Abstract
Inpainting with partial differential equations (PDEs) has been used successfully to reconstruct missing parts of an image, even for sparse data. On the other hand, sparse data interpolation is a rich field of its own with different methods such as scattered data interpolation with radial basis functions (RBFs).
The goal of this paper is to establish connections between inpainting with linear shift- and rotation-invariant differential operators and interpolation with radial basis functions. The bridge between these two worlds is built by generalising inpainting methods to handle pseudodifferential operators and by considering their Green’s functions. In this way, we find novel relations of various multiquadrics to pseudodifferential operators. Moreover, we show that many popular radial basis functions are related to processes from the diffusion and scale-space literature. We present a single numerical algorithm for all methods. It combines conjugate gradients with pseudodifferential operator evaluations in the Fourier domain. Our experiments show that the linear PDE- and the RBF-based communities have developed concepts of comparable quality.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Belhachmi, Z., Bucur, D., Burgeth, B., Weickert, J.: How to choose interpolation data in images. SIAM J. Appl. Math. 70(1), 333–352 (2009)
Bertalmío, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of SIGGRAPH 2000, New Orleans, LA, pp. 417–424, July 2000
Bucur, C.: Some observations on the Green function for the ball in the fractional Laplace framework. Commun. Pure Appl. Anal. 15(2), 657–699 (2016)
Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)
Burgeth, B., Didas, S., Weickert, J.: The Bessel scale-space. In: Olsen, O.F., Florack, L., Kuijper, A. (eds.) DSSCV 2005. LNCS, vol. 3753, pp. 84–95. Springer, Heidelberg (2005). https://doi.org/10.1007/11577812_8
Chen, Y., Ranftl, R., Pock, T.: A bi-level view of inpainting-based image compression. In: Kúkelová, Z., Heller, J. (eds.) Proceedings of 19th Computer Vision Winter Workshop, Křtiny, Czech Republic, pp. 19–26, February 2014
Duchon, J.: Interpolation des fonctions de deux variable suivant le principe de la flexion des plaques minces. Revue française d’automatique, informatique, recherche opérationelle. Analyse numérique 10(12), 5–12 (1976)
Duits, R., Florack, L., de Graaf, J., ter Haar Romeny, B.: On the axioms of scale space theory. J. Math. Imaging Vis. 20(3), 267–298 (2004)
Fasshauer, G.E., Ye, Q.: Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators. Numerische Mathematik 119(3), 585–611 (2011)
Felsberg, M., Sommer, G.: The monogenic scale-space: a unifying approach to phase-based image processing in scale-space. J. Math. Imaging Vis. 21(1), 5–26 (2004)
Franke, R., Nielson, G.M.: Scattered data interpolation and applications: A tutorial and survey. In: Hagen, H., Roller, D. (eds.) Geometric Modeling: Methods and Applications, pp. 131–160. Springer, Berlin (1991). https://doi.org/10.1007/978-3-642-76404-2_6
Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., Seidel, H.: Image compression with anisotropic diffusion. J. Math. Imaging Vis. 31(2–3), 255–269 (2008)
Hoeltgen, L., Setzer, S., Weickert, J.: An optimal control approach to find sparse data for Laplace interpolation. In: Heyden, A., Kahl, F., Olsson, C., Oskarsson, M., Tai, X.-C. (eds.) EMMCVPR 2013. LNCS, vol. 8081, pp. 151–164. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40395-8_12
Hoffmann, S., Plonka, G., Weickert, J.: Discrete Green’s functions for harmonic and biharmonic inpainting with sparse atoms. In: Tai, X.-C., Bae, E., Chan, T.F., Lysaker, M. (eds.) EMMCVPR 2015. LNCS, vol. 8932, pp. 169–182. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-14612-6_13
Iijima, T.: Basic theory on normalization of pattern (in case of typical one-dimensional pattern). Bull. Electrotechnical Lab. 26, 368–388 (1962). (in Japanese)
Iske, A.: Multiresolution Methods in Scattered Data Modelling. Lecture Notes in Computational Science and Engineering, vol. 37. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-642-18754-4
Kozhekin, N., Savchenko, V., Senin, M., Hagiwara, I.: An approach to surface retouching and mesh smoothing. Visual Comput. 19(8), 549–564 (2003)
Lehmann, T., Gönner, C., Spitzer, K.: Survey: Interpolation methods in medical image processing. IEEE Trans. Med. Imaging 18(11), 1049–1075 (1999)
Madych, W.R., Nelson, S.A.: Polyharmonic cardinal splines. J. Approximation Theory 60(2), 141–156 (1990)
Mainberger, M., Bruhn, A., Weickert, J., Forchhammer, S.: Edge-based image compression of cartoon-like images with homogeneous diffusion. Pattern Recogn. 44(9), 1859–1873 (2011)
Mainberger, M., Hoffmann, S., Weickert, J., Tang, C.H., Johannsen, D., Neumann, F., Doerr, B.: Optimising spatial and tonal data for homogeneous diffusion inpainting. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 26–37. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-24785-9_3
Masnou, S., Morel, J.: Level lines based disocclusion. In: Proceedings of 1998 IEEE International Conference on Image Processing, . Chicago, IL, vol. 3, pp. 259–263, October 1998
Peter, P., Hoffmann, S., Nedwed, F., Hoeltgen, L., Weickert, J.: Evaluating the true potential of diffusion-based inpainting in a compression context. Signal Process. Image Commun. 46, 40–53 (2016)
Schmaltz, C., Peter, P., Mainberger, M., Ebel, F., Weickert, J., Bruhn, A.: Understanding, optimising, and extending data compression with anisotropic diffusion. Int. J. Comput. Vis. 108(3), 222–240 (2014)
Schmidt, M., Weickert, J.: Morphological counterparts of linear shift-invariant scale-spaces. J. Math. Imaging Vis. 56(2), 352–366 (2016)
Schönlieb, C.B.: Partial Differential Equation Methods for Image Inpainting. Cambridge University Press, Cambridge (2015)
Stein, M.L.: Interpolation of Spatial Data - Some Theory for Kriging. Springer, New York (1999). https://doi.org/10.1007/978-1-4612-1494-6
Uhlir, K., Skala, V.: Radial basis function use for the restoration of damaged images. In: Wojciechowski, K., Smolka, B., Palus, H., Kozera, R., Skarbek, W., Noakes, L. (eds.) Computer Vision and Graphics, Computational Imaging and Vision, vol. 32, pp. 839–844. Springer, Dordrecht (2006). https://doi.org/10.1007/1-4020-4179-9_122
Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)
Yuille, A.L., Grzywacz, N.M.: A computational theory for the perception of coherent visual motion. Nature 333, 71–74 (1988)
Acknowledgement
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 741215, ERC Advanced Grant INCOVID).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Augustin, M., Weickert, J., Andris, S. (2019). Pseudodifferential Inpainting: The Missing Link Between PDE- and RBF-Based Interpolation. In: Lellmann, J., Burger, M., Modersitzki, J. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2019. Lecture Notes in Computer Science(), vol 11603. Springer, Cham. https://doi.org/10.1007/978-3-030-22368-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-22368-7_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22367-0
Online ISBN: 978-3-030-22368-7
eBook Packages: Computer ScienceComputer Science (R0)