Numerical Algebra For Image Restoration And Face Recognition

No time in human history has ever witnessed such explosive influence and impact of image processing on modern society, sciences and technology. It has become the object of study and research in many areas of engineering, computer science, information science, statistics, physics, chemistry, biology, medicine and even the social science. Image processing has helped mankind to see objects in various environments, to sense and communicate with the physical world, as well as to make optimal decisions and take right actions. It brings huge economic and social benefits to human beings. In the future, image processing will become a powerful tool of scientific research, social production and human life.In practical applications, image processing is usually considered as two aspects. The first aspect includes image denoising, image restoration, image enhancement, etc. The second aspect includes image segmentation, image classification, image recognition, e.g., face recognition. In this thesis, applications of numerical algebra methods to image restoration and face recognition are discussed. They are illustrated in detail as follows:Study on effective regularization methods for linear discrete ill-posed problems in image restoration. Tikhonov regularization method is one of the most common and well-studied regularization methods for discrete ill-posed problems in image restoration. Recently, Hochstenbach and Reichel propose a fractional Tikhonov regularization. It preserves more solution components with small singular values than Tikhonov regularization. Fuhry and Reichel propose a new Tikhonov regularization to dampen less solution components with big singular values than Tikhonov regularization. Motivated by good performances of fractional Tikhonov regularization and new Tikhonov regularization, in this thesis, we propose a modified Tikhonov regularization method. Our method preserves more solution components with small singular values and dampen less solution components with big singular values. Numerical experiments on image restoration demonstrate the effectiveness of our method.Study on effective methods for data representation and dimensionality reduction on face recognition. The effective approaches to represent high dimensional data in a low dimensional space and the methods of dimensionality reduction receive much attention. Locally linear embedding (LLE) algorithm is considered as a powerful method for dimensionality reduction. It preserves the information of the local neighborhood of the high dimensional data in low dimensional space. In this thesis, we research a recently proposed GPLLE algorithm. Thus, we propose a modified LLE algorithm. Our modified LLE algorithm not only preserves the local neighborhood, but also preserves the distant samples from nearing. Numerical results on face recognition show that our modified LLE algorithm achieves better recognition performances than LLE algorithm and GPLLE algorithm.