Research On Mathematical Models And High Performance Algorithms For Some Image Processing Problems

Image processing includes many important issues, e.g., image denoising and deblurring, image super-resolution and image inpainting, etc. These problems mainly arise from the procedure of image recording, transferring and storing, or other bad conditions. For instance, the shaking of camera will lead to noisy and blurred images, or the low-density sensors of camera may lead to the low-resolution images. For the arisen problems, how to found a reasonable model and design an efficient algorithm is the main goal of the dissertation.The dissertation mainly focuses on three important problems in image processing,i.e., image denoising and deblurring problem, image super-resolution problem and image inpainting problem. Actually, the “skeleton” of this dissertation is based on some mathematical optimization methods, then we apply the optimization methods to the three image problems. In the dissertation, we found different models and design the corresponding algorithms according to different image problems, then compare the proposed methods with some state-of-the-art methods by extensive experiments. In particular, the dissertation mainly includes the following points:1. Wavelet-based multilevel approach is a promising method for image restoration problems. We propose a method for image restoration based on combining the symmlets family with two-level technique. A key feature is that the symmlets which are used to construct transfer operators have many flexible attributes that lead to better stability and less artifacts. Experiments demonstrate the symmlets-based two-level method yields satisfied results.2. A multilevel method(MLM) combining multigrid method with Tikhonov regularization for signal restoration is proposed. The proposed method can transfer large-sized problems to small- or moderate-sized problems to make the SVD-based methods available. No pre-smoothers are implemented in the multilevel process to avoid damaging the parameter choice on the coarsest level. Furthermore, the soft-thresholding denoising technique is employed for the post-smoothers aiming to eliminate the high-frequency information(e.g., noise) due to the lack of pre-smoothers. Experiments demonstrate that the proposed method outperforms other SVD-based methods at a shorter CPU-time consumption.3. Image super-resolution has important applications in satellite imaging, high definition television, medical imaging, etc. We present an iterative scheme to solve single image super-resolution problems. It recovers a high quality high-resolution image from solely one low-resolution image. We solve the problem from image intensity function estimation perspective and assume the image contains smooth and edge components. We model the smooth components of an image using a thin-plate reproducing kernel Hilbert space(RKHS) and the edges using approximated Heaviside functions. The proposed method is applied to image patches, aiming to reduce computation and storage. Visual and quantitative comparisons with some competitive approaches show the superiority of the proposed method.4. In mathematics, there is a very simple function-Heaviside function. Researcher had proven that an image could be represented by approximated Heaviside function.We assume that an image is composed of smooth and non-smooth components, and we use two classes of approximated Heaviside functions(AHFs) to represent them respectively. Based on the representation, a sparse model that is solved by Alternating Direction Method of Multipliers(ADMM) is proposed. In addition, we apply the proposed iterative scheme to image patches to reduce computation and storage size. Extensive comparisons with some existing competitive methods show the effectiveness of the proposed method.5. Image inpainting aims to recover the scratches in photograph, repair the damaged regions of an image, remove the specify objects, etc. Exemplar-based algorithms are a popular technique for image inpainting. They mainly have two important phases: 1) deciding the filling-in order, 2) selecting good exemplars from source regions. Traditional exemplar-based algorithms do not consider image geometry and lead to unsatisfied results. To improve the problem, we introduce an independent strategy through investigating the process of patches propagation, and define a new separated priority definition that first propagating geometry and then synthesizing image textures, aiming to well recover image geometry and textures. This strategy can avoid destroying the image geometry.Extensive experiments and discussions demonstrate the proposed method is an excellent approach.