量积表示定理,本文提出一种新的掌纹提取方法。选取掌纹线上的一个点作为种子点,将该点的八邻域的点归一化后作为该点的8个特征,对图像进行遍历检测,设定阈值,判断每个点是否为掌纹线上的点。在实验中选取3个种子点,然后对3个结果进行融合。实验证明,该方法比传统方法提取的掌纹信息更精确。     

Coverage analysis for sensor networks based on Clifford algebra

The coverage performance is the foundation of information acquisition in distributed sensor networks. The previously proposed coverage work was mostly based on unit disk coverage model or ball coverage model in 2D or 3D space, respectively. However, most methods cannot give a homogeneous coverage model for targets with hybrid types. This paper presents a coverage analysis approach for sensor networks based on Clifford algebra and establishes a homogeneous coverage model for sensor networks with hybrid types of targets. The effectiveness of the approach is demonstrated with examples.     

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.     

Clifford代数3D人脸姿态矫正方法

非正面视图的3D人脸对特征提取及识别结果有不等程度的影响,因此在预处理阶段需要对3D姿态进行矫正,使其还原于正面视图.本文就其提出一种基于Clifford代数理论的3D人脸姿态矫正方法,即通过计算与检测3D人脸的特殊点,确定人脸正面所在平面及空间的几何位置;再构造Clifford代数的几何旋转算子对3D人脸顶点坐标向量进行旋转,使其3D人脸正面处于坐标正视图,眼睛保持水平位置以现实姿态矫正.实验结果表明该方法计算简单,几何意义直观,矫正精度高,并具有通用性.     

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.     

Clifford Algebraic Reduction Method for Automated Theorem Proving in Differential Geometry

In this paper a new method is proposed for mechanically proving theorems in the local theory of space curves. The method is based on Ritt-Wus well-ordering principle of ordinary differential polynomials, Clifford algebraic representation of Euclidean space and equation set solving in Clifford algebra formalism. It has been tested by various theorems and seems to be efficient.     

Analytical error analysis of Clifford gates by the fault-path tracer method

We estimate the success probability of quantum protocols composed of Clifford operations in the presence of Pauli errors. Our method is derived from the fault-point formalism previously used to determine the success rate of low-distance error correction codes. Here we apply it to a wider range of quantum protocols and identify circuit structures that allow for efficient calculation of the exact success probability and even the final distribution of output states. As examples, we apply our method to the Bernstein–Vazirani algorithm and the Steane [[7,1,3]] quantum error correction code and compare the results to Monte Carlo simulations.     

An algorithm for removing facial makeup disturbances based on Clifford algebras

The main goal of the paper is to show that Clifford algebras can be used to solve problems of removing facial makeup disturbance. After simulation and practical application experiments, the algorithm is a natural and effective manner.     

A Metric Approach to nD Images Edge Detection with Clifford Algebras

The aim of this paper is to perform edge detection in color-infrared images from the point of view of Clifford algebras. The main idea is that such an image can be seen as a section of a Clifford bundle associated to the RGBT-space (Red, Green, Blue, Temperature) of acquisition. Dealing with geometric calculus and covariant derivatives of appropriate sections with respect to well-chosen connections allows to get various color and temperature information needed for the segmentation. We show in particular how to recover the first fundamental form of the image embedded in a LSHT-space (Luminance, Saturation, Hue, Temperature) equipped with a metric tensor. We propose applications to color edge detection with some constraints on colors and to edge detection in color-infrared images with constraints on both colors and temperature. Other applications related to different choices of connections, sections and embedding spaces for nD images may be considered from this general theoretical framework.

基于超混沌映射和加法模运算的图像保密通信方案

提出了一种基于Clifford超混沌映射的图像加密方案,发送方利用Clifford映射快速地产生混沌序列,对图像在时域作加密预处理;然后通过简单的加法模运算对像素的灰度值进行替换与扩散,并且在每次迭代中采用不同的密钥.经通信双方同步之后,接收方即可解密出原始图像.实验结果表明本方案是可行的,并对系统密钥空间、密钥敏感性等密码学特性进行了分析.     

A multiexpert framework for character recognition: a novelapplication of Clifford networks

A novel multiple-expert framework for recognition of handwritten characters is presented. The proposed framework is composed of multiple classifiers (experts) put together in such a manner as to enhance the recognition capability of the combined network compared to the best performing individual expert participating in the framework. Each of these experts has been derived from a novel neural structure in which the weight values are derived from Clifford algebra. A Clifford algebra is a mathematical paradigm capable of capturing the interdimensional dependencies found in multidimensional data. It offers a technique for concise data storage and processing by representing dependencies between the component dimensions of the data which is otherwise difficult to encode and hence is often employed in analyzing multidimensional data. Results achieved by the proposed multiple-expert framework demonstrates significant improvement over alternative techniques.     

Explicit Solutions of Galois Embedding Problems by Means of Generalized Clifford Algebras

In this paper we study Galois embedding problems given by central extensions with cyclic kernel. We find a new expression for the obstruction to the solvability of these embedding problems in terms of Galois symbols. We also give a method to construct the solutions when these problems are solvable. We find a solution from a coordinate of the norm of an adequate element in a generalized Clifford algebra.     

基于Clifford半群上共轭搜索问题的密钥建立协议

本文采用更广泛的半群作为平台,推广了Iris Anshel等提出的代数密钥建立协议模型。在定义了Clifford半群上的多重同时共轭搜索问题（MSCSP）后,给出了基于此问题的密钥建立协议。在理论上证明了若Clifford半群上的多重同时共轭搜索问题（MSCSP）是困难的,那么可以利用MSCSP来构造密钥建立协议。从而说明利用半群作为平台构建密钥建立协议是可能的。本文也提供了一种新的利用辫群的思路,即考虑利用辫群上的强半格（许多辨群按照一定规则形成的无交并）构成的Clifford半群来构建密码协议,以弥补单个辫群可能存在的安全缺陷。     

Efficient underdetermined speech signal separation using encompassed Hammersley- Clifford algorithm and hardware implementation

Speech Separation is among the propelled advances for a wide range of uses in different sectors, where detachment from the Blind Source Separation Signal is a troublesome task. Blind source separation is a growing digital signal processing industry to separate the precise signal from the recorded dense. Exclusively, among the "Blind Source Separation," the "Under Determined Blind Source Separation" is considered as an Over Determined Blind Source Separation due to its wide range of usage. Nevertheless, it is seen that real implementation is very rarely done in existing researches because the real-time Implementation of UBSS (Underdetermined Blind Source Separation) exists to be a challenging one due to its lacking hardware characteristics of increased latency, reduced speed and consumption of more memory space. Consequently, an increasing need to implement an Underdetermined source signal separation in real-time with improved hardware utility. In this Unswerving framework, a Real-time feasible Source Signal separator formulated in which the source signals decomposed by Boosted Band-Limited VMD (Variational Mode Decomposition) "Multicomponent Signal”. The amount of "Band-Limited” Intrinsic Mode Function (BLIMF) was subjected to the Encompassed Hammersley–Clifford algorithm for source separation using Expectation-Maximization and Gibbs Sampling, an alternative to deterministic algorithms and to determine the exact estimated parameter from the E-M method. Subsequently, the source separation algorithm infers the best separation of source signals by exact estimation and determination from the decomposed signals. The iterations in E-M estimation reduced by the Gauss-Seidel Method. Thus, our novel source signal separates internally with a signal decomposer and a source separation algorithm with fewer iterations, which reduces memory consumption and yields better hardware realization with reduced latency and increased speed. The proposed implementation is done by utilizing Matlab for initial processing and the hardware analysis performed in Xilinx Platform.