arXiv subjects

Sebastian Neumayer

Publications and source records attributed to Sebastian Neumayer.

45 records · Page 3Linked to original sources

arXiv2020

Parseval Proximal Neural Networks

The aim of this paper is twofold. First, we show that a certain concatenation of a proximity operator with an affine operator is again a proximity operator on a suitable Hilbert space. Second, we use our findings to establish so-called proximal neural networks (PNNs) and stable tight frame proximal neural networks. Let $\mathcal H$ and $\mathcal K$ be real Hilbert spaces, $b\in\mathcal K$ and $T\in\mathcal{B}(\mathcal H,\mathcal K)$ have closed range and Moore-Penrose inverse $T^\dagger$. Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $\text{Prox}\colon\mathcal K\to\mathcal K$ the operator $T^\dagger\,\text{Prox} (T\cdot +b)$ is a proximity operator on $\mathcal H$ equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $\text{Prox} = S_λ\colon\ell_2 \rightarrow\ell_2$ and any frame analysis operator $T\colon\mathcal H\to\ell_2$ that the frame shrinkage operator $T^\dagger\, S_λ\,T$ is a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on $\mathbb R^d$ equipped with different norms establishes a PNN. If the network arises from tight frame analysis or synthesis operators, then it forms an averaged operator. Hence, it has Lipschitz constant 1 and belongs to the class of so-called Lipschitz networks, which were recently applied to defend against adversarial attacks. Moreover, due to its averaging property, PNNs can be used within so-called Plug-and-Play algorithms with convergence guarantee. In case of Parseval frames, we call the networks Parseval proximal neural networks (PPNNs). Then, the involved linear operators are in a Stiefel manifold and corresponding minimization methods can be applied for training. Finally, some proof-of-the concept examples demonstrate the performance of PPNNs.

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arXiv2020

Convergence of the Time Discrete Metamorphosis Model on Hadamard Manifolds

Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouvé, Younes and coworkers casts this problem in the frame of Riemannian geometry and geodesic paths between images. The associated metric in the space of images incorporates dissipation caused by a viscous flow transporting image intensities and its variations along motion paths. In many applications, images are maps from the image domain into a manifold (e.g. in diffusion tensor imaging (DTI) the manifold of symmetric positive definite matrices with a suitable Riemannian metric). In this paper, we propose a generalized metamorphosis model for manifold-valued images, where the range space is a finite-dimensional Hadamard manifold. A corresponding time discrete version was presented by Neumayer et al. based on the general variational time discretization proposed by Berkels et al. Here, we prove the Mosco--convergence of the time discrete metamorphosis functional to the proposed manifold-valued metamorphosis model, which implies the convergence of time discrete geodesic paths to a geodesic path in the (time continuous) metamorphosis model. In particular, the existence of geodesic paths is established. In fact, images as maps into Hadamard manifold are not only relevant in applications, but it is also shown that the joint convexity of the distance function which characterizes Hadamard manifolds is a crucial ingredient to establish existence of the metamorphosis model.

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arXiv2019

Frame Soft Shrinkage as Proximity Operator

Let $\mathcal H$ and $\mathcal K$ be real Hilbert spaces and $T \in \mathcal{B} (\mathcal H,\mathcal K)$ an injective operator with closed range and Moore-Penrose inverse $T^\dagger$. Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $\text{Prox} \colon \mathcal K \to \mathcal K$ the operator $T^\dagger \, \text{Prox} \, T$ is a proximity operator on the linear space $\mathcal H$ equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $\text{Prox} = S_λ\colon \ell_2 \rightarrow \ell_2$ and any frame analysis operator $T\colon \mathcal H \to \ell_2$, that the frame shrinkage operator $T^\dagger\, S_λ\, T$ is a proximity operator in a suitable Hilbert space.

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arXiv2019

On the Rotational Invariant $L_1$-Norm PCA

Principal component analysis (PCA) is a powerful tool for dimensionality reduction. Unfortunately, it is sensitive to outliers, so that various robust PCA variants were proposed in the literature. Among them the so-called rotational invariant $L_1$-norm PCA is rather popular. In this paper, we reinterpret this robust method as conditional gradient algorithm and show moreover that it coincides with a gradient descent algorithm on Grassmannian manifolds. Based on this point of view, we prove for the first time convergence of the whole series of iterates to a critical point using the Kurdyka-Łojasiewicz property of the energy functional.

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arXiv2019

Template-Based Image Reconstruction from Sparse Tomographic Data

We propose a variational regularisation approach for the problem of template-based image reconstruction from indirect, noisy measurements as given, for instance, in X-ray computed tomography. An image is reconstructed from such measurements by deforming a given template image. The image registration is directly incorporated into the variational regularisation approach in the form of a partial differential equation that models the registration as either mass- or intensity-preserving transport from the template to the unknown reconstruction. We provide theoretical results for the proposed variational regularisation for both cases. In particular, we prove existence of a minimiser, stability with respect to the data, and convergence for vanishing noise when either of the abovementioned equations is imposed and more general distance functions are used. Numerically, we solve the problem by extending existing Lagrangian methods and propose a multilevel approach that is applicable whenever a suitable downsampling procedure for the operator and the measured data can be provided. Finally, we demonstrate the performance of our method for template-based image reconstruction from highly undersampled and noisy Radon transform data. We compare results for mass- and intensity-preserving image registration, various regularisation functionals, and different distance functions. Our results show that very reasonable reconstructions can be obtained when only few measurements are available and demonstrate that the use of a normalised cross correlation-based distance is advantageous when the image intensities between the template and the unknown image differ substantially.

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arXiv2019

Minimal Lipschitz and $\infty$-Harmonic Extensions of Vector-Valued Functions on Finite Graphs

This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called lex and L-lex minimal extensions are actually the same and call them minimal Lipschitz extensions. Then we prove that the solution of the graph $p$-Laplacians converge to these extensions as $p\to \infty$. Furthermore, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to $\infty$-Laplacians for scalar-valued functions. A convergence proof for an iterative algorithm proposed by Elmoataz et al.~(2014) for finding the zero of the $\infty$-Laplacian is given. Finally, we present applications in image inpainting.

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arXiv2019

On the Robust PCA and Weiszfeld's Algorithm

Principal component analysis (PCA) is a powerful standard tool for reducing the dimensionality of data. Unfortunately, it is sensitive to outliers so that various robust PCA variants were proposed in the literature. This paper addresses the robust PCA by successively determining the directions of lines having minimal Euclidean distances from the data points. The corresponding energy functional is not differentiable at a finite number of directions which we call anchor directions. We derive a Weiszfeld-like algorithm for minimizing the energy functional which has several advantages over existing algorithms. Special attention is paid to the careful handling of the anchor directions, where we take the relation between local minima and one-sided derivatives of Lipschitz continuous functions on submanifolds of $\mathbb R^d$ into account. Using ideas for stabilizing the classical Weiszfeld algorithm at anchor points and the Kurdyka-Łojasiewicz property of the energy functional, we prove global convergence of the whole sequence of iterates generated by the algorithm to a critical point of the energy functional. Numerical examples demonstrate the very good performance of our algorithm.

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arXiv2018

Regularization of Inverse Problems via Time Discrete Geodesics in Image Spaces

This paper addresses the solution of inverse problems in imaging given an additional reference image. We combine a modification of the discrete geodesic path model for image metamorphosis with a variational model,actually the $L^2$-$TV$ model, for image reconstruction. We prove that the space continuous model has a minimizer which depends in a stable way from the input data. Two minimization procedures which alternate over the involved sequences of deformations and images in different ways are proposed. The updates with respect to the image sequence exploit recent algorithms from convex analysis to minimize the $L^2$-$TV$ functional. For the numerical computation we apply a finite difference approach on staggered grids together with a multilevel strategy. We present proof-of-the-concept numerical results for sparse and limited angle computerized tomography as well as for superresolution demonstrating the power of the method.

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arXiv2018

Morphing of Manifold-Valued Images inspired by Discrete Geodesics in Image Spaces

This paper addresses the morphing of manifold-valued images based on the time discrete geodesic paths model of Berkels, Effland and Rumpf 2015. Although for our manifold-valued setting such an interpretation of the energy functional is not available so far, the model is interesting on its own. We prove the existence of a minimizing sequence within the set of $L^2(Ω,\mathcal{H})$ images having values in a finite dimensional Hadamard manifold $\mathcal{H}$ together with a minimizing sequence of admissible diffeomorphisms. To this end, we show that the continuous manifold-valued functions are dense in $L^2(Ω,\mathcal{H})$. We propose a space discrete model based on a finite difference approach on staggered grids, where we focus on the linearized elastic potential in the regularizing term. The numerical minimization alternates between i) the computation of a deformation sequence between given images via the parallel solution of certain registration problems for manifold-valued images, and ii) the computation of an image sequence with fixed first (template) and last (reference) frame based on a given sequence of deformations via the solution of a system of equations arising from the corresponding Euler-Lagrange equation. Numerical examples give a proof of the concept of our ideas.

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