The cross-channel deblurring problem in color image processing is difficult to solve due to the complex coupling and structural blurring of color pixels. Until now, there are few efficient algorithms that can reduce color artifacts in deblurring process. To solve this challenging problem, we present a novel cross-space total variation (CSTV) regularization model for color image deblurring by introducing a quaternion blur operator and a cross-color space regularization functional. The existence and uniqueness of the solution is proved and a new L-curve method is proposed to find a balance of regularization terms on different color spaces. The Euler-Lagrange equation is derived to show that CSTV has taken into account the coupling of all color channels and the local smoothing within each color channel. A quaternion operator splitting method is firstly proposed to enhance the ability of color artifacts reduction of the CSTV regularization model. This strategy also applies to the well-known color deblurring models. Numerical experiments on color image databases illustrate the efficiency and effectiveness of the new model and algorithms. The color images restored by them successfully maintain the color and spatial information and are of higher quality in terms of PSNR, SSIM, MSE and CIEde2000 than the restorations of the-state-of-the-art methods.

The two-dimensional principal component analysis (2DPCA) has been one of the basic methods of developing artificial intelligent algorithms. To increase the feasibility, we propose a new general ridge regression model for 2DPCA and variations, with extracting low dimensional features under two projection subspaces. A new relaxed 2DPCA under the quaternion framework is proposed to utilize the label (if known) and color information to compute the essential features of generalization ability with optimization algorithms. The 2DPCA-based approaches for face recognition are also improved by weighting each principle component a scatter measure, which increases efficiently the rate of face recognition. In numerical experiments on well-known standard databases, the R2DPCA approach has high generalization ability and achieves a higher recognition rate than the state-of-the-art 2DPCA-like methods, and has better performance than the basic deep learning methods such as CNNs , DBNs, and DNNs in the small-sample case.