Abstract
We propose a partial differential equation to be used for interpolating M-channel data, such as digital color images. This equation is derived via a semi-group from a variational regularization method for minimizing displacement errors. For actual image interpolation, the solution of the PDE is projected onto a space of functions satisfying interpolation constraints. A comparison of the test results with standard and state-of-the-art interpolation algorithms shows the competitiveness of this approach.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
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
Belahmidi, A., Guichard, F.: A partial differential equation approach to image zoom. In: Proc. of the 2004 Int. Conf. on Image Processing, pp. 649–652 (2004)
Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: [13], pp. 417–424 (2000)
Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam (1973); North-Holland Mathematics Studies, No. 5. Notas de Matemática (50)
Burger, W., Burge, M.J.: Digitale Bildverarbeitung. Springer, Heidelberg (2005)
Chan, R., Setzer, S., Steidl, G.: Inpainting by flexible Haar wavelet shrinkage. Preprint, University of Mannheim (2008)
Chan, T., Kang, S., Shen, J.: Euler’s elastica and curvature based inpaintings. SIAM J. Appl. Math. 63(2), 564–592 (2002)
Dacorogna, B.: Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals. Lecture Notes in Mathematics, vol. 922. Springer, Heidelberg (1982)
Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, vol. 78. Springer, Berlin (1989)
Elbau, P., Grasmair, M., Lenzen, F., Scherzer, O.: Evolution by non-convex energy functionals. Reports of FSP S092 - Industrial Geometry 75, University of Innsbruck, Austria (submitted) (2008)
Grasmair, M., Lenzen, F., Obereder, A., Scherzer, O., Fuchs, M.: A non-convex PDE scale space. In: [15], pp. 303–315 (2005)
Guichard, F., Malgouyres, F.: Total variation based interpolation. In: Proceedings of the European Signal Processing Conference, vol. 3, pp. 1741–1744 (1998)
Hagen, H., Weickert, J. (eds.): Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Heidelberg (2006)
Hoffmeyer, S. (ed.): Proceedings of the Computer Graphics Conference 2000 (SIGGRAPH 2000). ACMPress, New York (2000)
Jähne, B.: Digitale Bildverarbeitung, 5th edn. Springer, Heidelberg (2002)
Kimmel, R., Sochen, N.A., Weickert, J. (eds.): Scale-Space 2005. LNCS, vol. 3459. Springer, Heidelberg (2005)
Malgouyres, F., Guichard, F.: Edge direction preserving image zooming: a mathematical and numerical analysis. SIAM J. Numer. Anal. 39, 1–37 (2001)
Nashed, M.Z. (ed.): Generalized inverses and applications. Academic Press/ Harcourt Brace Jovanovich Publishers, New York (1976)
Roussos, A., Maragos, P.: Vector-valued image interpolation by an anisotropic diffusion-projection pde. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 104–115. Springer, Heidelberg (2007)
Roussos, A., Maragos, P.: Reversible interpolation of vectorial images by an anisotropic diffusion-projection pde. In: Special Issue for the SSVM 2007 conference. Springer, Heidelberg (2007) (accepted for publication)
Scherzer, O., Weickert, J.: Relations between regularization and diffusion filtering. J. Math. Imaging Vision 12(1), 43–63 (2000)
Tschumperlé, D.: Fast anisotropic smoothing of multi-valued images using curvature-preserving pde’s. International Journal of Computer Vision (IJCV) 68, 65–82 (2006)
Tschumperlé, D., Deriche, R.: Vector valued image regularization with pdes: A common framework for different applications. IEEE Transactions on Pattern Analysis and Machine Intelligence 27 (2005)
Weickert, J., Welk, M.: Tensor field interpolation with pdes. In: [12], pp. 315–325 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lenzen, F., Scherzer, O. (2009). A Geometric PDE for Interpolation of M-Channel Data. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_35
Download citation
DOI: https://doi.org/10.1007/978-3-642-02256-2_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02255-5
Online ISBN: 978-3-642-02256-2
eBook Packages: Computer ScienceComputer Science (R0)