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We study inverse problems for the Poisson equation with source term the divergence of an $mathbf{R}^3$-valued measure, that is, the potential $Phi$ satisfies $$ Delta Phi= text{div} boldsymbol{mu}, $$ and $boldsymbol{mu}$ is to be reconstructed knowing (a component of) the field grad $Phi$ on a set disjoint from the support of $boldsymbol{mu}$. Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We develop methods for recovering $boldsymbol{mu}$ based on total variation regularization. We provide sufficient conditions for the unique recovery of $boldsymbol{mu}$, asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable. Numerical examples are provided to illustrate the main theoretical results.
We consider total variation minimization for manifold valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with $ell^p$-type data terms in the manifold case. These algorithms are
We propose a trust-region method that solves a sequence of linear integer programs to tackle integer optimal control problems regularized with a total variation penalty. The total variation penalty allows us to prove the existence of minimizers of
Various problems in computer vision and medical imaging can be cast as inverse problems. A frequent method for solving inverse problems is the variational approach, which amounts to minimizing an energy composed of a data fidelity term and a regulari
In order to determine the 3D structure of a thick sample, researchers have recently combined ptychography (for high resolution) and tomography (for 3D imaging) in a single experiment. 2-step methods are usually adopted for reconstruction, where the p
An overview is given of Bayesian inversion and regularization procedures. In particular, the conceptual basis of the maximum entropy method (MEM) is discussed, and extensions to positive/negative and complex data are highlighted. Other deconvolution