Color Image Denoising Algorithm Based On Quaternion Algebra

The traditional color image processing methods usually regard the color image as three independent gray-scale images.The color image is generally processed in three channels separately,ignoring the mutual connection among channels,which is difficult to achieve satisfactory processing results.So,it is of great theoretical and practical significance to find an effective representation of color images,which can take three color channels as a whole and preserve the correlation among the channels as much as possible.As the first found hyper-complex algebra,quaternion provides a practical mathematical tool for the three-dimensional color image signal reconstruction.Quaternion-based approaches mimic human perception of the visual environment and handle multi-channel information in a parallel way.As a new color image representation tool,quaternion has achieved excellent results in the color image recovery problems.To deal with the color image processing problems,a novel Quaternion Weighted nuclear Norm Minimization(QWNNM)method is proposed in this paper.The main research work and innovations of this paper can be concluded as follows.Firstly,the existing image denoising algorithms and the quaternion-based color image processing methods are systematically summarized.The basic operation of the quaternion,the quaternion representation of the color image is presented in detail.The Nuclear Norm Minimization(NNN)algorithm,Weighted Nuclear Norm Minimization(WNNM)algorithm and the Quaternion Singular Value Decomposition(QSVD)are studied in depth.Secondly,the most important innovation of this paper is that the WNNM model is extended to the quaternion domain and a novel Quaternion Weighted Nuclear Norm(QWNNM)method is proposed.According to the different importance of the singular values,a smaller weight is assigned to a larger singular value to decrease the shrinkage.An iterative reweighted algorithm is used to reconstruct the low-rank quaternion matrix and the brief proof of the optimal solution is given in this paper.Finally,based on the quaternion representation of color images and the Non-local Self-Similarity(NSS)of natural images,the QWNNM algorithm is applied to color image denoising.The denoising experiments are discussed in three cases,the low noise level,the high noise level and the unknown noise level.Especially,we applied the original WNNM algorithm in color image denoising in this paper.And for fairness,we add the pre-processing step with Gaussian Low Pass Filter(GLPF)in WNNM algorithm when the noise level is high.A large number of color image denoising experiments show that the proposed QWNNM method significantly outperforms the K-SVD and WNNM algorithm in terms of the subjective visual effect and objective denoising results.In the quaternion space,the inner structure of the color image can be well preserved during vector reconstruction.The QWNNM algorithm can also be used for other image processing problems,such as color image restoration,deblurring,and demosaicing.The idea of multi-dimensional low rank matrix reconstruction can also be extended for color image classification,machine learning and pattern recognition,which has a wide range of application and potential.