Two dimensional quaternion valued singular spectrum analysis with application to image denoising

Compared to the traditional single color plane based image denoising methods, the quaternion valued singular value decomposition (QSVD) exploits the relationship among different color planes. Hence, it has been applied to the color image denoising. On the other hand, compared to the non-overlapping based image denoising methods, the two dimensional real valued singular spectrum analysis (2DSSA) constructs the trajectory matrix with many elements in the matrix being overlapped. Since the 2DSSA exploits the local information within each color plane, it has also been applied to the single color plane based image denoising. However, neither these two image denoising methods can exploit the relationship among the color planes and the local information within each color plane simultaneously. Therefore, this paper proposes a two dimensional quaternion valued singular spectrum analysis (2DQSSA) based method for performing the color image denoising. Our proposed method can enjoy the advantages of both methods. However, the most critical issue for the 2DQSSA is on the selection of these 2DQSSA components. This paper finds that the optimal total number of the 2DQSSA components used for performing the reconstruction is monotonic decreasing with respect to the power of the noise in the image. Therefore, the polynomial fitting approach is proposed to model this relationship. Since the 2DQSSA based denoising method exploits the relationship among the red color plane, the green color plane and the blue color plane, the 2DQSSA based denoising method outperforms the conventional single color plane based denoising methods. Moreover, since the 2DQSSA based denoising method also exploits the local relationship within each color plane, the 2DQSSA based denoising method outperforms the non-overlapping based methods.