Total Variation Regularization for Manifold-Valued Data

Weinmann, Andreas; Demaret, Laurent; Storath, Martin

doi:10.1137/130951075

Cited by 83 publications

(154 citation statements)

References 78 publications

(120 reference statements)

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“…In this section we will describe a general framework for regularizing shape signals, which is similar to the one proposed by Rudin, Osher, and Fatemi in [20] as well as the one considered by Weinmann et al [29] and Lellmann et al [14]. It is important to notice that this formulation does not depend on the particular choice of the shape space.…”

Section: Regularization Of Shape Signalsmentioning

confidence: 99%

“…Lellmann et al [14] and Weinmann et al [29]. A typical example is pose tracking data, e.g., acquired with an optical tracking system, which can be represented as a series of rigid transformation matrices acquired at equally spaced points in time.…”

Section: Motivationmentioning

confidence: 99%

“…Algorithms for TV regularization for manifold-valued data have been developed by Lellmann et al [14] and by Weinmann et al [29]. Applications considered in [14] are the one and two-dimensional sphere as well as the three-dimensional group of rotations.…”

Section: Tv Regularization For Manifold-valued Datamentioning

confidence: 99%

“…Unfortunately, the label space grows rapidly with the dimension of the manifold which makes application to higher dimensional manifolds, such as shape spaces, hard. In contrast to this, the algorithms described by Weinmann et al [29], i.e., the cyclic and the parallel proximal point algorithm, do not require such a discretization. Furthermore, the computational times do not increase more than linearly with the dimension of the manifold.…”

Section: Tv Regularization For Manifold-valued Datamentioning

confidence: 99%

“…Furthermore, the computational times do not increase more than linearly with the dimension of the manifold. Application areas in [29] also include the six-dimensional space of diffusion tensor data. It should be noted that there are also some works on fitting curves on manifolds which share the same goal but essentially consider ℓ 2 -regularized scenarios, e.g., [8,21,24,23].…”

Section: Tv Regularization For Manifold-valued Datamentioning

confidence: 99%

See 4 more Smart Citations

Total variation regularization of shape signals

Baust

Demaret

Storath

et al. 2015

2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)

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Section: Regularization Of Shape Signalsmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

Section: Tv Regularization For Manifold-valued Datamentioning

confidence: 99%

Section: Tv Regularization For Manifold-valued Datamentioning

confidence: 99%

Section: Tv Regularization For Manifold-valued Datamentioning

confidence: 99%

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Total variation regularization of shape signals

Baust

Demaret

Storath

et al. 2015

2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)

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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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Denoising sphere‐valued data by relaxed total variation regularization

Beinert,

Bresch

2024

Proc Appl Math and Mech

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Circle‐ and sphere‐valued data play a significant role in inverse problems like magnetic resonance phase imaging and radar interferometry, in the analysis of directional information, and in color restoration tasks. In this paper, we aim to restore ‐sphere‐valued signals exploiting the classical anisotropic total variation on the surrounding ‐dimensional Euclidean space. For this, we propose a novel variational formulation, whose data fidelity is based on inner products instead of the usually employed squared norms. Convexifying the resulting non‐convex problem and using ADMM, we derive an efficient and fast numerical denoiser. In the special case of binary (0‐sphere‐valued) signals, the relaxation is provable tight, that is, the relaxed solution can be used to construct a solution of the original non‐convex problem. Moreover, the tightness can be numerically observed for barcode and QR code denoising as well as in higher dimensional experiments like the color restoration using hue and chromaticity and the recovery of SO(3)‐valued signals.

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Total Variation Regularization for Manifold-Valued Data

Cited by 83 publications

References 78 publications

Total variation regularization of shape signals

Total variation regularization of shape signals

Riemannian Structure and Flows for Smooth Geometric Image Labeling

Denoising sphere‐valued data by relaxed total variation regularization

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