A Coupled Variational Problem of Linear Growth Related to the Denoising and Inpainting of Images
We present some results conjectured by Bildhauer, Fuchs, and Weickert who investigated analytical aspects of coupled variational models with applications to mathematical imaging. We focus on variants of linear growth, which require a treatment within the framework of relaxation theory and convex analysis. We establish existence and regularity of (dual-) solutions.
Unable to display preview. Download preview PDF.
- 1.M. Bildhauer, M. Fuchs, and J. Weickert, “An alternative approach towards the higher order denoising of images. Analytical aspects” [in Russian], Zap. Nauchn. Sem. POMI 444, 47–88 (2016).Google Scholar
- 10.M. Fuchs and J. Müller, “A higher order TV-type variational problem related to the denoising and inpainting of images,” Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 154, 122–147 (2017).Google Scholar
- 11.M. Fuchs and C. Tietz, “Existence of generalized minimizers and of dual solutions for a class of variational problems with linear growth related to image recovery,” J. Math. Sci., New York 210, No. 4, 458–475 (2015).Google Scholar
- 12.J. Müller and C. Tietz, Existence and Almost Everywhere Regularity of Generalized Minimizers for a Class of Variational Problems with Linear Growth Related to Image Inpainting, Technical Report No. 363, Saarland University (2015).Google Scholar
- 13.I. Ekeland and R. T´emam. Convex Analysis and Variational Problems, SIAM, Philadelphia, PA (1999).Google Scholar
- 16.J. Frehse and G. Seregin, “Regularity for solutions of variational problems in the deformation theory of plasticity with logarithmic hardening,” Transl., Ser. 2, Am. Math. Soc. 193, 127–152 (1999).Google Scholar
- 17.M. Bildhauer, Convex Variational Problems: Linear, nearly Linear and Anisotropic Growth Conditions, Lect. Notes Math. 1818, Springer, Berlin etc. (2003).Google Scholar
- 19.M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univer. Press, Princeton, NJ (1983).Google Scholar