Abstract
In this paper we do a systematic investigation of continuous methods for pixel, line pixel and line dejittering. The basis for these investigations are the discrete line dejittering algorithm of Nikolova and the partial differential equation of Lenzen et al for pixel dejittering. To put these two different worlds in perspective we find infinite dimensional optimization algorithms linking to the finite dimensional optimization problems and formal flows associated with the infinite dimensional optimization problems. Two different kinds of optimization problems will be considered: Dejittering algorithms for determining the displacement and displacement error correction formulations, which correct the jittered image, without estimating the jitter. As a by-product we find novel variational methods for displacement error regularization and unify them into one family. The second novelty is a comprehensive comparison of the different models for different types of jitter, in terms of efficiency of reconstruction and numerical complexity.
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
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of inverse problems. Mathematics and its Applications, vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996)
Figueiredo, M.A.T., Dias, J.B., Oliveira, J.P., et al.: On total variation denoising: A new majorization-minimization algorithm and an experimental comparisonwith wavalet denoising. In: IEEE International Conference on Image Processing, pp. 2633–2636. IEEE (2006)
Grasmair, M., Lenzen, F., Obereder, A., Scherzer, O., Fuchs, M.: A non-convex PDE scale space. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 303–315. Springer, Heidelberg (2005)
Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston (1984)
Groetsch, C.W.: Inverse Problems. Mathematical Association of America, Washington, DC (1999). Activities for undergraduates
Kang, S.H., Shen, J.: Video dejittering by bake and shake. Image Vision Comput. 24(2), 143–152 (2006)
Kang, S.H., Shen, J.: Image dejittering based on slicing moments. In: Tai, X.C., Lie, K.A., Chan, T.F., Osher, S. (eds.) Image Processing Based on Partial Differential Equations. Mathematics and Visualization. Springer, New York (2007)
Lenzen, F., Scherzer, O.: A geometric pde for interpolation of \(m\)-channel data. In: Tai, X.C., Knut, M., Marius, L., Knut-Andreas, L. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 413–425. Springer-Verlag, Heidelberg (2009)
Lenzen, F., Scherzer, O.: Partial differential equations for zooming, deinterlacing and dejittering. Int. J. Comput. Vision 92(2), 162–176 (2011)
Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer Verlag, New York (1984)
Morozov, V.A.: Regularization Methods for Ill-Posed Problems. CRC Press, Boca Raton (1993)
Ndajah, P., Kikuchi, H., Yukawa, M., Watanabe, H., et al.: An investigation on the quality of denoised images. Int. J. Circ. Syst. Sign. Proc. 5, 423–434 (2011)
Nikolova, M.: Fast dejittering for digital video frames. In: Tai, X.C., Knut, M., Marius, L., Knut-Andreas, L. (eds.) SSVM 2009. LNCS, pp. 439–451. Springer, Heidelberg (2009)
Nikolova, M.: One-iteration dejittering of digital video images. J. Vis. Commun. Image Represent. 20, 254–274 (2009)
Scherzer, O.: Explicit versus implicit relative error regularization on the space of functions of bounded variation. In: Nashed, M.Z., Scherzer, O. (eds.) Inverse Problems, Image Analysis, and Medical Imaging. Contemporary Mathematics, vol. 313, pp. 171–198. AMC, Providence (2002)
Scherzer, O.: Scale space methods for denoising and inverse problem. Adv. Imaging Electron Phys. 128, 445–530 (2003)
Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational methods in imaging. Applied Mathematical Sciences, vol. 167. Springer, New York (2009)
Shen, J.: Bayesian video dejittering by bv image model. SIAM J. Appl. Math. 64(5), 1691–1708 (2004)
Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. John Wiley & Sons, Washington, D.C. (1977)
Tikhonov, A.N., Goncharsky, A., Stepanov, V., Yagola, A.: Numerical Methods for the Solution of Ill-Posed Problems. Kluwer, Dordrecht (1995)
Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Dong, G., Patrone, A.R., Scherzer, O., Öktem, O. (2015). Infinite Dimensional Optimization Models and PDEs for Dejittering. In: Aujol, JF., Nikolova, M., Papadakis, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science(), vol 9087. Springer, Cham. https://doi.org/10.1007/978-3-319-18461-6_54
Download citation
DOI: https://doi.org/10.1007/978-3-319-18461-6_54
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18460-9
Online ISBN: 978-3-319-18461-6
eBook Packages: Computer ScienceComputer Science (R0)