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Register a new userMumford-Shah and Potts Regularization for Manifold-Valued Data with Applications to DTI and Q-Ball Imaging
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Added by
Martin Storath
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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Mumford-Shah and Potts functionals are powerful variational models for regularization which are widely used in signal and image processing; typical applications are edge-preserving denoising and segmentation. Being both non-smooth and non-convex, they are computationally challenging even for scalar data. For manifold-valued data, the problem becomes even more involved since typical features of vector spaces are not available. In this paper, we propose algorithms for Mumford-Shah and for Potts regularization of manifold-valued signals and images. For the univariate problems, we derive solvers based on dynamic programming combined with (convex) optimization techniques for manifold-valued data. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging), we show that our algorithms compute global minimizers for any starting point. For the multivariate Mumford-Shah and Potts problems (for image regularization) we propose a splitting into suitable subproblems which we can solve exactly using the techniques developed for the corresponding univariate problems. Our method does not require any a priori restrictions on the edge set and we do not have to discretize the data space. We apply our method to diffusion tensor imaging (DTI) as well as Q-ball imaging. Using the DTI model, we obtain a segmentation of the corpus callosum.
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In this paper, we consider the variational regularization of manifold-valued data in the inverse problems setting. In particular, we consider TV and TGV regularization for manifold-valued data with indirect measurement operators. We provide results on the well-posedness and present algorithms for a numerical realization of these models in the manifold setup. Further, we provide experimental results for synthetic and real data to show the potential of the proposed schemes for applications.
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Minimizing the Mumford-Shah functional is frequently used for smoothing signals or time series with discontinuities. A significant limitation of the standard Mumford-Shah model is that linear trends -- and in general polynomial trends -- in the data are not well preserved. This can be improved by building on splines of higher order which leads to higher order Mumford-Shah models. In this work, we study these models in the univariate situation: we discuss important differences to the first order Mumford-Shah model, and we obtain uniqueness results for their solutions. As a main contribution, we derive fast minimization algorithms for Mumford-Shah models of arbitrary orders. We show that the worst case complexity of all proposed schemes is quadratic in the length of the signal. Remarkably, they thus achieve the worst case complexity of the fastest solver for the piecewise constant Mumford-Shah model (which is the simplest model of the class). Further, we obtain stability results for the proposed algorithms. We complement these results with a numerical study. Our reference implementation processes signals with more than 10,000 elements in less than one second.
The Ambrosio-Tortorelli functional is a phase-field approximation of the Mumford-Shah functional that has been widely used for image segmentation. The approximation has the advantages of being easy to implement, maintaining the segmentation ability, and $Gamma$-converging to the Mumford-Shah functional. However, it has been observed in actual computation that the segmentation ability of the Ambrosio-Tortorelli functional varies significantly with different values of the parameter and it even fails to $Gamma$-converge to the original functional for some cases. In this paper we present an asymptotic analysis on the gradient flow equation of the Ambrosio-Tortorelli functional and show that the functional can have different segmentation behavior for small but finite values of the regularization parameter and eventually loses its segmentation ability as the parameter goes to zero when the input image is treated as a continuous function. This is consistent with the existing observation as well as the numerical examples presented in this work. A selection strategy for the regularization parameter and a scaling procedure for the solution are devised based on the analysis. Numerical results show that they lead to good segmentation of the Ambrosio-Tortorelli functional for real images.
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