A new method for reconstructing the compressed sensing color image by solving an optimization problem based on total variation in the quaternion field is proposed, which can effectively improve the reconstructing ability of the color image. First, the color image is converted from RGB （red, green, blue） space to CMYK （cyan, magenta, yellow, black） space, which is assigned to a quaternion matrix. Meanwhile, the quaternion matrix is converted into the information of the phase and amplitude by the Euler form of the quatemion. Secondly, the phase and amplitude of the quatemion matrix are used as the smoothness constraints for the compressed sensing （CS） problem to make the reconstructing results more accurate. Finally, an iterative method based on gradient is used to solve the CS problem. Experimental results show that by considering the information of the phase and amplitude, the proposed method can achieve better performance than the existing method that treats the three components of the color image as independent parts.
To resolve the completeness and independence of an invariant set derived by the traditional method, a systematic method for deriving a complete set of pseudo-Zernike moment similarity （translation, scale and rotation） invariants is described. First, the relationship between pseudo-Zernike moments of the original image and those of the image having the same shape but distinct orientation and scale is established. Based on this relationship, a complete set of similarity invariants can be expressed as a linear combination of the original pseudo-Zernike moments of the same order and lower order. The problem of image reconstruction from a finite set of the pseudo-Zernike moment invariants （PZMIs） is also investigated. Experimental results show that the proposed PZMIs have better performance than complex moment invariants.
An algorithm for recovering the quaternion signals in both noiseless and noise contaminated scenarios by solving an L1-norm minimization problem is presented. The L1-norm minimization problem over the quaternion number field is solved by converting it to an equivalent second-order cone programming problem over the real number field, which can be readily solved by convex optimization solvers like SeDuMi. Numerical experiments are provided to illustrate the effectiveness of the proposed algorithm. In a noiseless scenario, the experimental results show that under some practically acceptable conditions, exact signal recovery can be achieved. With additive noise contamination in measurements, the experimental results show that the proposed algorithm is robust to noise. The proposed algorithm can be applied in compressed-sensing-based signal recovery in the quaternion domain.
