Abstract
Ill-posed linear equations are pervasive in computer vision. A popular way to solve an ill-posed problem is regularization. In this paper, we propose a new criterion for designing the regularizing filter. This criterion reveals the implicit assumption made by regularizing filters. Then with the help of the discrete Picard condition, we refine the exponential filter using our criterion. The effectiveness of our method is demonstrated on image restoration and interpolation.
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Bertero, M., Poggio, T.A., Torre, V.: Ill-posed problems in early vision. Proc. of the IEEE 76(8), 869–889 (1988)
Hadamard, J.: Lectures on Cauchy’s problem in linear partial differential equations. Courier Dover Publications (2003)
Tikhonov, A.N., Arsenin, V.Y.: Solutions of ill-posed problems. Wiley, New York (1977)
Kirsch, A.: An introduction to the mathematical theory of inverse problems. Springer, Heidelberg (1996)
Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. Journal of the ACM 9(1), 84–97 (1962)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B 58(1), 267–288 (1996)
Calvetti, D., Lewis, B., Reichel, L.: Smooth or abrupt: a comparison of regularization methods. In: Proc. SPIE, vol. 3461, pp. 286–295 (1998)
Li, G., Nashed, Z.: A modified Tikhonov regularization for linear operator equations. Numerical Functional Analysis and Optimization 26, 543–563 (2005)
Velipasaoglu, E.O., Sun, H., Zhang, F., Berrier, K.L., Khoury, D.S.: Spatial regularization of the electrocardiographic inverse problem and its application to endocardial mapping. IEEE Transactions on Biomedical Engineering 47(3), 327–337 (2000)
Hansen, P.C.: Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. SIAM, Philadelphia (1999)
Hanke, M., Hansen, P.C.: Regularized methods for large scale problems. Surv. Math. Ind. 3, 253–315 (1993)
Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, 215–223 (1979)
Groetsch, C.W.: The theory of Tikhonov regularization for Fredholm equations of the first kind. Pitman, Boston (1984)
Gilboa, G., Sochen, N., Zeevi, Y.Y.: Texture preserving variational denoising using an adaptive fidelity term. In: Proc. VLSM, pp. 137–144 (2003)
Biggs, D.S.C., Andrews, M.: Acceleration of iterative image restoration algorithms. Appl. Opt. 36(8), 1766–1775 (1997)
Chen, L., Yap, K.H.: Regularized interpolation using Kronecker product for still images. In: Proc. ICIP, vol. 2, pp. 1014–1017 (2005)
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Geng, Y., Lin, T., Lin, Z., Hao, P. (2010). Refined Exponential Filter with Applications to Image Restoration and Interpolation. In: Zha, H., Taniguchi, Ri., Maybank, S. (eds) Computer Vision – ACCV 2009. ACCV 2009. Lecture Notes in Computer Science, vol 5996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12297-2_4
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DOI: https://doi.org/10.1007/978-3-642-12297-2_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-12296-5
Online ISBN: 978-3-642-12297-2
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