The investigator, collaborators and students will develop new models for image processing, show the models are mathematically sound, determine geometric properties of their solutions, develop accurate and efficient numerical schemes, and directly apply these models to problems in the sciences. The models will address four fundamental problems in image processing: edge-preserving denoising, decomposition, deblurring, and image fusion. The proposed formulations will be based on variational methods and partial differential equations, which provide an appropriate framework for studying the mathematical foundations and physical interpretation of the models. These include models involving convex linear growth functionals, the Besov semi-norm, and negative Sobolov norms, which can retain desirable geometric image properties such as edges and textures, while avoiding the introduction of false artifacts. One of the challenges addressed here will be finding appropriate discrete representations of the continuous problem that retain desirable geometric features and are tractable.
Digital images are now used in almost every area of science and technology. The models developed in this project will be used to solve real world problems in areas such as medical imaging and material science. However, the models will be formulated in enough generality to potentially be applied to a wide array of applications in the sciences. Software developed in this grant will be made publicly available. The investigator regularly teaches courses on image processing, and leads workshops on image processing for middle school students and high school women and minorities. This project will also support undergraduate researchers. Thus this work will promote the training of young scientists, as well as provide educational opportunities to underrepresented groups.
