Quaternion Matrix Singular Value Decomposition And Its Application

Quaternion was found by Hamilton,a British mathematician,in 1843.The discovery of quaternion is an important event in the history of math.Quaternion is applied widely in many domains,such as Algebra,Geometry,Physies,Engineering and so on. At present,quaternion is applied more and more in computer science and engineering and gets more and more attention.Matrix decomposition is an important part of matrix theories.Matrix singular value decomposition is one of the elementary and essential implements in matrix theory and matrix computations,but also,can be directly applied in many domains, such as control theory,system recognition,signal processing,optimal problems,eigenproblems,least squares problems,statistics and so on.In this paper,the algorithems of quaternion matrix singular value decomposition and its application are mainly discussed.One side,by the virtue of the comple representation of quaternion matrix,the theory of quaternion matrix singular value decomposition is proved by the properties of quaternion companion vector,and the algorithem of quaternion matrix singular value decomposition is realized on it.;the other side,we reduce the complex representation of quaternion matrix to bidiagonal matrix by Householder transformation of complex matrix,then,we can obtain singular value of quaternion matrix,can construct singular value vectors by meaning of matrix construction between quaternion matrix and its complex representation,and give another algorithem of quaternion matrix singular value decomposition by conducting matrix singular decomposition to bidiagonal matrix reduced by Householder transformation. In application,the quaternion matrix model of color image is introduced in this paper,and a few problems on color image feature,color image reconstruction and color image de-noising are discussed.On the effect of color image de-noising,a method based on block quaternion matrix singular value decomposition is proposed in this paper,and singnal-to-noise ratio is improved by this method.Numerical examples are given to the algorithems proposed in this text,moreover, under programming environment of Matlab 7.0,procedures are compiled by this algorithem proposed in this text.The algorithems and procedures are applied in color image processing,which also confirm the algorithems proposed in this text are feasible and effective.