Dualization and Automatic Distributed Parameter Selection of Total Generalized Variation via Bilevel Optimization

References

• Adams, R. A., Fournier, J. (2003). Sobolev Spaces. 2nd ed. Cambridge, MA: Academic Press,
   Google Scholar
• Ambrosio, L., Fusco, N., Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. USA: Oxford University Press.
   Google Scholar
• Attouch, H., Brezis, H. (1986). Duality for the Sum of Convex Functions in General Banach Spaces. Vol. 34, North-Holland Mathematical Library, pp. 125–133.
   Google Scholar
• Barzilai, J., Borwein, J. M. (1988). Two-point step size gradient methods. IMA J Numer Anal. 8(1):141–148. DOI: 10.1093/imanum/8.1.141.
   Web of Science ®Google Scholar
• Benning, M., Brune, C., Burger, M., Müller, J. (2013). Higher-order TV methods: Enhancement via Bregman iteration. J Sci Comput. 54(2-3):269–310. DOI: 10.1007/s10915-012-9650-3.
   Web of Science ®Google Scholar
• Bergounioux, M., Papoutsellis, E., Stute, S., Tauber, C. (2018). Infimal convolution spatiotemporal PET reconstruction using total variation based priors, HAL preprint https://hal.archives-ouvertes.fr/hal-01694064.
   Google Scholar
• Borwein, J. M., Vanderwerff, J. D. (2010). Convex Functions. Cambridge: Cambridge University Press.
   Google Scholar
• Bredies, K., Dong, Y., Hintermüller, M. (2013). Spatially dependent regularization parameter selection in total generalized variation models for image restoration. Inter. J. Comput. Math. 90(1):109–123. DOI: 10.1080/00207160.2012.700400.
   Web of Science ®Google Scholar
• Bredies, K., Holler, M. (2012). Artifact-free JPEG decompression with total generalized variation. VISAP 2012: Proceedings of the International Conference on Computer Vision and Applications,
   Google Scholar
• Bredies, K., Holler, M. (2013). A TGV regularized wavelet based zooming model. Scale Space and Variational Methods in Computer Vision. Springer, DOI: 10.1007/978-3-642-38267-3_13., pp. 149–160.
   Google Scholar
• Bredies, K., Holler, M. (2014). Regularization of linear inverse problems with total generalized variation. J. Inverse Ill-Posed Prob. 22(6):871–913. DOI: 10.1515/jip-2013-0068.
   Web of Science ®Google Scholar
• Bredies, K., Holler, M. (2015). A TGV-based framework for variational image decompression, zooming, and reconstruction. Part I: Analytics. SIAM J. Imaging Sci. 8(4):2814–2850. DOI: 10.1137/15M1023865.
   Web of Science ®Google Scholar
• Bredies, K., Holler, M. (2015). A TGV-based framework for variational image decompression, zooming, and reconstruction. Part II: Numerics. SIAM J. Imaging Sci. 8(4):2851–2886. DOI: 10.1137/15M1023877.
   Web of Science ®Google Scholar
• Bredies, K., Holler, M., Storath, M., Weinmann, A. (2018). Total generalized variation for manifold-valued data. SIAM J. Imaging Sci. 11(3):1785–1848. DOI: 10.1137/17M1147597.
   Web of Science ®Google Scholar
• Bredies, K., Kunisch, K., Pock, T. (2010). Total generalized variation. SIAM J. Imaging Sci. 3(3):492–526. DOI: 10.1137/090769521.
   Web of Science ®Google Scholar
• Bredies, K., Kunisch, K., Valkonen, T. (2013). Properties of L1-TGV 2: The one-dimensional case. J. Math. Anal. Appl. 398(1):438–454. DOI: 10.1016/j.jmaa.2012.08.053.
   Web of Science ®Google Scholar
• Bredies, K., Valkonen, T. (2011). Inverse problems with second-order total generalized variation constraints. Proceedings of SampTA 2011 - 9th International Conference on Sampling Theory and Applications, Singapore,
   Google Scholar
• Burger, M., Papafitsoros, K., Papoutsellis, E., Schönlieb, C. B. (2016). Infimal convolution regularisation functionals of BV and [Formula: see text] Spaces: Part I: The Finite [Formula: see text] Case. J. Math. Imaging Vis. 55(3):343–369. DOI: 10.1007/s10851-015-0624-6.
   PubMed Web of Science ®Google Scholar
• Calatroni, L., Chung, C., Los Reyes, J. D., Schönlieb, C. B., Valkonen, T. (2017). Bilevel approaches for learning of variational imaging models. RADON Book Series on Computational and Applied Mathematics. Vol. 18, Berlin, Boston: De Gruyter, https://www.degruyter.com/view/product/458544.
   Google Scholar
• Calatroni, L., De Los Reyes, J. C., Schönlieb, C. B. (2017). Infimal convolution of data discrepancies for mixed noise removal. SIAM J. Imaging Sci. 10(3):1196–1233. DOI: 10.1137/16M1101684.
   Web of Science ®Google Scholar
• Calatroni, L., Papafitsoros, K. (2019). Analysis and automatic parameter selection of a variational model for mixed gaussian and salt-and-pepper noise removal. Inverse Prob. 35(11):114001. DOI: 10.1088/1361-6420/ab291a.
   Web of Science ®Google Scholar
• Caselles, V., Chambolle, A., Novaga, M. (2007). The discontinuity set of solutions of the TV denoising problem and some extensions. Multiscale Model. Simul. 6(3):879–894. DOI: 10.1137/070683003.
   Web of Science ®Google Scholar
• Chambolle, A., Duval, V., Peyré, G., Poon, C. (2017). Geometric properties of solutions to the total variation denoising problem. Inverse Prob. 33(1):015002. http://stacks.iop.org/0266-5611/33/i=1/a=015002. DOI: 10.1088/0266-5611/33/1/015002.
   Web of Science ®Google Scholar
• Chambolle, A., Lions, P. L. (1997). Image recovery via total variation minimization and related problems. Numerische Mathematik. 76(2):167–188. DOI: 10.1007/s002110050258.
   Web of Science ®Google Scholar
• Van Chung, C., De los Reyes, J. C., Schönlieb, C. B. (2017). Learning optimal spatially-dependent regularization parameters in total variation image denoising. Inverse Prob. 33(7):074005. DOI: 10.1088/1361-6420/33/7/074005.
   Web of Science ®Google Scholar
• De Los Reyes, J. C., Schönlieb, C. B., Valkonen, T. (2016). The structure of optimal parameters for image restoration problems. J. Math. Anal. Appl. 434(1):464–500. DOI: 10.1016/j.jmaa.2015.09.023.
   Web of Science ®Google Scholar
• De Los Reyes, J. C., Schönlieb, C. B., Valkonen, T. (2017). Bilevel parameter learning for higher-order Total Variation regularisation models. J. Math. Imaging Vis. 57(1):1–25. DOI: 10.1007/s10851-016-0662-8.
   PubMed Web of Science ®Google Scholar
• Demengel, F., Temam, R. (1984). Convex functions of a measure and applications. Indiana Univ. Math. J. 33(5):673–709. DOI: 10.1512/iumj.1984.33.33036.
   Web of Science ®Google Scholar
• Ekeland, I., Temam, R. (1999). Convex analysis and variational problems. Classics Appl. Mathem. Soci. Indust. Appl. Math.
   Google Scholar
• Girault, V., Raviart, P. A. (1986). Finite Element Method for Navier-Stokes Equation. Berlin, Germany: Springer.
   Google Scholar
• Hintermüller, M., Holler, M., Papafitsoros, K. (2018). A function space framework for structural total variation regularization with applications in inverse problems. Inverse Prob. 34(6):064002. http://stacks.iop.org/0266-5611/34/i=6/a=064002. DOI: 10.1088/1361-6420/aab586.
   Web of Science ®Google Scholar
• Hintermüller, M., Kunisch, K. (2006). Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17(1):159–187. DOI: 10.1137/040611598.
   Web of Science ®Google Scholar
• Hintermüller, M., Papafitsoros, K. (2019). Generating structured nonsmooth priors and associated primal-dual methods. Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2. In: Ron K. and Xue-Cheng T., eds. Handbook of Numerical Analysis, Vol. 20, DOI: 10.1016/bs.hna.2019.08.001., pp. 437–502.
   Google Scholar
• Hintermüller, M., Papafitsoros, K., Rautenberg, C. N. (2017). Analytical aspects of spatially adapted total variation regularisation. J. Math. Anal. Appl. 454(2):891–935. DOI: 10.1016/j.jmaa.2017.05.025.
   Web of Science ®Google Scholar
• Hintermüller, M., Papafitsoros, K., Rautenberg, C. N. (2020). Variable step mollifiers and applications. Integr. Equ. Oper. Theory. 92(6). DOI: 10.1007/s00020-020-02608-2.
   Web of Science ®Google Scholar
• Hintermüller, M., Rautenberg, C. N. (2015). On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces. J. Math. Anal. Appl. 426(1):585–593. DOI: 10.1016/j.jmaa.2015.01.060.
   Web of Science ®Google Scholar
• Hintermüller, M., Rautenberg, C. N. (2017). Optimal selection of the regularization function in a weighted total variation model. Part I: Modelling and theory. J. Math. Imaging Vis. 59(3):498–514. DOI: 10.1007/s10851-017-0744-2.
   Web of Science ®Google Scholar
• Hintermüller, M., Rautenberg, C. N., Rösel, S. (2017). Density of convex intersections and applications. Proc. Royal Soci. London A Math. Phys. Engng Sci. 473(2205) DOI: 10.1098/rspa.2016.0919.
   Google Scholar
• Hintermüller, M., Rautenberg, C. N., Wu, T., Langer, A. (2017). Optimal selection of the regularization function in a weighted total variation model. Part II: Algorithm, its analysis and numerical tests. J. Math. Imaging Vis. 59(3):515–533. DOI: 10.1007/s10851-017-0736-2.
   Web of Science ®Google Scholar
• Hintermüller, M., Stadler, G. (2006). An infeasible primal-dual algorithm for total bounded variation–based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28(1):1–23. DOI: 10.1137/040613263.
   Web of Science ®Google Scholar
• Holler, M., Kunisch, K. (2014). On infimal convolution of TV-type functionals and applications to video and image reconstruction. SIAM J. Imaging Sci. 7(4):2258–2300. DOI: 10.1137/130948793.
   Web of Science ®Google Scholar
• Huber, R., Haberfehlner, G., Holler, M., Kothleitner, G., Bredies, K. (2019). Total generalized variation regularization for multi-modal electron tomography. Nanoscale. 11(12):5617–5632. DOI: 10.1039/c8nr09058k.
   PubMed Web of Science ®Google Scholar
• Jalalzai, K. Discontinuities of the minimizers of the weighted or anisotropic total variation for image reconstruction, arXiv preprint 1402.0026 (2014), http://arxiv.org/abs/1402.0026.
   Google Scholar
• Jalalzai, K. (2016). Some remarks on the staircasing phenomenon in total variation-based image denoising. J. Math. Imaging Vis. 54(2):256–268. DOI: http://dx.doi.org/10.1007/s10851-015-0600-1.
   Web of Science ®Google Scholar
• Knoll, F., Bredies, K., Pock, T., Stollberger, R. (2011). Second order total generalized variation (TGV) for MRI. Magn. Reson. Med. 65(2):480–491. DOI: 10.1002/mrm.22595.
   PubMed Web of Science ®Google Scholar
• Knoll, F., Holler, M., Koesters, T., Otazo, R., Bredies, K., Sodickson, D. K. (2017). Joint MR-PET reconstruction using a multi-channel image regularizer. IEEE Trans. Med. Imaging. 36(1):1–16. DOI: 10.1109/TMI.2016.2564989.
   PubMed Web of Science ®Google Scholar
• Kunisch, K., Hintermüller, M. (2004). Total bounded variation regularization as a bilaterally constrained optimization problem. SIAM J. Appl. Math. 64(4):1311–1333. DOI: 10.1137/S0036139903422784.
   Web of Science ®Google Scholar
• Nesterov, Y. E. (1983). A method for solving the convex programming problem with convergence rate O(1/k2). Soviet Math. Dokl. 27:367–372.
   Google Scholar
• Papafitsoros, K. (2014). Novel higher order regularisation methods for image reconstruction. Ph.D. thesis. University of Cambridge. https://www.repository.cam.ac.uk/handle/1810/246692.
   Google Scholar
• Papafitsoros, K., Bredies, K. (2015). Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, Cambridge A study of the one dimensional total generalised variation regularisation problem. Inverse Prob. Imaging. 9(2):511–550. DOI: 10.3934/ipi.2015.9.511.
   Web of Science ®Google Scholar
• Papafitsoros, K., Valkonen, T. (2015). Asymptotic behaviour of total generalised variation. Scale Space and Variational Methods in Computer Vision: 5th International Conference, SSVM 2015, Proceedings. In: Jean-François A., Mila N., and Nicolas P., eds. Springer International Publishing, pp. 702–714. DOI: 10.1007/978-3-319-18461-6_56.
   Google Scholar
• Pöschl, C., Scherzer, O. (2015). Exact solutions of one-dimensional total generalized variation. Comm. Math. Sci. 13(1):171–202. DOI: 10.4310/CMS.2015.v13.n1.a9.
   Web of Science ®Google Scholar
• Ring, W. (2000). Structural properties of solutions to total variation regularization problems. Esaim: M2an. 34(4):799–810. DOI: 10.1051/m2an:2000104.
   Web of Science ®Google Scholar
• Rudin, L. I., Osher, S., Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena. 60(1-4):259–268. DOI: 10.1016/0167-2789(92)90242-F.
   Web of Science ®Google Scholar
• Schloegl, M., Holler, M., Schwarzl, A., Bredies, K., Stollberger, R. (2017). Infimal convolution of total generalized variation functionals for dynamic MRI. Magn Reson Med. 78(1):142–155. DOI: 10.1002/mrm.26352.
   PubMed Web of Science ®Google Scholar
• Temam, R. (1985). Mathematical Problems in Plasticity. Vol. 15, Gauthier-Villars Paris.
   Google Scholar
• Temam, R., Strang, G. (1980). Functions of bounded deformation. Arch. Rational Mech. Anal. 75(1):7–21. DOI: 10.1007/BF00284617.
   Web of Science ®Google Scholar
• Valkonen, T. (2017). The jump set under geometric regularisation. Part 2: Higher-order approaches. J. Math. Anal. Appl. 453(2):1044–1085. DOI: 10.1016/j.jmaa.2017.04.037.
   Web of Science ®Google Scholar
• Valkonen, T., Bredies, K., Knoll, F. (2013). Total generalized variation in diffusion tensor imaging. SIAM J. Imaging Sci. 6(1):487–525. DOI: 10.1137/120867172.
   Web of Science ®Google Scholar
• Zowe, J., Kurcyusz, S. (1979). Regularity and stability for the mathematical programming problem in Banach spaces. Appl Math Optim. 5(1):49–62. DOI: 10.1007/BF01442543.
   Web of Science ®Google Scholar