A Graph Framework for Manifold-Valued Data

Bergmann, Ronny; Tenbrinck, Daniel

doi:10.1137/17m1118567

Cited by 18 publications

(13 citation statements)

References 74 publications

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“…These models are able to nicely reconstruct the linear parts in the ellipsoid and the edges of the boxes. Compared to the nonlocal methods [45] and [15] shown in (e) and (f), respectively, the TGV models have nearly the same error. However, looking at the paraboloid in the bottom right corner, they outperform the nonlocal methods visually.…”

Section: S 1 -Valued Datamentioning

confidence: 89%

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

Bergmann¹,

Fitschen²,

Persch³

et al. 2018

J Math Imaging Vis

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We generalize discrete variational models involving the infimal convolution (IC) of first and second order differences and the total generalized variation (TGV) to manifold-valued images. We propose both extrinsic and intrinsic approaches. The extrinsic models are based on embedding the manifold into an Euclidean space of higher dimension with manifold constraints. An alternating direction methods of multipliers can be employed for finding the minimizers. However, the components within the extrinsic IC or TGV decompositions live in the embedding space which makes their interpretation difficult. Therefore we investigate two intrinsic approaches: for Lie groups, we employ the group action within the models; for more general manifolds our IC model is based on recently developed absolute second order differences on manifolds, while our TGV approach uses an approximation of the parallel transport by the pole ladder. For computing the minimizers of the intrinsic models we apply gradient descent algorithms. Numerical examples demonstrate that our approaches work well for certain manifolds.

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Section: S 1 -Valued Datamentioning

confidence: 89%

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

Bergmann¹,

Fitschen²,

Persch³

et al. 2018

J Math Imaging Vis

Self Cite

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“…In this method, we have taken the domain, Ω, to be a Euclidean set. It would be very interesting to consider the case when Ω is a graph and the energy (4) is formulated using the analogous graph operators [Gen+14;BT18].…”

Section: Discussionmentioning

confidence: 99%

Diffusion generated methods for denoising target-valued images

Osting¹,

Wang²

2020

Inverse Problems &Amp; Imaging

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We consider the inverse problem of denoising an image where each point (pixel) is an element of a target set, which we refer to as a target-valued image. The target sets considered are either (i) a closed convex set of Euclidean space or (ii) a closed subset of the sphere such that the closest point mapping is defined almost everywhere. The energy for the denoising problem consists of an L 2 -fidelity term which is regularized by the Dirichlet energy. A relaxation of this energy, based on the heat kernel, is introduced and the associated minimization problem is proven to be well-posed. We introduce a diffusion generated method which can be used to efficiently find minimizers of this energy. We prove results for the stability and convergence of the method for both types of target sets. The method is demonstrated on a variety of synthetic and test problems, with associated target sets given by the semi-positive definite matrices, the cube, spheres, the orthogonal matrices, and the real projective line.

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“…Similar to [5] and [6], spatial regularization is introduced by adding the squared Riemannian distances d S between assignments in a spatial neighborhood N (i), controlled by the parameter ρ > 0. Similar to [5] and [6], spatial regularization is introduced by adding the squared Riemannian distances d S between assignments in a spatial neighborhood N (i), controlled by the parameter ρ > 0.…”

Section: Alternative Variational Modelmentioning

confidence: 99%

“…The function E ρ,α has as data-dependent term W i , D i which selects the best label fit for every pixel i depending on the distance information D i given f . Similar to [5] and [6], spatial regularization is introduced by adding the squared Riemannian distances d S between assignments in a spatial neighborhood N (i), controlled by the parameter ρ > 0. Since we deal with fully probabilistic assignments, an additional entropy term H(W i ) = − n j=1 W ij log(W ij ) is added for enforcing (approximately) Section 21: Mathematical signal and image processing integral assignments, depending on an integrality parameter α > 0.…”

Section: Alternative Variational Modelmentioning

confidence: 99%

Riemannian Structure and Flows for Smooth Geometric Image Labeling

Savarino

Schnörr

2019

Proc Appl Math and Mech

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The image labeling problem can be described as assigning to each pixel a single element from a finite set of predefined labels. Recently, a smooth geometric approach for inferring such label assignments was proposed by following the Riemannian gradient flow of a given objective function on the so-called assignment manifold. Due to the specific Riemannian structure, this results in a coupled replicator dynamic incorporating local spatial geometric averages of lifted data-dependent distances. However, in this framework an approximation of the flow is necessary in order to arrive at explicit formulas. We discuss preliminary results of an alternative model where lifting and averaging is decoupled in the objective function so as to stay closer to established approaches and at the same time preserve the ingredients of the original approach. As a consequence the resulting flow is explicitly given, without the need for any approximation, while still exploiting the underlying Riemannian structure. Motivation and PreliminariesSuppose f : V → F is a given image on a set of pixels V = {1, . . . , m} with values in some feature space F which might be a manifold. A labeling on V given f assigns to every pixel i ∈ V a label l(i) ∈ L from a predefined set of labels L = {l 1 , . . . , l n }. In [1] a new smooth geometric approach for image labeling was proposed. The basic idea is to encode (relaxed) labelings as points on the assignment manifold given by the set of row-stochastic matrices with full support, which is turned into a Riemannian manifold by choosing the Fisher-Rao (information) metric. In this geometric setting, every row W i := (W i1 , . . . , W in ) of an assignment matrix W ∈ W represents a probability assignment of labels L at pixel i ∈ V. Furthermore, a distance function d : F × L → R measuring the fit of labels to the data is assumed to be given. All the distance information is collected into a global matrix D := (d(f i , l j )) i,j ∈ R m×n and lifted onto W depending on the current assignment W . The Riemannian mean of these lifted distances in a spatial neighborhood around every pixel i ∈ V is used to define the similarity matrix S ∈ W. A labeling is inferred by maximizing the correlation between the current assignment W and the spatial geometric averages S(W, D) following the Riemannian gradient ascent flow on W. After ignoring the slow dynamics in the model and approximating the Riemannian mean to first order by the geometric mean, the resulting dynamical model in [1] is given by spatially coupled replicator equations called assignment flow.While this new approach results in fast numerical updates with good performance, there are some open mathematical aspects. It can be shown that the resulting assignment flow is not variational anymore, i.e. there exists no potential in the Fisher-Rao geometry. This is a consequence of the mentioned simplifying assumption and approximation, which are unavoidable for achieving explicit formulas and efficient numerical schemes. In the following we briefly present an alternat...

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A Graph Framework for Manifold-Valued Data

Cited by 18 publications

References 74 publications

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

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

Diffusion generated methods for denoising target-valued images

Riemannian Structure and Flows for Smooth Geometric Image Labeling

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