Variational and PDE Models, and their Computation for Image Inpainting

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

DMS Award Abstract Award #: 0202565 PI: Shen, Jianhong Institution: University of Minnesota, Twin Cities Program: Applied Mathematics Program Manager: Catherine Mavriplis Title: Variational and Partial Differential Equation Models, and their Computation for Image Inpainting This project is intended to discover and develop the most fundamental and crucial mathematical principles and frameworks for highly diversified applications of digital inpainting. These frameworks will allow us to further construct many universally applicable inpainting models, and design their efficient and robust computational algorithms. Our proposed approach is to employ several high level mathematical tools for the modeling and computation of inpainting, which include the Bayesian decision theory, nonlinear partial differential equations (e.g. mean curvature motions, nonlinear transport and diffusion), variational methods in the space of Bounded Variations and for free boundary problems, a variety of state-of-the-art tools from harmonics analysis such as wavelets and multiresolution analysis, and many efficient schemes in numerical analysis and computational partial differential equations. The project is also highlighted by the fact that we are proposing for the first time to integrate visually important curve, surface, and image geometry into the traditionally statistical models and dynamic processes. Digital inpainting is to develop an automatic process to intelligently recover and complete the missing, unavailable, or purposely disguised image information. Such loss of information occurs ubiquitously in a variety of important fields including computer vision, network (especially wireless) communication, robust image coding and transmission (from the Hubble Space Telescope for example), three-dimensional volumetric organ reconstruction from two-dimensional medical images, disguise of enemy weapons and personnel in the battlefields, and the digital restoration of cracked ancient paintings in digitized fine art museums. This project will develop a mathematical framework for image inpainting. Besides the broad impact on the numerous important fields mentioned above, the project will also strengthen the integration of high level pure mathematics into the contemporary digital, computer, and artificial intelligence technology, and in return, create numerous opportunities for mathematical modeling, analysis, and computation. It will also help the principal investigator develop new curricula and train new graduate students in this booming fresh field of applied mathematics. Date: May 22, 2002