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- ItemPatch-Wise Adaptive Weights Smoothing in R(Los Angeles, Calif. : UCLA, Dept. of Statistics, 2020) Polzehl, Jörg; Papafitsoros, Kostas; Tabelow, KarstenImage reconstruction from noisy data has a long history of methodological development and is based on a variety of ideas. In this paper we introduce a new method called patch-wise adaptive smoothing, that extends the propagation-separation approach by using comparisons of local patches of image intensities to define local adaptive weighting schemes for an improved balance of reduced variability and bias in the reconstruction result. We present the implementation of the new method in an R package aws and demonstrate its properties on a number of examples in comparison with other state-of-the art image reconstruction methods. © 2020, American Statistical Association. All rights reserved.
- ItemA function space framework for structural total variation regularization with applications in inverse problems(Bristol [u.a.] : Inst., 2018) Hintermüller, Michael; Holler, Martin; Papafitsoros, KostasIn this work, we introduce a function space setting for a wide class of structural/weighted total variation (TV) regularization methods motivated by their applications in inverse problems. In particular, we consider a regularizer that is the appropriate lower semi-continuous envelope (relaxation) of a suitable TV type functional initially defined for sufficiently smooth functions. We study examples where this relaxation can be expressed explicitly, and we also provide refinements for weighted TV for a wide range of weights. Since an integral characterization of the relaxation in function space is, in general, not always available, we show that, for a rather general linear inverse problems setting, instead of the classical Tikhonov regularization problem, one can equivalently solve a saddle-point problem where no a priori knowledge of an explicit formulation of the structural TV functional is needed. In particular, motivated by concrete applications, we deduce corresponding results for linear inverse problems with norm and Poisson log-likelihood data discrepancy terms. Finally, we provide proof-of-concept numerical examples where we solve the saddle-point problem for weighted TV denoising as well as for MR guided PET image reconstruction.
- ItemVariable Step Mollifiers and Applications(Berlin ; Heidelberg : Springer, 2020) Hintermüller, Michael; Papafitsoros, Kostas; Rautenberg, Carlos N.We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections. © 2020, The Author(s).
- ItemOptimization with learning-informed differential equation constraints and its applications(Les Ulis : EDP Sciences, 2022) Dong, Guozhi; Hintermüller, Michael; Papafitsoros, KostasInspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.
- ItemA physically oriented method for quantitative magnetic resonance imaging(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Dong, Guozhi; Hintermüller, Michael; Papafitsoros, KostasQuantitative magnetic resonance imaging (qMRI) denotes the task of estimating the values of magnetic and tissue parameters, e.g., relaxation times T1, T2, proton density p and others. Recently in [Ma et al., Nature, 2013], an approach named Magnetic Resonance Fingerprinting (MRF) was introduced, being capable of simultaneously recovering these parameters by using a two step procedure: (i) a series of magnetization maps are created and then (ii) these are matched to parameters with the help of a pre-computed dictionary (Bloch manifold). In this paper, we initially put MRF and its variants in the perspective of optimization and inverse problems, providing some mathematical insights into these methods. Motivated by the fact that the Bloch manifold is non-convex, and the accuracy of the MRF type algorithms is limited by the discretization size of the dictionary, we propose here a novel physically oriented method for qMRI. In contrast to the conventional two step models, our model is dictionary-free and it is described by a single nonlinear equation, governed by an operator for which we prove differentiability and other properties. This non-linear equation is efficiently solved via robust Newton type methods. The effectiveness of our method for noisy and undersampled data is shown both analytically and via numerical examples where also improvement over MRF and its variants is observed.
- ItemVariable step mollifiers and applications(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Hintermüller, Michael; Papafitsoros, Kostas; Rautenberg, Carlos N.We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections.
- ItemFirst-order conditions for the optimal control of learning-informed nonsmooth PDEs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Dong, Guozhi; Hintermüller, Michael; Papafitsoros, Kostas; Völkner, KathrinIn this paper we study the optimal control of a class of semilinear elliptic partial differential equations which have nonlinear constituents that are only accessible by data and are approximated by nonsmooth ReLU neural networks. The optimal control problem is studied in detail. In particular, the existence and uniqueness of the state equation are shown, and continuity as well as directional differentiability properties of the corresponding control-to-state map are established. Based on approximation capabilities of the pertinent networks, we address fundamental questions regarding approximating properties of the learning-informed control-to-state map and the solution of the corresponding optimal control problem. Finally, several stationarity conditions are derived based on different notions of generalized differentiability.
- ItemOptimization with learning-informed differential equation constraints and its applications(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Dong, Guozhi; Hintermüller, Michael; Papafitsoros, KostasInspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.
- ItemGenerating structured non-smooth priors and associated primal-dual methods(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Hintermüller, Michael; Papafitsoros, KostasThe purpose of the present chapter is to bind together and extend some recent developments regarding data-driven non-smooth regularization techniques in image processing through the means of a bilevel minimization scheme. The scheme, considered in function space, takes advantage of a dualization framework and it is designed to produce spatially varying regularization parameters adapted to the data for well-known regularizers, e.g. Total Variation and Total Generalized variation, leading to automated (monolithic), image reconstruction workflows. An inclusion of the theory of bilevel optimization and the theoretical background of the dualization framework, as well as a brief review of the aforementioned regularizers and their parameterization, makes this chapter a self-contained one. Aspects of the numerical implementation of the scheme are discussed and numerical examples are provided.
- ItemDualization and automatic distributed parameter selection of total generalized variation via bilevel optimization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Hintermüller, Michael; Papafitsoros, Kostas; Rautenberg, Carlos N.; Sun, HongpengTotal Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, two Newton type methods for the solution of the lower level problem, i.e. the image reconstruction problem, and two bilevel TGV algorithms are introduced, respectively. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.