Research Of Low-Rank And Dual For Image Sequence Processing

In recent years, the image(sequence) processing technologies have been widely used in the fields of computer vision, information science, engineering modeling, film and television special effects, etc. The image(sequence) processing tools based on the mathematical models not only be able to provide beneficial theoretical basis to solve the problems, but also facilitate the scholars to analysis the performance and improve the innovation of the models.This dissertation mainly researches on the foreground-background separation and the image(sequence) inpainting problems. We propose some novel models based on the low-rank or approximate low-rank of the image(sequence) data matrix, and design the corresponding fast algorithms for those models. The main contributions can be summarized as follows:1. A novel “low-rank + dual”model based on the Schatten p-norms is proposed, which is used for the dimension reduction problems of the huge data matrix. By using the submultiplicative and unitarily invariant properties of the Schatten p-norms, we prove that the solution of the proposed model can be obtained by an “l∞+ l1”minimization problem in the vector version. We deduce and proof the relevant theory in details, and provide a simple fast algorithm to solve the new model. Numerical experiments demonstrate that the proposed data decomposition model based on the Schatten p-norms performs well than other methods. We apply our model both on the simulation data for the matrix restoration or completion problems, and the real image sequence data for the foreground-background separation or light-shadow removal problems.2. To overcome the disadvantages of the examplar-based inpainting methods in the practical application, we propose a new method based on low-rank and dual approximation for the image(sequence) inpainting problems. The new method firstly uses a salient-based ranking approach based on higher-order statistics to ensure the priority of the target patch with obvious visual structural edges; then searches the similar examplar-patches to the target patch according to image Euclidean distance; and finally extracts the useful information data by the low-rank and dual approximation of the matrix with similar examplar-patches to inpaint the missing part of the target patch. Numerical experiments demonstrate that the proposed new method satisfies: 1) preferentially inpaint the salient structural edges more accurate than the classical examplar-based inpainting methods, and the salient image of our inpainting re-sult also has good visual connectivity; 2) deal better with a variety of types of texture or structure damages than other existing inpainting methods, and our inpainting result not only has better visual effect but also has higher peak signal to noise ratio and structural similarity.3. Two video foreground-background separation models with dual norm regularization have been proposed, including the preliminary dual norm model and the improved dual functional model. To overcome the limitations of the “low-rank + sparse”priori restraint widely used in the common video foreground-background separation methods, the new models only keep the low-rank presumption and hypothesis of the background, then extract the foreground image from background and restrain the correlation between the data by using the property of the dual norms. In the dual norm model, we apply the dual norm of the nuclear norm to replace the l1-norm used in the existing low-rank and sparse relaxation model, and restrain the correlation between foreground-background image and simplify the prior conditions. In the dual functional model, we further apply the re-weighting nuclear norm to replace the general nuclear norm to get better low-rank approximation, and find the dual functional of the re-weighting nuclear norm to restrain the correlation between the foreground-background image. Numerical experiments demonstrate that the proposed two new models not only performance well in the foreground-background separation problems, but also have lower time complexity and better data expression than the existing methods for the low-rank and sparse relaxation model.4. Two low-rank and sparse decomposition model with Max-norm for high-dimensional data have been proposed, including the Max minimal model and the Max constraint model.Such models introduce the Max-norm as the convex relaxation of the rank function to keep approximation low-rank, which is better than the general low-rank and sparse decomposition model introduce the nuclear norm as the convex relaxation of the rank function. In the Max minimal model, we adopt the alternating direction method of multipliers to divide the model into sub-problems. In the respectively processing step, the Max minimization problem should be transformed into the equivalent l22,∞-norm minimization problem for solving,which cause the high computational complexity and the unsatisfied time consumption of the Max minimal model. In the Max constraint model, we set the Max-norm as a constraint condition of the optimization problem, which make the corresponding sub-problem can be directly solved by the projected gradient algorithm. Thus, the Max constraint model spends less time than the Max minimal model. Numerical experiments demonstrate that the proposed two Max-norm based models performance well in the separation problems of the video sequence and the facial collection images.