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Total Variation Regularization by Iteratively Reweighted Least Squares on Hadamard Spaces and the Sphere

by P. Grohs and M. Sprecher

(Report number 2014-39)

Abstract
We consider the problem of reconstructing an image from noisy and/or incomplete data, where the image/data take values in a metric space \(X\) (e.g. \(\mathbb{R}\) for grayscale, \(S^2\) for the chromaticity component of RGB-images or \(SPD(3)\), the set of positive definite \(3\times 3\)-Matrices, for Diffusion Tensor Magnetic Resonance Imaging (DT-MRI)). We use the common technique of minimizing a total variation (TV) functional \(J\). After having defined \(J\) for arbitrary metric spaces \(X\) we will propose an adaption of the Iteratively Reweighted Least Squares (IRLS) algorithm to minimize \(J\). For the case of \(X\) being a Hadamard space, such as \(SPD(n)\), we prove existence and uniqueness of a minimizer of a regularized functional \(J^\epsilon\) where \(\epsilon>0\) and show that these minimizers convergence to a minimizer of \(J\) when the regularization parameter \(\epsilon\) tends to zero. We show that IRLS can also be applied for \(X\) being a half-sphere. For the case of \(X\) being a Riemannian manifold we propose to use Newton's method on Manifolds to numerically compute the minimizer of \(J^\epsilon\). To demonstrate our algorithm we present some numerical experiments where we denoise and/or inpaint sphere-valued and SPD-valued images.

Keywords: Iteratively reweighted least squares, total variation, regularization, manifold-valued data

@Techreport{GS14_589,
  author = {P. Grohs and M. Sprecher},
  title = {Total Variation Regularization by Iteratively Reweighted Least Squares on Hadamard Spaces and the Sphere},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-39},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-39.pdf },
  year = {2014}
}

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