Normal Multi-scale Transforms for Curves

Harizanov, Stanislav; Oswald, Peter; Shingel, Tatiana

doi:10.1007/s10208-011-9104-6

Cited by 6 publications

(4 citation statements)

References 19 publications

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“…Similar conditions also appear in the convergence analysis of nonlinear subdivision schemes for manifold-valued data in [47,48]. In the context of nonlinear subdivision schemes, even more severe restrictions such as 'almost equally spaced data' are frequently required [26]. This imposes additional conditions on the second order differences to make the data almost lie on a 'line'.…”

Section: Convergencementioning

confidence: 83%

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Second Order Differences of Cyclic Data and Applications in Variational Denoising

Bergmann¹,

Laus²,

Steidl³

et al. 2014

SIAM J. Imaging Sci.

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In many image and signal processing applications, such as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis, ground-based astronomy, and color image restoration, in HSV or LCh spaces the data has its range on the one-dimensional sphere S 1 . Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edge-preserving image restoration , such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the so-called staircasing effect. This effect can be avoided by involving higher order derivatives into the functional. This is the first paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration. We introduce absolute higher order differences for S 1 -valued data in a sound way which is independent of the chosen representation system on the circle. Our absolute cyclic first order difference is just the geodesic distance between points. Similar to the geodesic distances, the absolute cyclic second order differences have only values in [0, π]. We update the cyclic variational TV approach by our new cyclic second order differences. To minimize the corresponding functional we apply a cyclic proximal point method which was recently successfully proposed for Hadamard manifolds. Choosing appropriate cycles this algorithm can be implemented in an efficient way. The main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions. Under certain conditions we prove the convergence of our algorithm. Various numerical examples with artificial as well as real-world data demonstrate the advantageous performance of our algorithm.

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Section: Convergencementioning

confidence: 83%

“…Then we can apply Remark 4.5 and conclude that the minimizer y * of J fulfills y * − f 2 ≤ ε < π 16 . By (26) we obtain…”

Section: Convergencementioning

confidence: 97%

Second Order Differences of Cyclic Data and Applications in Variational Denoising

Bergmann¹,

Laus²,

Steidl³

et al. 2014

SIAM J. Imaging Sci.

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“…This means that the data has to be locally nearby which does not mean that circular data components are (globally) restricted to certain sectors -the data in these components my wrap around. Similar restrictions on the nearness of data and even more severe restrictions requiring almost equidistant-data have been imposed in the analysis of nonlinear subdivision schemes; see, e.g., [46,82,84]. As it is also pointed out in these references, the analysis is qualitative in the sense that empirically convergence is observed for a significantly wider range of input data.…”

Section: Convergence Analysismentioning

confidence: 83%

“…As already pointed out, such restrictions on the nearness of data are typical for the analysis of algorithms of nonlinear data in general; cf. [6,46,82,84]. We note that d Ω ∞ (f ) depends on the chosen covering and that we suppress this dependence in the notation.…”

Section: Convergence Analysismentioning

confidence: 99%

A Second-Order TV-Type Approach for Inpainting and Denoising Higher Dimensional Combined Cyclic and Vector Space Data

Bergmann

Weinmann

2016

J Math Imaging Vis

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Abstract. In this paper we consider denoising and inpainting problems for higher dimensional combined cyclic and linear space valued data. These kind of data appear when dealing with nonlinear color spaces such as HSV, and they can be obtained by changing the space domain of, e.g., an optical flow field to polar coordinates. For such nonlinear data spaces, we develop algorithms for the solution of the corresponding second order total variation (TV) type problems for denoising, inpainting as well as the combination of both. We provide a convergence analysis and we apply the algorithms to concrete problems.

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Normal Multi-scale Transforms for Surfaces

Oswald

2012

Curves and Surfaces

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Normal Multi-scale Transforms for Curves

Cited by 6 publications

References 19 publications

Second Order Differences of Cyclic Data and Applications in Variational Denoising

Second Order Differences of Cyclic Data and Applications in Variational Denoising

A Second-Order TV-Type Approach for Inpainting and Denoising Higher Dimensional Combined Cyclic and Vector Space Data

Normal Multi-scale Transforms for Surfaces

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