Abstract
Deblurring with a spatially invariant kernel of arbitrary shape is a frequent problem in image processing. We address this task by studying nonconvex variational functionals that lead to diffusion-reaction equations of Perona–Malik type. Further we consider novel deblurring PDEs with anisotropic diffusion tensors. In order to improve deblurring quality we propose a continuation strategy in which the diffusion weight is reduced during the process. To evaluate our methods, we compare them to two established techniques: Wiener filtering which is regarded as the best linear filter, and a total variation based deconvolution which is the most widespread deblurring PDE. The experiments confirm the favourable performance of our methods, both visually and in terms of signal-to-noise ratio.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bar, L., Sochen, N., Kiryati, N.: Variational pairing of image segmentation and blind restoration. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3022, pp. 166–177. Springer, Heidelberg (2004)
Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. IoP Publishing, Bristol (1998)
Bertero, M., Poggio, T.A., Torre, V.: Ill-posed problems in early vision. Proceedings of the IEEE 76(8), 869–889 (1988)
Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987)
Chan, T., Chan, R., Zhou, H.: A continuation method for total variation denoising problems. In: Luk, F. (ed.) Proceedings of the SPIE Conference on Advanced Signal Processing Algorithms, Algorithms, Architectures, and Implementations, San Diego, vol. 2563, pp. 314–325 (1995)
Chan, T.F., Wong, C.K.: Total variation blind deconvolution. IEEE Transactions on Image Processing 7, 370–375 (1998)
Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 2nd edn. Addison–Wesley, Reading (2002)
Marquina, A., Osher, S.: A new time dependent model based on level set motion for nonlinear deblurring and noise removal. In: Nielsen, M., Johansen, P., Fogh Olsen, O., Weickert, J. (eds.) Scale-Space 1999. LNCS, vol. 1682, pp. 429–434. Springer, Heidelberg (1999)
Osher, S., Rudin, L.: Shocks and other nonlinear filtering applied to image processing. In: Tescher, A.G. (ed.) Applications of Digital Image Processing XIV Proceedings of SPIE, vol. 1567, pp. 414–431. SPIE Press, Bellingham (1991)
Osher, S., Rudin, L.: Total variation based image restoration with free local constraints. In: Proceedings of the IEEE ICIP, Austin, pp. 31–35 (1994)
Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 629–639 (1990)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Weickert, J.: A review of nonlinear diffusion filtering. In: ter Haar Romeny, B., Florack, L., Koenderink, J., Viergever, M. (eds.) Scale-Space 1997. LNCS, vol. 1252, pp. 3–28. Springer, Heidelberg (1997)
Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)
Weickert, J., ter Haar Romeny, B.M., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Transactions on Image Processing 7(3), 398–410 (1998)
Wiener, N.: Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications. The MIT Press, Cambridge (1949)
You, Y.-L., Kaveh, M.: Anisotropic blind image restoration. In: Proc. 1996 IEEE International Conference on Image Processing, Lausanne, Switzerland, September 1996, vol. 2, pp. 461–464 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Welk, M., Theis, D., Brox, T., Weickert, J. (2005). PDE-Based Deconvolution with Forward-Backward Diffusivities and Diffusion Tensors. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds) Scale Space and PDE Methods in Computer Vision. Scale-Space 2005. Lecture Notes in Computer Science, vol 3459. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11408031_50
Download citation
DOI: https://doi.org/10.1007/11408031_50
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25547-5
Online ISBN: 978-3-540-32012-8
eBook Packages: Computer ScienceComputer Science (R0)