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Iterative TV-regularization of grey-scale images. (English. Russian original) Zbl 1427.94013
J. Math. Sci., New York 242, No. 2, 323-336 (2019); translation from Probl. Mat. Anal. 99, 127-137 (2019).
Summary: The TV-regularization method due to L. I. Rudin et al. [Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)] is widely used in mathematical image analysis. We consider a nonstationary and iterative variant of this approach and provide a mathematical theory that extends the results of E. Radmoser et al. to the BV setting [“Scale-space properties of nonstationary iterative regularization methods”, J. Vis. Commun. Image Represent. 11, No. 2, 96–114 (2000; doi:10.1006/jvci.1999.0437)]. While existence and uniqueness, a maximum-minimum principle, and preservation of the average grey value are not hard to prove, we also establish the convergence to a constant steady state and consider a large family of Lyapunov functionals. These properties allow us to interpret the iterated TV-regularization as a time-discrete scale-space representation of the original image.
MSC:
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
68U10 Computing methodologies for image processing
49N60 Regularity of solutions in optimal control
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