arXiv subjects

Gabriele Steidl

Publications and source records attributed to Gabriele Steidl.

At least 91 records · Page 5Linked to original sources

arXiv2020

A New Constrained Optimization Model for Solving the Nonsymmetric Stochastic Inverse Eigenvalue Problem

The stochastic inverse eigenvalue problem aims to reconstruct a stochastic matrix from its spectrum. While there exists a large literature on the existence of solutions for special settings, there are only few numerical solution methods available so far. Recently, Zhao et al. (2016) proposed a constrained optimization model on the manifold of so-called isospectral matrices and adapted a modified Polak-Ribière-Polyak conjugate gradient method to the geometry of this manifold. However, not every stochastic matrix is an isospectral one and the model from Zhao et al. is based on the assumption that for each stochastic matrix there exists a (possibly different) isospectral, stochastic matrix with the same spectrum. We are not aware of such a result in the literature, but will see that the claim is at least true for $3 \times 3$ matrices. In this paper, we suggest to extend the above model by considering matrices which differ from isospectral ones only by multiplication with a block diagonal matrix with $2 \times 2$ blocks from the special linear group $SL(2)$, where the number of blocks is given by the number of pairs of complex-conjugate eigenvalues. Every stochastic matrix can be written in such a form, which was not the case for the form of the isospectral matrices. We prove that our model has a minimizer and show how the Polak-Ribière-Polyak conjugate gradient method works on the corresponding more general manifold. We demonstrate by numerical examples that the new, more general method performs similarly as the one from Zhao et al.

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arXiv2020

Alternatives to the EM Algorithm for ML-Estimation of Location, Scatter Matrix and Degree of Freedom of the Student-$t$ Distribution

In this paper, we consider maximum likelihood estimations of the degree of freedom parameter $ν$, the location parameter $μ$ and the scatter matrix $Σ$ of the multivariate Student-$t$ distribution. In particular, we are interested in estimating the degree of freedom parameter $ν$ that determines the tails of the corresponding probability density function and was rarely considered in detail in the literature so far. We prove that under certain assumptions a minimizer of the negative log-likelihood function exists, where we have to take special care of the case $ν\rightarrow \infty$, for which the Student-$t$ distribution approaches the Gaussian distribution. As alternatives to the classical EM algorithm we propose three other algorithms which cannot be interpreted as EM algorithm. For fixed $ν$, the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fix $ν$, we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step for $ν$. We show how the objective function behaves for the different updates of $ν$ and prove for all three algorithms that it decreases in each iteration step. We compare the algorithms as well as some accelerated versions by numerical simulation and apply one of them for estimating the degree of freedom parameter in images corrupted by Student-$t$ noise.

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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.

math.OC↗

arXiv2019

Linkage between piecewise constant Mumford-Shah model and ROF model and its virtue in image segmentation

The piecewise constant Mumford-Shah (PCMS) model and the Rudin-Osher-Fatemi (ROF) model are two important variational models in image segmentation and image restoration, respectively. In this paper, we explore a linkage between these models. We prove that for the two-phase segmentation problem a partial minimizer of the PCMS model can be obtained by thresholding the minimizer of the ROF model. A similar linkage is still valid for multiphase segmentation under specific assumptions. Thus it opens a new segmentation paradigm: image segmentation can be done via image restoration plus thresholding. This new paradigm, which circumvents the innate non-convex property of the PCMS model, therefore improves the segmentation performance in both efficiency (much faster than state-of-the-art methods based on PCMS model, particularly when the phase number is high) and effectiveness (producing segmentation results with better quality) due to the flexibility of the ROF model in tackling degraded images, such as noisy images, blurry images or images with information loss. As a by-product of the new paradigm, we derive a novel segmentation method, called thresholded-ROF (T-ROF) method, to illustrate the virtue of managing image segmentation through image restoration techniques. The convergence of the T-ROF method is proved, and elaborate experimental results and comparisons are presented.

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arXiv2019

Multivariate Myriad Filters based on Parameter Estimation of Student-$t$ Distributions

The contribution of this study is twofold: First, we propose an efficient algorithm for the computation of the (weighted) maximum likelihood estimators for the parameters of the multivariate Student-$t$ distribution, which we call generalized multivariate myriad filter. Second, we use the generalized multivariate myriad filter in a nonlocal framework for the denoising of images corrupted by different kinds of noise. The resulting method is very flexible and can handle heavy-tailed noise such as Cauchy noise, as well as the other extreme, namely Gaussian noise. Furthermore, we detail how the limiting case $ν\rightarrow 0$ of the projected normal distribution in two dimensions can be used for the robust denoising of periodic data, in particular for images with circular data corrupted by wrapped Cauchy noise.

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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

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

Recent Advances in Denoising of Manifold-Valued Images

Modern signal and image acquisition systems are able to capture data that is no longer real-valued, but may take values on a manifold. However, whenever measurements are taken, no matter whether manifold-valued or not, there occur tiny inaccuracies, which result in noisy data. In this chapter, we review recent advances in denoising of manifold-valued signals and images, where we restrict our attention to variational models and appropriate minimization algorithms. The algorithms are either classical as the subgradient algorithm or generalizations of the half-quadratic minimization method, the cyclic proximal point algorithm, and the Douglas-Rachford algorithm to manifolds. An important aspect when dealing with real-world data is the practical implementation. Here several groups provide software and toolboxes as the Manifold Optimization (Manopt) package and the manifold-valued image restoration toolbox (MVIRT).

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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

On the Atomic Decomposition of Coorbit Spaces with Non-Integrable Kernel

This paper ist concerned with recent progress in the context of coorbit space theory. Based on a square integrable group representation, the coorbit theory provides new families of associated smoothness spaces, where the smoothness of a function is measured by the decay of the associated voice transform. Moreover, by discretizing the representation, atomic decomposi- tions and Banach frames can be constructed. Usually, the whole machinery works well if the associated reproducing kernel is integrable with respect to a weighted Haar measure on the group. In recent studies, it has turned out that to some extent coorbit spaces can still be established if this condition is violated. In this paper, we clarify in which sense atomic decompositions and Banach frames for these generalized coorbit spaces can be obtained.

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arXiv2018

Priors with Coupled First and Second Order Differences for Manifold-Valued Image Processing

Recently variational models with priors involving first and second order derivatives resp. differences were successfully applied for image restoration. There are several ways to incorporate the derivatives of first and second order into the prior, for example additive coupling or using infimal convolution (IC), as well as the more general model of total generalized variation (TGV). The later two methods give also decompositions of the restored images into image components with distinct "smoothness" properties which are useful in applications. This paper is the first attempt to generalize these models to manifold-valued images. We propose both extrinsic and intrinsic approaches. The extrinsic approach is based on embedding the manifold into an Euclidean space of higher dimension. Models following this approach can be formulated within the Euclidean space with a constraint restricting them to the manifold. Then alternating direction methods of multipliers can be employed for finding minima. However, the components within the infimal convolution or total generalized variation decomposition live in the embedding space rather than on the manifold which makes their interpretation difficult. Therefore we also investigate two intrinsic approaches. For manifolds which are Lie groups we propose three priors which exploit the group operation, an additive one, another with IC coupling and a third TGV like one. For computing the minimizers of the intrinsic models we apply gradient descent algorithms. For general Riemannian manifolds we further define a model for infimal convolution based on the recently developed second order differences. We demonstrate by numerical examples that our approaches works well for the circle, the 2-sphere, the rotation group, and the manifold of positive definite matrices with the affine invariant metric.

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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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arXiv2018

Nonlocal Myriad Filters for Cauchy Noise Removal

The contribution of this paper is two-fold. First, we introduce a generalized myriad filter, which is a method to compute the joint maximum likelihood estimator of the location and the scale parameter of the Cauchy distribution. Estimating only the location parameter is known as myriad filter. We propose an efficient algorithm to compute the generalized myriad filter and prove its convergence. Special cases of this algorithm result in the classical myriad filtering, respective an algorithm for estimating only the scale parameter. Based on an asymptotic analysis, we develop a second, even faster generalized myriad filtering technique. Second, we use our new approaches within a nonlocal, fully unsupervised method to denoise images corrupted by Cauchy noise. Special attention is paid to the determination of similar patches in noisy images. Numerical examples demonstrate the excellent performance of our algorithms which have moreover the advantage to be robust with respect to the parameter choice.

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arXiv2018

Strain Analysis by a Total Generalized Variation Regularized Optical Flow Model

In this paper we deal with the important problem of estimating the local strain tensor from a sequence of micro-structural images realized during deformation tests of engineering materials. Since the strain tensor is defined via the Jacobian of the displacement field, we propose to compute the displacement field by a variational model which takes care of properties of the Jacobian of the displacement field. In particular we are interested in areas of high strain. The data term of our variational model relies on the brightness invariance property of the image sequence. As prior we choose the second order total generalized variation of the displacement field. This prior splits the Jacobian of the displacement field into a smooth and a non-smooth part. The latter reflects the material cracks. An additional constraint is incorporated to handle physical properties of the non-smooth part for tensile tests. We prove that the resulting convex model has a minimizer and show how a primal-dual method can be applied to find a minimizer. The corresponding algorithm has the advantage that the strain tensor is directly computed within the iteration process. Our algorithm is further equipped with a coarse-to-fine strategy to cope with larger displacements. Numerical examples with simulated and experimental data demonstrate the very good performance of our algorithm. In comparison to state-of-the-art engineering software for strain analysis our method can resolve local phenomena much better.

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arXiv2017

Convergence of Functions and their Moreau Envelopes on Hadamard Spaces

A well known result of H. Attouch states that the Mosco convergence of a sequence of proper convex lower semicontinuous functions defined on a Hilbert space is equivalent to the pointwise convergence of the associated Moreau envelopes. In the present paper we generalize this result to Hadamard spaces. More precisely, while it has already been known that the Mosco convergence of a sequence of convex lower semicontinuous functions on a Hadamard space implies the pointwise convergence of the corresponding Moreau envelopes, the converse implication was an open question. We now fill this gap. Our result has several consequences. It implies, for instance, the equivalence of the Mosco and Frolík-Wijsman convergences of convex sets. As another application, we show that there exists a complete metric on the cone of proper convex lower semicontinuous functions on a separable Hadamard space such that a sequence of functions converges in this metric if and only if it converges in the sense of Mosco.

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arXiv2017

Examplar-Based Face Colorization Using Image Morphing

Colorization of gray-scale images relies on prior color information. Examplar-based methods use a color image as source of such information. Then the colors of the source image are transferred to the gray-scale image. In the literature, this transfer is mainly guided by texture descriptors. Face images usually contain few texture so that the common approaches frequently fail. In this paper we propose a new method based on image morphing. This technique is able to compute a correspondence map between images with similar shapes. It is based on the geometric structure of the images rather than textures which is more reliable for faces. Our numerical experiments show that our morphing based approach clearly outperforms state-of-the-art methods.

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arXiv2017

Unsupervised Multi Class Segmentation of 3D Images with Intensity Inhomogeneities

Intensity inhomogeneities in images constitute a considerable challenge in image segmentation. In this paper we propose a novel biconvex variational model to tackle this task. We combine a total variation approach for multi class segmentation with a multiplicative model to handle the inhomogeneities. Our method assumes that the image intensity is the product of a smoothly varying part and a component which resembles important image structures such as edges. Therefore, we penalize in addition to the total variation of the label assignment matrix a quadratic difference term to cope with the smoothly varying factor. A critical point of our biconvex functional is computed by a modified proximal alternating linearized minimization method (PALM). We show that the assumptions for the convergence of the algorithm are fulfilled by our model. Various numerical examples demonstrate the very good performance of our method. Particular attention is paid to the segmentation of 3D FIB tomographical images which was indeed the motivation of our work.

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