四元数矩阵正交特征向量系的求解方法及其在彩色人脸识别中的应用

Obtain Method of Quaternion Matrix Orthogonal Eigenvector Set and Its Application in Color Face Recognition

针对四元数矩阵正交特征矢量系求解困难的缺点,本文提出一种获取四元数矩阵正交特征矢量集等效、便捷的方法,其基本思路为:首先,构造四元数矩阵定义于复数域的导出阵,并利用该导出阵特征矢量空间的一种特殊的等价空间间接获取相应特征值所对应的特征矢量.然后,将复数矢量转换为四元数矢量,按如此方式获取的对应所有特征值的非零特征矢量则构成原始四元数矩阵的正交特征矢量系.同时,本文将定义于实数域的主成分分析方法(Principal component algorithm,PCA)向四元数体作合理的推广,给出详细的数学推导过程,证明该方法的合理性及其在统计模式识别领域得以应用的可能性.最后,作者将彩色图像像素的R、G、B三分量作为四元数的三个虚数部分,首次在人脸识别中引入基于四元数的彩色人脸识别方法.较传统的基于灰度图像的识别方法,本文方法不仅利用了人脸图像灰度值的空间分布信息,而且充分利用不同人脸之间的色彩差异,从而得到更多的鉴别信息.在四川大学人工智能研究所的彩色人脸库上进行的实验表明,所提出的基于四元数的识别方法不仅大幅度提高了识别率,而且具有较高的鲁棒性.

Considering the difficulty of obtaining orthogonal eigenvector set of quaternion matrix, a novel obtaining method is proposed in this paper. The main idea of this method can be described as follows. Firstly, construct the educing matrix of quaternion matrix, which is defined in complex field, secondly get complex eigenvector by using a specific space, which is similar to the eigenvector space of the educing matrix, then the orthogonai eigenvector set can be obtained through transforming the eigenvectors of all eigenvalues to quaternion eigenvectors. Simultaneously, quaternionbased principal component algorithm （PCA） method is proposed and detailed mathematical calculations are also given to explain its rationality and practicability in pattern recognition field. Finally, quaternion-based color face recognition method, which uses R, G, B as the three imaginary numbers of quaternion, is proposed in the paper. Compared with the traditional method, our algorithm uses face grayscale information and color information at the same time in order to get more discrimination information. Experiments performed on color face database of TianSi brainpower graduate school of Sichuan University indicate that the recognition rates are improved significantly and the proposed method is superior to the traditional one in the mass.