45 research outputs found
Total Generalized Variation for Manifold-valued Data
In this paper we introduce the notion of second-order total generalized
variation (TGV) regularization for manifold-valued data in a discrete setting.
We provide an axiomatic approach to formalize reasonable generalizations of TGV
to the manifold setting and present two possible concrete instances that
fulfill the proposed axioms. We provide well-posedness results and present
algorithms for a numerical realization of these generalizations to the manifold
setup. Further, we provide experimental results for synthetic and real data to
further underpin the proposed generalization numerically and show its potential
for applications with manifold-valued data
A First Derivative Potts Model for Segmentation and Denoising Using ILP
Unsupervised image segmentation and denoising are two fundamental tasks in
image processing. Usually, graph based models such as multicut are used for
segmentation and variational models are employed for denoising. Our approach
addresses both problems at the same time. We propose a novel ILP formulation of
the first derivative Potts model with the data term, where binary
variables are introduced to deal with the norm of the regularization
term. The ILP is then solved by a standard off-the-shelf MIP solver. Numerical
experiments are compared with the multicut problem.Comment: 6 pages, 2 figures. To appear at Proceedings of International
Conference on Operations Research 2017, Berli
Disparity and optical flow partitioning using extended Potts priors
This paper addresses the problems of disparity and optical flow partitioning based on the brightness invariance assumption. We investigate new variational approaches to these problems with Potts priors and possibly box constraints. For the optical flow partitioning, our model includes vector-valued data and an adapted Potts regularizer. Using the notion of asymptotically level stable (als) functions, we prove the existence of global minimizers of our functionals. We propose a modified alternating direction method of multipliers. This iterative algorithm requires the computation of global minimizers of classical univariate Potts problems which can be done efficiently by dynamic programming. We prove that the algorithm converges both for the constrained and unconstrained problems. Numerical examples demonstrate the very good performance of our partitioning method
Simple, Accurate, and Robust Nonparametric Blind Super-Resolution
This paper proposes a simple, accurate, and robust approach to single image
nonparametric blind Super-Resolution (SR). This task is formulated as a
functional to be minimized with respect to both an intermediate super-resolved
image and a nonparametric blur-kernel. The proposed approach includes a
convolution consistency constraint which uses a non-blind learning-based SR
result to better guide the estimation process. Another key component is the
unnatural bi-l0-l2-norm regularization imposed on the super-resolved, sharp
image and the blur-kernel, which is shown to be quite beneficial for estimating
the blur-kernel accurately. The numerical optimization is implemented by
coupling the splitting augmented Lagrangian and the conjugate gradient (CG).
Using the pre-estimated blur-kernel, we finally reconstruct the SR image by a
very simple non-blind SR method that uses a natural image prior. The proposed
approach is demonstrated to achieve better performance than the recent method
by Michaeli and Irani [2] in both terms of the kernel estimation accuracy and
image SR quality
Amplitude and sign decompositions by complex wavelets - Theory and applications to image analysis.
The present thesis deals with amplitude and sign decompositions of images via complex wavelets. We develop the monogenic curvelet transform for the directional decomposition of images into amplitude and sign. We use the amplitude information for the development and the analysis of an algorithm for the separation of edges in images, and the sign information for a new edge detection method. Eventually, we draw a connection between amplitude and sign representations and the perception of optical illusions
Wavelet sparse regularization for manifold-valued data.
n this paper, we consider the sparse regularization of manifold-valued data with respect to an interpolatory wavelet/multiscale transform. We propose and study variational models for this task and provide results on their well-posedness. We present algorithms for a numerical realization of these models in the manifold setup. Further, we provide experimental results to show the potential of the proposed schemes for applications
Iterative Potts and Blake-Zisserman minimization for the recovery of functions with discontinuities from indirect measurements.
Signals with discontinuities appear in many problems in the applied sciences ranging from mechanics, electrical engineering to biology and medicine. The concrete data acquired are typically discrete, indirect and noisy measurements of some quantities describing the signal under consideration. The task is to restore the signal and, in particular, the discontinuities. In this respect, classical methods perform rather poor, whereas non-convex non-smooth variational methods seem to be the correct choice. Examples are methods based on Mumford-Shah and piecewise constant Mumford-Shah functionals and discretized versions which are known as Blake- Zisserman and Potts functionals. Owing to their non-convexity, minimization of such functionals is challenging. In this paper, we propose a new iterative minimization strategy for Blake-Zisserman as well as Potts functionals and a related jump-sparsity problem dealing with indirect, noisy measurements. We provide a convergence analysis and underpin our findings with numerical experiments