We present an approach for variational regularization of inverse and imaging problems for recovering functions with values in a set of vectors. We introduce regularization functionals, which are derivative-free double integrals of such functions. These regularization functionals are motivated from double integrals, which approximate Sobolev semi-norms of intensity functions. These were introduced in Bourgain et al. (Another look at Sobolev spaces. In: Menaldi, Rofman, Sulem (eds) Optimal control and partial differential equations-innovations and applications: in honor of professor Alain Bensoussan’s 60th anniversary, IOS Press, Amsterdam, pp 439–455,
2001
). For the proposed regularization functionals, we prove existence of minimizers as well as a stability and convergence result for functions with values in a set of vectors.
We present a family of non-local variational regularization methods for solving tomographic problems,
where the solutions are functions with range in a closed subset of the Euclidean space, for example if the
solution only attains values in an embedded sub-manifold. Recently, in [R. Ciak, M. Melching and O. Scherzer,
Regularization with metric double integrals of functions with values in a set of vectors,
J. Math. Imaging Vision 61 2019, 6, 824–848], such regularization methods
have been investigated analytically and their efficiency has been tested for basic imaging tasks such as denoising and
inpainting. In this paper we investigate solving complex vector tomography problems with non-local variational methods both analytically and numerically.
We present a family of non-local variational regularization methods for solving tomographic problems, where the solutions are functions with range in a closed subset of the Euclidean space, for example if the solution only attains values in an embedded sub-manifold. Recently, in [8], such regularization methods have been investigated analytically and their efficiency has been tested for basic imaging tasks such as denoising and inpainting. In this paper we investigate solving complex vector tomography problems with non-local variational methods both analytically and numerically.
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