Browsing by Author "Osher, Stanley"
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Item A convex model for matrix factorization and dimensionality reduction on physical space and its application to blind hyperspectral unmixing(University of Minnesota. Institute for Mathematics and Its Applications, 2010-10) Möller, Michael; Esser, Ernie; Osher, Stanley; Sapiro, Guillermo; Xin, JackItem Simultaneous structure and texture image inpainting(2002-07) Bertalmio, Marcelo; Vese, Luminita; Sapiro, Guillermo; Osher, StanleyAn algorithm for the simultaneous filling-in of texture and structure in regions of missing image information is presented in this paper. The basic idea is to first decompose the image into the sum of two functions with different basic characteristics, and then reconstruct each one of these functions separately with structure and texture filling-in algorithms. The first function used in the decomposition is of bounded variation, representing the underlying image structure, while the second function captures the texture and possible noise. The region of missing information in the bounded variation image is reconstructed using image inpainting algorithms, while the same region in the texture image is filled-in with texture synthesis techniques. The original image is then reconstructed adding back these two sub-images. The novel contribution of this paper is then in the combination of these three previously developed components, image decomposition with inpainting and texture synthesis, which permits the simultaneous use of filling-in algorithms that are suited for different image characteristics. Examples on real images show the advantages of this proposed approach.Item Solving variational problems and partial differential equations mapping into general target manifolds(2002-01) Memoli, Facundo; Sapiro, Guillermo; Osher, StanleyA framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper. We discuss the framework for arbitrary target manifolds, while the domain manifold problem was addressed in [3]. The key idea is to implicitly represent the target manifold as the level-set of a higher dimensional function, and then implement the equations in the Cartesian coordinate system of this new embedding function. In the case of variational problem, we restrict the search of the minimizing map to the class of maps whose target is the level-set of interest. In the case of partial differential equations, we implicitly represent all the equation characteristics. We then obtain a set of equations that while defined on the whole Euclidean space, they are intrinsic to the implicit target manifold and map into it. This permits the use of classical numerical techniques in Cartesian grids, regardless of the geometry of the target manifold. The extension to open surfaces and submanifolds is addressed in this paper as well. In the latter case, the submanifold is defined as the intersection of two higher dimensional surfaces, and all the computations are restricted to this intersection. Examples of the applications of the framework here described include harmonic maps in liquid crystals, where the target manifold is an hypersphere; probability maps, where the target manifold is an hyperplane; chroma enhancement; texture mapping; and general geometric mapping between high dimensional surfaces.Item Variational problems and partial differential equations on implicit surfaces: The framework and examples in image processing and pattern formation(2000-06) Bertalmio, Marcelo; Cheng, Li-Tien; Osher, Stanley; Guillermo, Sapiro