Abstract
Recently infimal convolution type functions were used in regularization terms of variational models for restoring and decomposing images. This is the first attempt to generalize the infimal convolution of first and second order differences to manifold-valued images. We propose both an extrinsic and an intrinsic approach. Our focus is on the second one since the summands arising in the infimal convolution lie on the manifold themselves and not in the higher dimensional embedding space. We demonstrate by numerical examples that the approach works well on the circle, the 2-sphere, the rotation group, and the manifold of positive definite matrices with the affine invariant metric.
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References
Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2009)
Adams, B.L., Wright, S.I., Kunze, K.: Orientation imaging: the emergence of a new microscopy. J. Metall. Mater. Trans. A 24, 819–831 (1993)
Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-Lojasiewicz inequality. Math. Oper. Res. 35(2), 438–457 (2010)
Aujol, J.-F., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition – modelling, algorithms and parameter selection. Int. J. Comput. Vision 67(1), 111–136 (2006)
Azagra, R.D., Ferrera, C.J.: Inf-convolution and regularization of convex functions on Riemannian manifolds of nonpositive curvature. Revista matemática Complutense 19(2), 323–345 (2006)
Bačák, M., Bergmann, R., Steidl, G., Weinmann, A.: A second order non-smooth variational model for restoring manifold-valued images. SIAM J. Sci. Comput. 38(1), A567–A597 (2016)
Bachmann, F., Hielscher, R.: MTEX - MATLAB toolbox for quantitative texture analysis (2005–2016). http://mtex-toolbox.github.io/
Bachmann, F., Hielscher, R., Jupp, P.E., Pantleon, W., Schaeben, H., Wegert, E.: Inferential statistics of electron backscatter diffraction data from within individual crystalline grains. J. Appl. Crystallogr. 43, 1338–1355 (2010)
Balle, F., Eifler, D., Fitschen, J.H., Schuff, S., Steidl, G.: Computation and visualization of local deformation for multiphase metallic materials by infimal convolution of TV-type functionals. In: Aujol, J.-F., Nikolova, M., Papadakis, N. (eds.) SSVM 2015. LNCS, vol. 9087, pp. 385–396. Springer, Cham (2015). doi:10.1007/978-3-319-18461-6_31
Bergmann, R., Chan, R.H., Hielscher, R., Persch, J., Steidl, G.: Restoration of manifold-valued images by half-quadratic minimization. Inverse Prob. Imaging 10(2), 281–304 (2016)
Bergmann, R., Weinmann, A.: Inpainting of cyclic data using first and second order differences. In: Tai, X.-C., Bae, E., Chan, T.F., Lysaker, M. (eds.) EMMCVPR 2015. LNCS, vol. 8932, pp. 155–168. Springer, Cham (2015). doi:10.1007/978-3-319-14612-6_12
Bergmann, R., Weinmann, A.: A second order TV-type approach for inpainting and denoising higher dimensional combined cyclic and vector space data. J. Math. Imaging Vis. 55(3), 401–427 (2016)
Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)
Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)
Donoho, D.L., Kutyniok, G.: Geometric separation using a wavelet-shearlet dictionary. In: SampTA 2009 (2009)
Giaquinta, M., Mucci, D.: Maps of bounded variation with values into a manifold: total variation and relaxed energy. Pure Appli. Math. Q. 3(2), 513–538 (2007)
Holler, M., Kunisch, K.: On infimal convolution of TV-type functionals and applications to video and image reconstruction. SIAM J. Imaging Sci. 7(4), 2258–2300 (2014)
Lellmann, J., Strekalovskiy, E., Koetter, S., Cremers, D.: Total variation regularization for functions with values in a manifold. In: IEEE ICCV, pp. 2944–2951 (2013)
Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 63(1), 20–63 (1956)
Papafitsoros, K., Schönlieb, C.B.: A combined first and second order variational approach for image reconstruction. J. Math. Imaging Vis. 2(48), 308–338 (2014)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rosman, G., Tai, X.-C., Kimmel, R., Bruckstein, A.M.: Augmented-Lagrangian regularization of matrix-valued maps. Methods Appl. Anal. 21(1), 121–138 (2014)
Rosman, G., Wang, Y., Tai, X.-C., Kimmel, R., Bruckstein, A.M.: Fast regularization of matrix-valued images. In: Bruhn, A., Pock, T., Tai, X.-C. (eds.) Efficient Algorithms for Global Optimization Methods in Computer Vision. LNCS, vol. 8293, pp. 19–43. Springer, Heidelberg (2014). doi:10.1007/978-3-642-54774-4_2
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1), 259–268 (1992)
Setzer, S., Steidl, G.: Variational methods with higher order derivatives in image processing. In: Approximation XII: San Antonio 2007, pp. 360–385 (2008)
Setzer, S., Steidl, G., Teuber, T.: Infimal convolution regularizations with discrete \(\ell _1\)-type functionals. Commun. Math. Sci. 9(3), 797–827 (2011)
Starck, J.-L., Elad, M., Donoho, D.L.: Image decomposition via the combination of sparse representations and a variational approach. IEEE Trans. Image Process. 14(10), 1570–1582 (2005)
Steidl, G., Setzer, S., Popilka, B., Burgeth, B.: Restoration of matrix fields by second order cone programming. Computing 81, 161–178 (2007)
Strekalovskiy, E., Cremers, D.: Total variation for cyclic structures: Convex relaxation and efficient minimization. In: IEEE CVPR , pp. 1905–1911. IEEE (2011)
Sun, S., Adams, B., King, W.: Observation of lattice curvature near the interface of a deformed aluminium bicrystal. Philos. Mag. A 80, 9–25 (2000)
Valkonen, T., Bredies, K., Knoll, F.: Total generalized variation in diffusion tensor imaging. SIAM J. Imaging Sci. 6(1), 487–525 (2013)
Wang, Y., Yin, W., Zeng, J.: Global convergence of ADMM in nonconvex nonsmooth optimization. ArXiv preprint arXiv:1511.06324 (2015)
Weinmann, A., Demaret, L., Storath, M.: Total variation regularization for manifold-valued data. SIAM J. Imaging Sci. 7(4), 2226–2257 (2014)
Whitney, H.: Differentiable manifolds. Ann. Math. 37(3), 645–680 (1936)
Acknowledgements
Funding by the German Research Foundation (DFG) within the project STE 571/13-1 & BE 5888/2-1 and within the Research Training Group 1932, project area P3, is gratefully acknowledged. Furthermore, G. Steidl acknowledges the support by the German Federal Ministry of Education and Research (BMBF) through grant 05M13UKA (AniS).
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Bergmann, R., Fitschen, J.H., Persch, J., Steidl, G. (2017). Infimal Convolution Coupling of First and Second Order Differences on Manifold-Valued Images. In: Lauze, F., Dong, Y., Dahl, A. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2017. Lecture Notes in Computer Science(), vol 10302. Springer, Cham. https://doi.org/10.1007/978-3-319-58771-4_36
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