Abstract
Signals and images with discontinuities appear in many problems in such diverse areas as biology, medicine, mechanics and electrical engineering. The concrete data are often discrete, indirect and noisy measurements of some quantities describing the signal under consideration. A frequent task is to find the segments of the signal or image which corresponds to finding the discontinuities or jumps in the data. Methods based on minimizing the piecewise constant Mumford–Shah functional—whose discretized version is known as Potts energy—are advantageous in this scenario, in particular, in connection with segmentation. However, due to their non-convexity, minimization of such energies is challenging. In this paper, we propose a new iterative minimization strategy for the multivariate Potts energy dealing with indirect, noisy measurements. We provide a convergence analysis and underpin our findings with numerical experiments.
Highlights
Problems involving reconstruction tasks for functions with discontinuities appear in various biological and medical applications
Contributions The contributions of this paper are threefold: (i) We propose a new iterative minimization strategy for multivariate piecewise constant Mumford– Shah/Potts objective functions as well as a quadratic penalty relaxation. (ii) We provide a convergence analysis of the proposed schemes. (iii) We show the applicability of our schemes in several experiments
In contrast to the approaches in [9,33] and [60,61] for sparsity problems which lead to thresholding algorithms, our approach leads to non-separable yet computationally tractable problems in the backward step
Summary
Problems involving reconstruction tasks for functions with discontinuities appear in various biological and medical applications. They show that their method converges to a local minimizer in the univariate case Their approach principally carries over to the piecewise constant Mumford–Shah functional as explained in [84,90] and results in a 0 sparsity problem. Methods for restoring piecewise constant images without restricting the range space are proposed in Nikolova et al [68,69] They use non-convex regularizers which are algorithmically approached using a graduated non-convexity approach. Contributions The contributions of this paper are threefold: (i) We propose a new iterative minimization strategy for multivariate piecewise constant Mumford– Shah/Potts objective functions as well as a (still NP-hard) quadratic penalty relaxation.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
