100 research outputs found
Multiscale Representations for Manifold-Valued Data
We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere , the special orthogonal group , the positive definite matrices , and the Grassmann manifolds . The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the and maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as , , , where the and maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper
Definability and stability of multiscale decompositions for manifold-valued data
We discuss multiscale representations of discrete manifold-valued data. As it
turns out that we cannot expect general manifold-analogues of biorthogonal
wavelets to possess perfect reconstruction, we focus our attention on those
constructions which are based on upscaling operators which are either
interpolating or midpoint-interpolating. For definable multiscale
decompositions we obtain a stability result
Inpainting of Cyclic Data using First and Second Order Differences
Cyclic data arise in various image and signal processing applications such as
interferometric synthetic aperture radar, electroencephalogram data analysis,
and color image restoration in HSV or LCh spaces. In this paper we introduce a
variational inpainting model for cyclic data which utilizes our definition of
absolute cyclic second order differences. Based on analytical expressions for
the proximal mappings of these differences we propose a cyclic proximal point
algorithm (CPPA) for minimizing the corresponding functional. We choose
appropriate cycles to implement this algorithm in an efficient way. We further
introduce a simple strategy to initialize the unknown inpainting region.
Numerical results both for synthetic and real-world data demonstrate the
performance of our algorithm.Comment: accepted Converence Paper at EMMCVPR'1
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
Total variation regularization for manifold-valued data
We consider total variation minimization for manifold valued data. We propose
a cyclic proximal point algorithm and a parallel proximal point algorithm to
minimize TV functionals with -type data terms in the manifold case.
These algorithms are based on iterative geodesic averaging which makes them
easily applicable to a large class of data manifolds. As an application, we
consider denoising images which take their values in a manifold. We apply our
algorithms to diffusion tensor images, interferometric SAR images as well as
sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds
(which includes the data space in diffusion tensor imaging) we show the
convergence of the proposed TV minimizing algorithms to a global minimizer
A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images
We introduce a new non-smooth variational model for the restoration of
manifold-valued data which includes second order differences in the
regularization term. While such models were successfully applied for
real-valued images, we introduce the second order difference and the
corresponding variational models for manifold data, which up to now only
existed for cyclic data. The approach requires a combination of techniques from
numerical analysis, convex optimization and differential geometry. First, we
establish a suitable definition of absolute second order differences for
signals and images with values in a manifold. Employing this definition, we
introduce a variational denoising model based on first and second order
differences in the manifold setup. In order to minimize the corresponding
functional, we develop an algorithm using an inexact cyclic proximal point
algorithm. We propose an efficient strategy for the computation of the
corresponding proximal mappings in symmetric spaces utilizing the machinery of
Jacobi fields. For the n-sphere and the manifold of symmetric positive definite
matrices, we demonstrate the performance of our algorithm in practice. We prove
the convergence of the proposed exact and inexact variant of the cyclic
proximal point algorithm in Hadamard spaces. These results which are of
interest on its own include, e.g., the manifold of symmetric positive definite
matrices
Quasi-interpolation in Riemannian manifolds
We consider quasi-interpolation operators for functions assuming their values in a Riemannian manifold. We construct such operators from corresponding linear quasi-interpolation operators by replacing affine averages with the Riemannian centre of mass. As a main result, we show that the approximation rate of such a nonlinear operator is the same as for the linear operator it has been derived from. In order to formulate this result in an intrinsic way, we use the Sasaki metric to compare the derivatives of the function to be approximated with the derivatives of the nonlinear approximant. Numerical experiments confirm our theoretical finding
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