The investigators will study new variational PDE-based mathematical models and fast computational algorithms for solving them. These models have emerged recently as a powerful addition to the variety of methodologies for image processing and computer vision. They are distinguished by their use of powerful mathematical tools from differential geometry, PDE and functional analysis to handle geometry regularities and sharp feature boundaries. A particularly popular example of this class of models is the total variation based image restoration model, first introduced by Rudin, Osher, and Fatemi (ROF). The ROF model has proven to be extremely successful, both in practice and in theory, and has created a lot of interest and extensions. At this point in time, there is quite a complete understanding of its theoretical properties and both its advantages and limitations. For example, the model has been extended to include deblurring, to vector-valued images, to higher order regularization involving curvatures of the underlying level sets of the image intensity function, to deal with the "stair-casing" effect, Meyer's recent "image decomposition" (as opposed to image denoising), introduction of the dual norm of TV to extract texture information, and the idea of an "inverse scale space". Taken all together, there has truly been an explosion of exciting and novel ideas recently that were inspired by the original ROF model, and the field has been enjoying a revival of interest and activities. Despite all the new developments mentioned above, there remains a major drawback of nonlinear PDE-based models: their computational efficiency. There are many reasons for this difficulty: their nonlinearity (which are designed on purpose to allow the models to recover discontinuous solutions), non-smooth objective functions (making it difficult to construct Newton-type methods to achieve quadratic convergence), and globally spatial coupling (resulting in a spatial stiffness that presents a major challenge to any solution algorithm). In this proposal, the investigators will study systematically the design of robust, scalable and efficient computational algorithms for this class models. They will make use of new paradigms involving the use of a dual formulation to recover sharp discontinuities without the usual edge-smearing numerical regularization of the primal TV formulation, the use of novel Newton-type linearizations (Newton on the primal-dual system, and use of non-smooth Newton) to obtain faster than linear convergence, and the use of novel multigrid methods to deal with the spatial stiffness. The investigators will also study the TVL1 model. This seemingly simple extension of the ROF model (replacing the L2 fidelity term by a L1 term) turns out to produce fundamentally new, and often desirable, properties: contrast invariance, better scale separation, better multiscale image decomposition, and intrinsic geometric properties which allow its use to derive powerful but simple convex optimization algorithms which can find the global optimums of several non-convex shape optimization problems. The final part of the proposal is on the application of these variational PDE-based models to image processing on manifolds. This class of problems has many important applications, especially in medical imaging (e.g. brain mapping) and computer graphics. The particular approach proposed here is a novel one based on some new conformal mapping techniques developed for brain mapping that are particularly simple to use in conjunction with PDE-based image processing models.
Imaging sciences has emerged as a powerful paradigm in computational and applied mathematics. It has attracted a lot of interested from mathematicians from other fields, especially among students and young researchers. There are many applications to science, engineering, medicine and even in the entertainment industry. This proposal is on novel new ideas that are at the forefront of this relatively new field and that have aroused much interest in the field. These new developments are mathematically based and make use of powerful and subtle concepts from other parts of computational mathematics, such as duality, non-smooth optimization and multigrid methods. The focus is on key issues, such as computational efficiency, feature extraction, multiscale decomposition, global optimization, which are keys to further advances. It also leverages powerful new techniques from computational biology and combine them with PDE-based image models for new applications to the general class of problems of solving PDEs on manifolds. The ultimate goal is to produce an efficient set of algorithms for a general class of nonlinear PDE-based models that are robust, scalable, accurate, and fast.
