Abstract
Partial differential equations (PDEs) have been successfully applied to many computer vision and image processing problems. However, designing PDEs requires high mathematical skills and good insight into the problems. In this paper, we show that the design of PDEs could be made easier by borrowing the learning strategy from machine learning. In our learning-based PDE (L-PDE) framework for image restoration, there are two terms in our PDE model: (i) a regularizer which encodes the prior knowledge of the image model and (ii) a linear combination of differential invariants, which is data-driven and can effectively adapt to different problems and complex conditions. The L-PDE is learnt from some input/output pairs of training samples via an optimal control technique. The effectiveness of our L-PDE framework for image restoration is demonstrated with two exemplary applications: image denoising and inpainting, where the PDEs are obtained easily and the produced results are comparable to or better than those of traditional PDEs, which were elaborately designed.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Gabor, D.: Information theory in electron microscopy. Laboratory Investigation 14, 801–807 (1965)
Jain, A.: Partial differential equations and finite-difference methods in image processing, part 1. Journal of Optimization Theory and Applications 23, 65–91 (1977)
Koenderink, J.: The structure of images. Biological Cybernetics 50, 363–370 (1984)
Witkin, A.: Scale-space filtering. In: International Joint Conference on Artificial Intelligence, IJCAI (1983)
Pietro, P., Jitendra, M.: Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 629–639 (1990)
Alvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal on Numerical Analysis 29, 845–866 (1992)
Chen, T., Shen, J.: Image processing and analysis: variational, PDE, wavelet, and stochastic methods. SIAM Publisher, Philadelphia (2005)
Tikhonov, A., Arsenin, V.: Solutions of ill-posed problems. Halsted Press (1977)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Osher, S., Rudin, L.: Feature-oriented image enhancement using shock filters. SIAM Journal on Numerical Analysis 27, 919–940 (1990)
Olver, P.: Applications of Lie groups to differential equations. Springer, Heidelberg (1993)
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE contraints. Springer, Heidelberg (2009)
Papadakis, N., Corpetti, T., Memin, E.: Dynamically consistent optical flow estimation. In: International Conference on Computer Vision, ICCV (2007)
Papadakis, N., Memin, E.: Variational optimal control technique for the tracking of deformable objects. In: International Conference on Computer Vision, ICCV (2007)
Florack, L., Romeny, B., Koenderink, J., Viergever, M.: Scale and the differential structure of image. Image and Vision Computing 10, 376–388 (1992)
Lindeberg, T.: Discrete derivative approximations with scale-space properties: a basis for low-level feature extraxtion. Journal of Mathematical Imaging and Vision 3, 349–376 (1993)
Chan, T., Shen, J.: Mathematical models for local nontexture inpaintings. SIAM Journal on Applied Mathematics 62, 1019–1043 (2002)
Stoer, J., Bulirsch, R.: Introduction to numerical analysis, 2nd edn. Springer, Heidelberg (1998)
Lions, J.: Optimal control systems governed by partial differential equations. Springer, Heidelberg (1971)
Lin, Z., Zhang, W., Tang, X.: Learning partial differential equations for computer vision. Technical report, Microsoft Research, MSR-TR-2008-189 (2008)
Lin, Z., Zhang, W., Tang, X.: Designing partial differential equations for image processing by combining differental invariants. Technical report, Microsoft Research, MSR-TR-2009-192 (2009)
Chan, T., Esedoglu, S.: Aspects of total variation regularizaed L 1 function approximation. SIAM Journal on Applied Mathematics 65, 1817–1837 (2005)
Martin, D.R., Fowlkes, C., Tal, D., Malik, J.: A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: International Conference on Computer Vision, ICCV (2001)
Roth, S., Black, M.J.: Fields of experts: a framework for learning image priors. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR (2005)
Weickert, J., Hagen, H.: Visualization and processing of tensor fields. Springer, Heidelberg (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
1 Electronic Supplementary Material
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Liu, R., Lin, Z., Zhang, W., Su, Z. (2010). Learning PDEs for Image Restoration via Optimal Control. In: Daniilidis, K., Maragos, P., Paragios, N. (eds) Computer Vision – ECCV 2010. ECCV 2010. Lecture Notes in Computer Science, vol 6311. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15549-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-15549-9_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15548-2
Online ISBN: 978-3-642-15549-9
eBook Packages: Computer ScienceComputer Science (R0)