Abstract
Finding optimal data for inpainting is a key problem for image-compression with partial differential equations. Not only the location of important pixels but also their values should be optimal to maximise the quality gain. The position of important data is usually encoded in a binary mask. Recent studies have shown that allowing non-binary masks may lead to tremendous speedups but comes at the expense of higher storage costs and yields prohibitive memory requirements for the design of competitive image compression codecs. We show that a recently suggested heuristic to eliminate the additional storage costs of the non-binary mask has a strong theoretical foundation in finite dimension. Binary and non-binary masks are equivalent in the sense that they can both give the same reconstruction error if the binary mask is supplemented with optimal data which does not increase the memory footprint. Further, we suggest two fast numerical schemes to obtain this optimised data. This provides a significant building block in the conception of efficient data compression schemes with partial differential equations.
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
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Masnou, S., Morel, J.M.: Level lines based disocclusion. In: Proc. of the International Conference on Image Processing, vol. 3, pp. 259–263. IEEE (1998)
Bertalmío, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proc. SIGGRAPH, pp. 417–424. ACM Press/Addison-Wesley Publishing Company, New Orleans, LI (2000)
Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., Seidel, H.-P.: Towards PDE-based image compression. In: Paragios, N., Faugeras, O., Chan, T., Schnörr, C. (eds.) VLSM 2005. LNCS, vol. 3752, pp. 37–48. Springer, Heidelberg (2005)
Liu, D., Sun, X., Wu, F., Li, S., Zhang, Y.Q.: Image compression with edge-based inpainting. IEEE Transactions on Circuits, Systems and Video Technology 7(10), 1273–1286 (2007)
Belhachmi, Z., Bucur, D., Burgeth, B., Weickert, J.: How to choose interpolation data in images. SIAM Journal on Applied Mathematics 70(1), 333–352 (2009)
Schmaltz, C., Weickert, J., Bruhn, A.: Beating the quality of JPEG 2000 with anisotropic diffusion. In: Denzler, J., Notni, G., Süße, H. (eds.) DAGM 2009. LNCS, vol. 5748, pp. 452–461. Springer, Heidelberg (2009)
Mainberger, M., Bruhn, A., Weickert, J., Forchhammer, S.: Edge-based compression of cartoon-like images with homogeneous diffusion. Pattern Recognition 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)
Bourquard, A., Unser, M.: Anisotropic interpolation of sparse generalized image samples. IEEE Transactions on Image Processing 22(2), 459–472 (2013)
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)
Chen, Y., Ranftl, R., Pock, T.: A bi-level view of inpainting - based image compression. Computing Research Repository (2014), http://arxiv.org/abs/1401.4112v2
Gomathi, R., Kumar, A.V.A.: A multiresolution image completion algorithm for compressing digital color images. Journal of Applied Mathematics 2014, Article ID 757318 (2014)
Köstler, H., Stürmer, M., Freundl, C., Rüde, U.: PDE based video compression in real time. Lehrstuhlbericht 07-11, Friedrich-Alexander-Universität Erlangen-Nürnberg (2011)
Ochs, P., Chen, Y., Brox, T., Pock, T.: iPiano: Inertial proximal algorithm for non-convex optimization. SIAM Journal on Imaging Sciences (to appear, 2014)
Ochs, P., Brox, T., Pock, T.: iPiasco: Inertial proximal algorithm for strongly convex optimization. Technical report, Universität Freiburg (2014)
Zaremba, S.: Sur un problème mixte relatif à l’équation de Laplace. Bulletin de l’Académie des Sciences de Cracovie, 313–344 (1910)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer (2001)
Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., Seidel, H.: Image compression with anisotropic diffusion. Journal of Mathematical Imaging and Vision 31(2-3), 255–269 (2008)
Paige, C.C., Saunders, M.A.: Algorithm 583; LSQR: Sparse linear equations and least-squares problems. ACM Transactions on Mathematical Software 8(2), 195–209 (1982)
Paige, C.C., Saunders, M.A.: LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Transactions on Mathematical Software 8(1), 43–71 (1982)
Golub, G.H., Kahan, W.: Calculating the singular values and pseudoinverse of a matrix. Journal of the Society for Industrial and Applied Mathematics 2(2), 205–224 (1965)
Davis, T.A.: Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Transactions on Mathematical Software 30(2), 196–199 (2004)
Davis, T.A., Duff, I.S.: An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM Journal on Matrix Analysis and Applications 18(1), 104–158 (1997)
Davis, T.A., Duff, I.S.: A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Transactions on Mathematical Software 25(1), 1–19 (1999)
Chambolle, A., Pock, T.: A first order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40(1), 120–145 (2011)
Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization. In: Metaxas, D., Quan, L., Sanfeliu, A., Van Gool, L. (eds.) 2011 International Conference on Computer Vision (ICCV 2011), pp. 1762–1769. IEEE (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Hoeltgen, L., Weickert, J. (2015). Why Does Non-binary Mask Optimisation Work for Diffusion-Based Image Compression?. In: Tai, XC., Bae, E., Chan, T.F., Lysaker, M. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2015. Lecture Notes in Computer Science, vol 8932. Springer, Cham. https://doi.org/10.1007/978-3-319-14612-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-14612-6_7
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-14611-9
Online ISBN: 978-3-319-14612-6
eBook Packages: Computer ScienceComputer Science (R0)