Models currently used in image processing are discrete. They are piecewise constant approximations of the true models - integral equation models. The discrete models impose bottleneck model errors which cannot be compensated from the numerical methods developed based upon them. To overcome this drawback, the PIs propose to directly use the integral equation models for image processing. The integral equation models will offer us much greater flexibility for the in-depth analysis of the corresponding images. An ideal method for image processing should be sensitive to geometric features of images and computationally efficient. Presently, the total variation method and the multiscale method are two main mathematical approaches for image processing. They are complementary to each other in their strength and weakness. The total variation method is sensitive to geometric features of images but it is computationally inefficient. The standard multiscale method is convenient for computation due to its multiscale structure but it is not very sensitive to the geometric features of images. Aiming at designing computationally efficient algorithms that are sensitive to geometric features of images, the PIs propose to develop multiscale total variation methods which combine the strengths from both of these two methods. The PIs study the following four mathematical problems related to image processing: (1) Multiscale approximation of images based on integral equation models; (2) Multiscale total variation regularization; (3) Missing data recovery with redundant systems; and (4) Design of application-driven wavelet and framelet filter banks.
Image processing arises in a variety of scientific, medical and engineering applications. Specifically, applications in medical sciences and technologies range from computer tomography to diagnoses of diseases, applications in environmental sciences include natural resources and pollution control via satellite imaging, applications in art sciences have vision analysis and digital restorations of cracked ancient paintings in digitized fine art museums and applications in security identification include weapon, fingerprints and face identifications. In these applications, a key issue is restorating images from available data. This is an ill-posed problem. Solving this problem needs advanced mathematical models and efficient computational algorithms. The main objective of this proposal directly addresses this issue by proposing multiscale total variation methods for integral equation models in image processing. The projects in the proposal will enhance the integration of high level pure mathematics with the contemporary digital and computer technology. These projects will train graduate studetns in this important area to prepare them to face the mathematical and computational challenge in future scientifical and technological development. Moreover, th PIs will develop a multidisciplinary course for upper level undergraduate students based on research results of these projects.
