2 research outputs found
Second Order Differences of Cyclic Data and Applications in Variational Denoising
In many image and signal processing applications, as interferometric
synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis or
color image restoration in HSV or LCh spaces the data has its range on the
one-dimensional sphere . 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 -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,{\pi}]. 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.Comment: 32 pages, 16 figures, shortened version of submitted manuscrip
A Second Order TV-type Approach for Inpainting and Denoising Higher Dimensional Combined Cyclic and Vector Space Data
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.Comment: revised submitted versio