Theory of reduced biquaternion sparse representation and its applications

Traditional sparse representation models treat color image either represent color channels independently using the monochromatic model or concatenate color channels using the concatenation model. However, these two strategies cannot make full use of the correlation among the three color channels. In this paper, we propose a novel sparse representation model for color image based on reduced biquaternion. The advances of the proposed model are in two aspects: 1) compared with quaternion algebra, reduced biquaternion algebra is commutative algebra which has a low computation cost; 2) benefited from reduced biquaternion representation, the proposed model can treat the color image as a whole. We firstly extend the singular value decomposition (SVD) to the reduced biquaternion domain (RBSVD). The framework of the proposed model contains two major stages: first, reduced biquaternion orthogonal matching pursuit (RBOMP) is proposed in the stage of sparse coding; second, generalized K-means clustering for RBSVD (K-RBSVD) is proposed in the stage of dictionary training. Afterward, a new color image denoising algorithm is developed to demonstrate the effectiveness of the proposed model. The experimental results demonstrate that the proposed sparse representation model achieves state-of-the-art denoising performance in terms of both quantitative metrics and visual quality.