Total variation regularization for manifold-valued data

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 based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images, interferometric SAR images as well as sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer

Total variation regularization of multi-material topology optimization

This work is concerned with the determination of the diffusion coefficient from distributed data of the state. This problem is related to homogenization theory on the one hand and to regularization theory on the other hand. An approach is proposed which involves total variation regularization combined with a suitably chosen cost functional that promotes the diffusion coefficient assuming prespecified values at each point of the domain. The main difficulty lies in the delicate functional-analytic structure of the resulting nondifferentiable optimization problem with pointwise constraints for functions of bounded variation, which makes the derivation of useful pointwise optimality conditions challenging. To cope with this difficulty, a novel reparametrization technique is introduced. Numerical examples using a regularized semismooth Newton method illustrate the structure of the obtained diffusion coefficient.

Image Reconstruction from Undersampled Confocal Microscopy Data using Multiresolution Based Maximum Entropy Regularization

We consider the problem of reconstructing 2D images from randomly under-sampled confocal microscopy samples. The well known and widely celebrated total variation regularization, which is the L1 norm of derivatives, turns out to be unsuitable for this problem; it is unable to handle both noise and under-sampling together. This issue is linked with the notion of phase transition phenomenon observed in compressive sensing research, which is essentially the break-down of total variation methods, when sampling density gets lower than certain threshold. The severity of this breakdown is determined by the so-called mutual incoherence between the derivative operators and measurement operator. In our problem, the mutual incoherence is low, and hence the total variation regularization gives serious artifacts in the presence of noise even when the sampling density is not very low. There has been very few attempts in developing regularization methods that perform better than total variation regularization for this problem. We develop a multi-resolution based regularization method that is adaptive to image structure. In our approach, the desired reconstruction is formulated as a series of coarse-to-fine multi-resolution reconstructions; for reconstruction at each level, the regularization is constructed to be adaptive to the image structure, where the information for adaption is obtained from the reconstruction obtained at coarser resolution level. This adaptation is achieved by using maximum entropy principle, where the required adaptive regularization is determined as the maximizer of entropy subject to the information extracted from the coarse reconstruction as constraints. We demonstrate the superiority of the proposed regularization method over existing ones using several reconstruction examples

Finite element approximation of source term identification with TV-regularization

In this paper we investigate the problem of recovering the source term in an elliptic system from a measurement of the state on a part of the boundary. For the particular interest in reconstructing probably discontinuous sources, we use the standard least squares method with the total variation regularization. The finite element method is then applied to discretize the minimization problem, we show the stability and the convergence of this technique. Furthermore, we have proposed an algorithm to stably solve the minimization problem. We prove the iterate sequence generated by the derived algorithm converging to a minimizer of the regularization problem, and that convergence measurement is also established. Finally, a numerical experiment is presented to illustrate our theoretical findings.Comment: Inverse source problem, boundary observation, total variation regularization, ill-posedness, finite element method, stability and convergence, elliptic boundary value proble