Skip to main content

Refined Exponential Filter with Applications to Image Restoration and Interpolation

  • Conference paper
Computer Vision – ACCV 2009 (ACCV 2009)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5996))

Included in the following conference series:

  • 1638 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bertero, M., Poggio, T.A., Torre, V.: Ill-posed problems in early vision. Proc. of the IEEE 76(8), 869–889 (1988)

    Article  Google Scholar 

  2. Hadamard, J.: Lectures on Cauchy’s problem in linear partial differential equations. Courier Dover Publications (2003)

    Google Scholar 

  3. Tikhonov, A.N., Arsenin, V.Y.: Solutions of ill-posed problems. Wiley, New York (1977)

    MATH  Google Scholar 

  4. Kirsch, A.: An introduction to the mathematical theory of inverse problems. Springer, Heidelberg (1996)

    MATH  Google Scholar 

  5. 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)

    Article  MATH  Google Scholar 

  6. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MATH  Google Scholar 

  7. Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B 58(1), 267–288 (1996)

    MATH  MathSciNet  Google Scholar 

  8. Calvetti, D., Lewis, B., Reichel, L.: Smooth or abrupt: a comparison of regularization methods. In: Proc. SPIE, vol. 3461, pp. 286–295 (1998)

    Google Scholar 

  9. Li, G., Nashed, Z.: A modified Tikhonov regularization for linear operator equations. Numerical Functional Analysis and Optimization 26, 543–563 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. Hansen, P.C.: Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. SIAM, Philadelphia (1999)

    Google Scholar 

  12. Hanke, M., Hansen, P.C.: Regularized methods for large scale problems. Surv. Math. Ind. 3, 253–315 (1993)

    MATH  MathSciNet  Google Scholar 

  13. Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, 215–223 (1979)

    Google Scholar 

  14. Groetsch, C.W.: The theory of Tikhonov regularization for Fredholm equations of the first kind. Pitman, Boston (1984)

    MATH  Google Scholar 

  15. Gilboa, G., Sochen, N., Zeevi, Y.Y.: Texture preserving variational denoising using an adaptive fidelity term. In: Proc. VLSM, pp. 137–144 (2003)

    Google Scholar 

  16. Biggs, D.S.C., Andrews, M.: Acceleration of iterative image restoration algorithms. Appl. Opt. 36(8), 1766–1775 (1997)

    Article  Google Scholar 

  17. Chen, L., Yap, K.H.: Regularized interpolation using Kronecker product for still images. In: Proc. ICIP, vol. 2, pp. 1014–1017 (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

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

Download citation

  • 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

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics