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.
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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)
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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
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DOI: https://doi.org/10.1007/11408031_50
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25547-5
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