Applications of Mathematics, Vol. 64, No. 3, pp. 281-300, 2019


Theoretical foundation of the weighted Laplace inpainting problem

Laurent Hoeltgen, Andreas Kleefeld, Isaac Harris, Michael Breuss

Received July 31, 2018.   Published online April 24, 2019.

Abstract:  Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed, where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers the corresponding weak formulation and aims at using the Theorem of Lax-Milgram to assert the existence of a solution. To this end we have to resort to weighted Sobolev spaces. Our analysis shows that solutions do not exist unconditionally. The weights need some regularity and must fulfil certain growth conditions. The results from this work complement findings which were previously only available for a discrete setup.
Keywords:  image inpainting; image reconstruction; Laplace equation; Laplace interpolation; mixed boundary condition; partial differential equation; weighted Sobolev space
Classification MSC:  35J15, 35J70, 46E35, 94A08


References:
[1] K. Atkinson, W. Han: Theoretical Numerical Analysis. A Functional Analysis Framework. Texts in Applied Mathematics 39, Springer, Berlin (2009). DOI 10.1007/978-1-4419-0458-4 | MR 2511061 | Zbl 1181.47078
[2] A. Azzam, E. Kreyszig: On solutions of elliptic equations satisfying mixed boundary conditions. SIAM J. Math.Anal. 13 (1982), 254-262. DOI 10.1137/0513018 | MR 0647124 | Zbl 0485.35041
[3] Z. Belhachmi, D. Bucur, B. Burgeth, J. Weickert: How to choose interpolation data in images. SIAM J. Appl. Math. 70 (2009), 333-352. DOI 10.1137/080716396 | MR 2521220 | Zbl 1190.94006
[4] M. Bertalmío, G. Sapiro, V. Caselles, C. Ballester: Image inpainting. Proc. 27th Annual Conf. Computer Graphics and Interactive Techniques. ACM Press/Addison-Wesley Publishing Company, New York, 2000, pp. 417-424. DOI 10.1145/344779.344972
[5] M. I. G. Bloor, M. J. Wilson: Generating blend surfaces using partial differential equations. Comput.-Aided Des. 21 (1989), 165-171. DOI 10.1016/0010-4485(89)90071-7 | Zbl 0669.65006
[6] K. Bredies: A variational weak weighted derivative: Sobolev spaces and degenerate elliptic equations. Available at https://imsc.uni-graz.at/bredies/publications_de.html (2008).
[7] R. Brown: The mixed problem for Laplace's equation in a class of Lipschitz domains. Commun. Partial Differ. Equations 19 (1994), 1217-1233. DOI 10.1080/03605309408821052 | MR 1284808 | Zbl 0831.35043
[8] R. S. Cantrell, C. Cosner: Spatial Ecology via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology, Wiley, Chichester (2003). DOI 10.1002/0470871296 | MR 2191264 | Zbl 1059.92051
[9] V. Caselles, J.-M. Morel, C. Sbert: An axiomatic approach to image interpolation. IEEE Trans. Image Process. 7 (1998), 376-386. DOI 10.1109/83.661188 | MR 1669524 | Zbl 0993.94504
[10] J. Chabrowski: The Dirichlet Problem with $L^2$ Boundary Data for Elliptic Linear Equations. Lecture Notes in Mathematics 1482, Springer, Berlin (1991). DOI 10.1007/BFb0095750 | MR 1165533 | Zbl 0734.35024
[11] T. F. Chan, S. H. Kang: Error analysis for image inpainting. J. Math. Imaging Vis. 26 (2006), 85-103. DOI 10.1007/s10851-006-6865-7 | MR 2283872
[12] T. F. Chan, J. Shen: Mathematical models for local non-texture inpaintings. SIAM J. Appl. Math. 62 (2002), 1019-1043. DOI 10.1137/S0036139900368844 | MR 1897733 | Zbl 1050.68157
[13] I. K. Crain: Computer interpolation and contouring of two-dimensional data: A review. Geoexploration 8 (1970), 71-86. DOI 10.1016/0016-7142(70)90021-9
[14] G. Dal Maso, U. Mosco: Wiener's criterion and $\Gamma$-convergence. Appl. Math. Optimization 15 (1987), 15-63. DOI 10.1007/BF01442645 | MR 0866165 | Zbl 0644.35033
[15] D. E. Edmunds, B. Opic: Weighted Poincaré and Friedrichs inequalities. J. Lon. Math. Soc., II. Ser. 47 (1993), 79-96. DOI 10.1112/jlms/s2-47.1.79 | MR 1200980 | Zbl 0797.46027
[16] A. Ern, J.-L. Guermond: Theory and Practice of Finite Elements. Applied Mathematical Sciences 159, Springer, New York (2004). DOI 10.1007/978-1-4757-4355-5 | MR 2050138 | Zbl 1059.65103
[17] G. Fichera: Analisi esistenziale per le soluzioni dei problemi al contorno misti, relativi all'equazione e ai sistemi di equazioni del secondo ordine di tipo ellittico, autoaggiunti. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 1 (1949), 75-100. (In Italian.) MR 0035370 | Zbl 0035.18603
[18] I. Galić, J. Weickert, M. Welk, A. Bruhn, A. Belyaev, H.-P. Seidel: Towards PDE-based image compression. Variational, Geometric, and Level Set Methods in Computer Vision (N. Paragios et al., eds.). Lecture Notes in Computer Science 3752, Springer, Berlin, 2005, pp. 37-48. DOI 10.1007/11567646 | Zbl 1159.68589
[19] I. Galić, J. Weickert, M. Welk, A. Bruhn, A. Belyaev, H.-P. Seidel: Image compression with anisotropic diffusion. J. Math. Imaging Vis. 31 (2008), 255-269. DOI 10.1007/s10851-008-0087-0 | MR 2407524
[20] V. Gol'dshtein, A. Ukhlov: Weighted Sobolev spaces and embedding theorems. Trans. Am. Math. Soc. 361 (2009), 3829-3850. DOI 10.1090/S0002-9947-09-04615-7 | MR 2491902 | Zbl 1180.46022
[21] C. Guillemot, O. Le Meur: Image inpainting: Overview and recent advances. IEEE Signal Processing Magazine 31 (2014), 127-144. DOI 10.1109/msp.2013.2273004
[22] L. Hoeltgen: Optimal interpolation data for image reconstructions. Ph.D. Thesis, Saarland University, Saarbrücken (2014).
[23] L. Hoeltgen: Understanding image inpainting with the help of the Helmholtz equation. Math. Sci., Springer 11 (2017), 73-77. DOI 10.1007/s40096-017-0207-3 | MR 3612257 | Zbl 06781216
[24] L. Hoeltgen, I. Harris, M. Breuß, A. Kleefeld: Analytic existence and uniqueness results for PDE-based image reconstruction with the Laplacian. International Conference on Scale Space and Variational Methods in Computer Vision (F. Lauze et al., eds.). Lecture Notes in Computer Science 10302, Springer, Cham, 2017, pp. 66-79. DOI 10.1007/978-3-319-58771-4_6
[25] L. Hoeltgen, M. Mainberger, S. Hoffmann, J. Weickert, C. H. Tang, S. Setzer, D. Johannsen, F. Neumann, B. Doerr: Optimizing spatial and tonal data for PDE-based inpainting. Variational Methods, In Imaging and Geometric Control (M. Bergounioux et al., eds.). Radon Series on Computational and Applied Mathematics 18, De Gruyter, Berlin, 2017, pp. 35-83. DOI 10.1515/9783110430394 | MR 3618249 | Zbl 06984290
[26] L. Hoeltgen, S. Setzer, J. Weickert: An optimal control approach to find sparse data for Laplace interpolation. Energy Minimization Methods in Computer Vision and Pattern Recognition (A. Heyden et al., eds.). Lecture Notes in Computer Science 8081, Springer, Berlin, 2013, pp. 151-164. DOI 10.1007/978-3-642-40395-8_12
[27] L. Hoeltgen, J. Weickert: Why does non-binary mask optimisation work for diffusion-based image compression? Energy Minimization Methods in Computer Vision and Pattern Recognition (X.-C. Tai et al., eds.). Lecture Notes in Computer Science 8932, Springer, Cham, 2015, pp. 85-98. DOI 10.1007/978-3-319-14612-6_7
[28] A. Kufner: Weighted Sobolev Spaces. Teubner-Texte zur Mathematik 31, BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1980). MR 0664599 | Zbl 0455.46034
[29] A. Kufner, B. Opic: The Dirichlet problem and weighted spaces. I. Čas. Pěst. Mat. 108 (1983), 381-408. MR 0727537 | Zbl 0589.35016
[30] A. Kufner, B. Opic: How to define reasonably weighted Sobolev spaces. Commentat. Math. Univ. Carol. 25 (1984), 537-554. MR 0775568 | Zbl 0557.46025
[31] A. Kufner, B. Opic: Some remarks on the definition of weighted Sobolev spaces. Partial Differential Equations, 1983 "Nauka" Sibirsk. Otdel, Novosibirsk (1986), 119-126. (In Russian.) MR 0851604
[32] A. Kufner, B. Opic: The Dirichlet problem and weighted spaces. II. Čas. Pěstování Mat. 111 (1986), 242-253. MR 0853789 | Zbl 0654.35039
[33] A. Kufner, A.-M. Sändig: Some Applications of Weighted Sobolev Spaces. Teubner-Texte zur Mathematik 100, BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1987). DOI 10.1007/978-3-663-11385-0 | MR 0926688 | Zbl 0662.46034
[34] M. Mainberger, A. Bruhn, J. Weickert, S. Forchhammer: Edge-based compression of cartoon-like images with homogeneous diffusion. Pattern Recognition 44 (2011), 1859-1873. DOI 10.1016/j.patcog.2010.08.004
[35] M. Mainberger, S. Hoffmann, J. Weickert, C. H. Tang, D. Johannsen, F. Neumann, B. Doerr: Optimising spatial and tonal data for homogeneous diffusion inpainting. Scale Space and Variational Methods in Computer Vision (A. M. Bruckstein et al., eds.). Lecture Notes in Computer Science 6667, Springer, Berlin, 2012, pp. 26-37. DOI 10.1007/978-3-642-24785-9_3
[36] B. Martinet: Régularisation d'inéquations variationnelles par approximations successives. Rev. Franç. Inform. Rech. Opér. 4 (1970), 154-158. (In French.) MR 0298899 | Zbl 0215.21103
[37] S. Masnou, J.-M. Morel: Level lines based disocclusion. Proceedings 1998 International Conference on Image Processing. ICIP98. IEEE, 2002, pp. 259-263. DOI 10.1109/icip.1998.999016
[38] C. Miranda: Sul problema misto per le equazioni lineari ellittiche. Ann. Mat. Pura Appl., IV. Ser. 39 (1955), 279-303 (In Italian.) DOI 10.1007/BF02410775 | MR 0078561 | Zbl 0066.34301
[39] R. H. Nochetto, E. Otárola, A. J. Salgado: Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications. Numer. Math. 132 (2016), 85-130. DOI 10.1007/s00211-015-0709-6 | MR 3439216 | Zbl 1334.65030
[40] A. A. Noma, M. G. Misulia: Programming topographic maps for automatic terrain model construction. Surveying and Mapping 19 (1959), 355-366.
[41] O. A. Oleĭnik, E. V. Radkevič: Second-Order Equations with Nonnegative Characteristic Form. American Mathematical Society, Providence (1973). DOI 10.1007/978-1-4684-8965-1 | MR 0457908 | Zbl 0217.41502
[42] B. Opic, A. Kufner: Hardy-type Inequalities. Pitman Research Notes in Mathematics 219, Longman Scientific & Technical, Harlow; John Wiley & Sons, New York (1990). MR 1069756 | Zbl 0698.26007
[43] P. Peter, S. Hoffmann, F. Nedwed, L. Hoeltgen, J. Weickert: Evaluating the true potential of diffusion-based inpainting in a compression context. Signal Processing: Image Communication 46 (2016), 40-53. DOI 10.1016/j.image.2016.05.002
[44] P. Peter, S. Hoffmann, F. Nedwed, L. Hoeltgen, J. Weickert: From optimised inpainting with linear PDEs towards competitive image compression codecs. Image and Video Technology (T. Bräunl et al., eds.). Lecture Notes in Computer Science 9431, Springer, Cham, 2016, pp. 63-74. DOI 10.1007/978-3-319-29451-3_6
[45] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery: Numerical Recipes in C++. The Art of Scientific Computing. Cambridge University Press, Cambridge (2002). MR 1880993 | Zbl 1078.65500
[46] E. T. Sawyer, R. L. Wheeden: Degenerate Sobolev spaces and regularity of subelliptic equations. Trans. Am. Math. Soc. 362 (2010), 1869-1906. DOI 10.1090/S0002-9947-09-04756-4 | MR 2574880 | Zbl 1191.35085
[47] C. Schmaltz, J. Weickert, A. Bruhn: Beating the quality of JPEG 2000 with anisotropic diffusion. Pattern Recognition (J. Denzler et al., eds.). Lecture Notes in Computer Science 5748, Springer, Berlin, 2009, pp. 452-461. DOI 10.1007/978-3-642-03798-6_46
[48] C.-B. Schönlieb: Partial Differential Equation Methods for Image Inpainting. Cambridge Monographs on Applied and Computational Mathematics 29, Cambridge University Press, Cambridge (2015). DOI 10.1017/CBO9780511734304 | MR 3558995 | Zbl 1335.94002
[49] C. Solomon, T. Breckon: Fundamentals of Digital Image Processing. A Practical Approach with Examples in Matlab. Wiley-Blackwell, Chichester (2014). DOI 10.1002/9780470689776
[50] B. O. Turesson: Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics 1736, Springer, Berlin (2000). DOI 10.1007/BFb0103908 | MR 1774162 | Zbl 0949.31006
[51] M. I. Višik, V. V. Grušin: Boundary value problems for elliptic equations degenerate on the boundary of a domain. Math. USSR, Sb. 9 (1969), 423-454. DOI 10.1070/sm1969v009n04abeh002055
[52] W. Wang, J. Sun, Z. Zheng: Poincaré inequalities in weighted Sobolev spaces. Appl. Math. Mech., Engl. Ed. 27 (2006), 125-132. DOI 10.1007/s10483-006-0116-1 | MR 2213423 | Zbl 1160.46315
[53] A. Weber: The USC-SIPI image database, 2014. Available at http://sipi.usc.edu/database/. sw 15845
[54] S. Zaremba: Sur un problème mixte relatif à l'équation de Laplace. Bulletin international de l'Académie des sciences de Cracovie (1910), 313-344 (In French.) JFM 41.0854.12

Affiliations:   Laurent Hoeltgen, Institute for Mathematics, Brandenburg Technical University, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany, e-mail: hoeltgen@b-tu.de; Andreas Kleefeld, Forschungszentrum Jülich GmbH, Jülich Supercomputing Centre, Wilhelm-Johnen-Straße, 52425 Jülich, Germany, e-mail: a.kleefeld@fz-juelich.de; Isaac Harris, Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A., e-mail: harri814@purdue.edu; Michael Breuss, Institute for Mathematics, Brandenburg Technical University, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany, e-mail: breuss@b-tu.de


 
PDF available at: