Abstract

We are concerned with the performance evaluation of the ridgelet bi-frame for image denoising application. The ridgelet bi-frame is a new (as far as we know) bi-frame system that can efficiently deal with straight singularities in two dimensions. We show that, for images dominated by straight edges, the ridgelet bi-frame can obtain much better restoration results than wavelet systems. We also investigate the statistical properties of the ridgelet bi-frame coefficients of these images. Results indicate that the marginal distribution of ridgelet bi-frame coefficients has higher kurtosis than that of wavelet coefficients of the same images. We describe a simple method through which statistical denoising algorithms previously developed in the wavelet domain can be conveniently introduced into the ridgelet bi-frame domain. In addition, we use the ridgelet bi-frame to construct another new bi-frame system referred to as the curvelet bi-frame, which can be viewed as a generalized version of the curvelet. Experiment results show that the simple hard-threshold procedure in the curvelet bi-frame domain produces restoration results comparable with those due to the state-of-the-art denoising methods.

© 2006 Optical Society of America

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  1. E. J. Candès, "Harmonic analysis of neural networks," Appl. Comput. Harmon. Anal. 6, 197-218 (1999).
    [Crossref]
  2. M. N. Do and M. Vetterli, "The finite ridgelet transform for image representation," IEEE Trans. Image Process. 12, 16-28 (2003).
    [Crossref]
  3. D. L. Donoho, "Orthonormal ridgelet and linear singularities," SIAM J. Math. Anal. 31, 1062-1099 (2000).
    [Crossref]
  4. E. J. Candès and D. L. Donoho, "Curvelets—a surprisingly effective nonadaptive representation for objects with edges," in Curve and Surface Fitting: Saint-Malo 1999, A.Cohen, C.Rabut, and L.L.Schumaker, eds. (Vanderbilt U. Press, 1999).
  5. M. N. Do and M. Vetterli, "The contourlet transform: an efficient directional multiresolution image representation," IEEE Trans. Image Process. 14, 2091-2106 (2005).
    [Crossref] [PubMed]
  6. E. J. Candès and D. L. Donoho, "New tight frames of curvelets and optimal representations of objects with piecewise-C2 singularities," Commun. Pure Appl. Math. 57, 219-266 (2004).
    [Crossref]
  7. E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, "Fast discrete curvelet transforms," (2005), http://www.curvelet.org/papers/FDCT.pdf.
  8. E. J. Candès, "On the representation of mutilated Sobolev functions," SIAM J. Math. Anal. 1, 2495-2509 (1999).
  9. S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999).
  10. A. G. Flesia, H. Hel-Or, A. Averbuch, E. J. Candès, R. R. Coifman, and D. L. Donoho, "Digital implementation of ridgelet packets," in Beyond Wavelets, J.Stoeckler and G.V.Welland, eds. (Academic, 2003).
    [Crossref]
  11. S. Tan and L. Jiao, "Ridgelet bi-frame," Appl. Comput. Harmon. Anal. 20, 391-402 (2006).
    [Crossref]
  12. J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Process. 12, 1338-1351 (2003).
    [Crossref]
  13. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, 1983).
  14. A. Averbuch, R. R. Coifman, D. L. Donoho, and M. Israeli, "Fast Slant Stack: A notion of Radon transform for data in a Cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible," Tech. Rep. (Stanford University, 2003).
  15. E. P. Simoncelli and E. H. Adelson, "Noise removal via Bayesian wavelet coring," in Proceedings of IEEE Conference on Image Processing (IEEE Press, 1996).
    [Crossref]
  16. S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
    [Crossref]
  17. M. S. Crouse, R. D. Nowak, and R. C. Baraniuk, "Wavelet-based statistical signal processing using hidden Markov models," IEEE Trans. Signal Process. 46, 886-902 (1998).
    [Crossref]
  18. H. Chipman, E. Kolaczyk, and R. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Am. Stat. Assoc. 92, 1413-1421 (1997).
    [Crossref]
  19. A. N. Netravali and B. G. Haskell, Digital Pictures (Plenum, 1988).
  20. I. M. Joinstone, "Wavelets and the theory of non-parametric function estimation," Philos. Trans. R. Soc. London, Ser. A 357, 2475-2493 (1999).
    [Crossref]
  21. E. J. Candès, "Monoscale ridgelet for the representation of images with edges," Tech. Rep. (Department of Statististics, Stanford University, 1999).
  22. D. L. Donoho and M. R. Duncan, "Digital curvelet transform: strategy, implementation and experiments," Tech. Rep. (Stanford University, 1999).
  23. J. L. Starck, E. J. Candès, and D. L. Donoho, "The curvelet transform for image denoising," IEEE Trans. Image Process. 11, 670-684 (2002).
    [Crossref]
  24. M. J. Shensa, U. Center, and C. S. Diego, "The discrete wavelet transform: wedding the àtrous and Mallat algorithms," IEEE Trans. Signal Process. 40, 2464-2482 (1992).
    [Crossref]
  25. B. A. Olshausen and D. J. Field, "Emergence of simple-cell receptive field properties by learning a sparse code for natural images," Nature 381, 607-609 (1996).
    [Crossref] [PubMed]
  26. B. A. Olshausen and D. J. Field, "Sparse coding with an overcomplete basis set: a strategy employed by V1?" Vision Res. 37, 3311-3325 (1997).
    [Crossref]
  27. S. Tan and L. Jiao, "New evidences for sparse coding strategy employed in visual neurons: from the image processing and nonlinear approximation viewpoint," presented at the Thirteenth European Symposium on Artificial Neural Networks, Bruges, Belgium, April 27-29, 2005.
  28. A. Cohen, I. Daubechies, and J.-C. Feauveau, "Biorthogonal bases of compactly supported wavelets," Commun. Pure Appl. Math. 45, 485-560 (1992).
    [Crossref]
  29. J. Villasenor, B. Belzer, and J. Liao, "Wavelet filter evaluation for image compression," IEEE Trans. Image Process. 2, 1053-1060 (1995).
    [Crossref]
  30. J. L. Starck, D. L. Donoho, and E. Candès, "Very high quality image restoration by combining wavelets and curvelets," in Proc. SPIE 4478, 9-19 (2001).
    [Crossref]
  31. A. L. Cunha, J. Zhou, and M. N. Do, "The nonsubsampled contourlet transform: theory, design, and applications," IEEE Trans. Image Process. (to be published).
  32. D. D.-Y. Po and M. N. Do, "Directional multiscale modeling of images using the contourlet transform," IEEE Trans. Image Process. (to be published).

2006 (1)

S. Tan and L. Jiao, "Ridgelet bi-frame," Appl. Comput. Harmon. Anal. 20, 391-402 (2006).
[Crossref]

2005 (1)

M. N. Do and M. Vetterli, "The contourlet transform: an efficient directional multiresolution image representation," IEEE Trans. Image Process. 14, 2091-2106 (2005).
[Crossref] [PubMed]

2004 (1)

E. J. Candès and D. L. Donoho, "New tight frames of curvelets and optimal representations of objects with piecewise-C2 singularities," Commun. Pure Appl. Math. 57, 219-266 (2004).
[Crossref]

2003 (2)

M. N. Do and M. Vetterli, "The finite ridgelet transform for image representation," IEEE Trans. Image Process. 12, 16-28 (2003).
[Crossref]

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Process. 12, 1338-1351 (2003).
[Crossref]

2002 (1)

J. L. Starck, E. J. Candès, and D. L. Donoho, "The curvelet transform for image denoising," IEEE Trans. Image Process. 11, 670-684 (2002).
[Crossref]

2001 (1)

J. L. Starck, D. L. Donoho, and E. Candès, "Very high quality image restoration by combining wavelets and curvelets," in Proc. SPIE 4478, 9-19 (2001).
[Crossref]

2000 (1)

D. L. Donoho, "Orthonormal ridgelet and linear singularities," SIAM J. Math. Anal. 31, 1062-1099 (2000).
[Crossref]

1999 (3)

E. J. Candès, "On the representation of mutilated Sobolev functions," SIAM J. Math. Anal. 1, 2495-2509 (1999).

E. J. Candès, "Harmonic analysis of neural networks," Appl. Comput. Harmon. Anal. 6, 197-218 (1999).
[Crossref]

I. M. Joinstone, "Wavelets and the theory of non-parametric function estimation," Philos. Trans. R. Soc. London, Ser. A 357, 2475-2493 (1999).
[Crossref]

1998 (1)

M. S. Crouse, R. D. Nowak, and R. C. Baraniuk, "Wavelet-based statistical signal processing using hidden Markov models," IEEE Trans. Signal Process. 46, 886-902 (1998).
[Crossref]

1997 (2)

H. Chipman, E. Kolaczyk, and R. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Am. Stat. Assoc. 92, 1413-1421 (1997).
[Crossref]

B. A. Olshausen and D. J. Field, "Sparse coding with an overcomplete basis set: a strategy employed by V1?" Vision Res. 37, 3311-3325 (1997).
[Crossref]

1996 (1)

B. A. Olshausen and D. J. Field, "Emergence of simple-cell receptive field properties by learning a sparse code for natural images," Nature 381, 607-609 (1996).
[Crossref] [PubMed]

1995 (1)

J. Villasenor, B. Belzer, and J. Liao, "Wavelet filter evaluation for image compression," IEEE Trans. Image Process. 2, 1053-1060 (1995).
[Crossref]

1992 (2)

A. Cohen, I. Daubechies, and J.-C. Feauveau, "Biorthogonal bases of compactly supported wavelets," Commun. Pure Appl. Math. 45, 485-560 (1992).
[Crossref]

M. J. Shensa, U. Center, and C. S. Diego, "The discrete wavelet transform: wedding the àtrous and Mallat algorithms," IEEE Trans. Signal Process. 40, 2464-2482 (1992).
[Crossref]

1989 (1)

S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
[Crossref]

Adelson, E. H.

E. P. Simoncelli and E. H. Adelson, "Noise removal via Bayesian wavelet coring," in Proceedings of IEEE Conference on Image Processing (IEEE Press, 1996).
[Crossref]

Averbuch, A.

A. Averbuch, R. R. Coifman, D. L. Donoho, and M. Israeli, "Fast Slant Stack: A notion of Radon transform for data in a Cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible," Tech. Rep. (Stanford University, 2003).

A. G. Flesia, H. Hel-Or, A. Averbuch, E. J. Candès, R. R. Coifman, and D. L. Donoho, "Digital implementation of ridgelet packets," in Beyond Wavelets, J.Stoeckler and G.V.Welland, eds. (Academic, 2003).
[Crossref]

Baraniuk, R. C.

M. S. Crouse, R. D. Nowak, and R. C. Baraniuk, "Wavelet-based statistical signal processing using hidden Markov models," IEEE Trans. Signal Process. 46, 886-902 (1998).
[Crossref]

Belzer, B.

J. Villasenor, B. Belzer, and J. Liao, "Wavelet filter evaluation for image compression," IEEE Trans. Image Process. 2, 1053-1060 (1995).
[Crossref]

Candès, E.

J. L. Starck, D. L. Donoho, and E. Candès, "Very high quality image restoration by combining wavelets and curvelets," in Proc. SPIE 4478, 9-19 (2001).
[Crossref]

Candès, E. J.

E. J. Candès and D. L. Donoho, "New tight frames of curvelets and optimal representations of objects with piecewise-C2 singularities," Commun. Pure Appl. Math. 57, 219-266 (2004).
[Crossref]

J. L. Starck, E. J. Candès, and D. L. Donoho, "The curvelet transform for image denoising," IEEE Trans. Image Process. 11, 670-684 (2002).
[Crossref]

E. J. Candès, "Harmonic analysis of neural networks," Appl. Comput. Harmon. Anal. 6, 197-218 (1999).
[Crossref]

E. J. Candès, "On the representation of mutilated Sobolev functions," SIAM J. Math. Anal. 1, 2495-2509 (1999).

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, "Fast discrete curvelet transforms," (2005), http://www.curvelet.org/papers/FDCT.pdf.

E. J. Candès and D. L. Donoho, "Curvelets—a surprisingly effective nonadaptive representation for objects with edges," in Curve and Surface Fitting: Saint-Malo 1999, A.Cohen, C.Rabut, and L.L.Schumaker, eds. (Vanderbilt U. Press, 1999).

A. G. Flesia, H. Hel-Or, A. Averbuch, E. J. Candès, R. R. Coifman, and D. L. Donoho, "Digital implementation of ridgelet packets," in Beyond Wavelets, J.Stoeckler and G.V.Welland, eds. (Academic, 2003).
[Crossref]

E. J. Candès, "Monoscale ridgelet for the representation of images with edges," Tech. Rep. (Department of Statististics, Stanford University, 1999).

Center, U.

M. J. Shensa, U. Center, and C. S. Diego, "The discrete wavelet transform: wedding the àtrous and Mallat algorithms," IEEE Trans. Signal Process. 40, 2464-2482 (1992).
[Crossref]

Chipman, H.

H. Chipman, E. Kolaczyk, and R. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Am. Stat. Assoc. 92, 1413-1421 (1997).
[Crossref]

Cohen, A.

A. Cohen, I. Daubechies, and J.-C. Feauveau, "Biorthogonal bases of compactly supported wavelets," Commun. Pure Appl. Math. 45, 485-560 (1992).
[Crossref]

Coifman, R. R.

A. G. Flesia, H. Hel-Or, A. Averbuch, E. J. Candès, R. R. Coifman, and D. L. Donoho, "Digital implementation of ridgelet packets," in Beyond Wavelets, J.Stoeckler and G.V.Welland, eds. (Academic, 2003).
[Crossref]

A. Averbuch, R. R. Coifman, D. L. Donoho, and M. Israeli, "Fast Slant Stack: A notion of Radon transform for data in a Cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible," Tech. Rep. (Stanford University, 2003).

Crouse, M. S.

M. S. Crouse, R. D. Nowak, and R. C. Baraniuk, "Wavelet-based statistical signal processing using hidden Markov models," IEEE Trans. Signal Process. 46, 886-902 (1998).
[Crossref]

Cunha, A. L.

A. L. Cunha, J. Zhou, and M. N. Do, "The nonsubsampled contourlet transform: theory, design, and applications," IEEE Trans. Image Process. (to be published).

Daubechies, I.

A. Cohen, I. Daubechies, and J.-C. Feauveau, "Biorthogonal bases of compactly supported wavelets," Commun. Pure Appl. Math. 45, 485-560 (1992).
[Crossref]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, 1983).

Demanet, L.

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, "Fast discrete curvelet transforms," (2005), http://www.curvelet.org/papers/FDCT.pdf.

Diego, C. S.

M. J. Shensa, U. Center, and C. S. Diego, "The discrete wavelet transform: wedding the àtrous and Mallat algorithms," IEEE Trans. Signal Process. 40, 2464-2482 (1992).
[Crossref]

Do, M. N.

M. N. Do and M. Vetterli, "The contourlet transform: an efficient directional multiresolution image representation," IEEE Trans. Image Process. 14, 2091-2106 (2005).
[Crossref] [PubMed]

M. N. Do and M. Vetterli, "The finite ridgelet transform for image representation," IEEE Trans. Image Process. 12, 16-28 (2003).
[Crossref]

A. L. Cunha, J. Zhou, and M. N. Do, "The nonsubsampled contourlet transform: theory, design, and applications," IEEE Trans. Image Process. (to be published).

D. D.-Y. Po and M. N. Do, "Directional multiscale modeling of images using the contourlet transform," IEEE Trans. Image Process. (to be published).

Donoho, D. L.

E. J. Candès and D. L. Donoho, "New tight frames of curvelets and optimal representations of objects with piecewise-C2 singularities," Commun. Pure Appl. Math. 57, 219-266 (2004).
[Crossref]

J. L. Starck, E. J. Candès, and D. L. Donoho, "The curvelet transform for image denoising," IEEE Trans. Image Process. 11, 670-684 (2002).
[Crossref]

J. L. Starck, D. L. Donoho, and E. Candès, "Very high quality image restoration by combining wavelets and curvelets," in Proc. SPIE 4478, 9-19 (2001).
[Crossref]

D. L. Donoho, "Orthonormal ridgelet and linear singularities," SIAM J. Math. Anal. 31, 1062-1099 (2000).
[Crossref]

A. Averbuch, R. R. Coifman, D. L. Donoho, and M. Israeli, "Fast Slant Stack: A notion of Radon transform for data in a Cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible," Tech. Rep. (Stanford University, 2003).

A. G. Flesia, H. Hel-Or, A. Averbuch, E. J. Candès, R. R. Coifman, and D. L. Donoho, "Digital implementation of ridgelet packets," in Beyond Wavelets, J.Stoeckler and G.V.Welland, eds. (Academic, 2003).
[Crossref]

E. J. Candès and D. L. Donoho, "Curvelets—a surprisingly effective nonadaptive representation for objects with edges," in Curve and Surface Fitting: Saint-Malo 1999, A.Cohen, C.Rabut, and L.L.Schumaker, eds. (Vanderbilt U. Press, 1999).

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, "Fast discrete curvelet transforms," (2005), http://www.curvelet.org/papers/FDCT.pdf.

D. L. Donoho and M. R. Duncan, "Digital curvelet transform: strategy, implementation and experiments," Tech. Rep. (Stanford University, 1999).

Duncan, M. R.

D. L. Donoho and M. R. Duncan, "Digital curvelet transform: strategy, implementation and experiments," Tech. Rep. (Stanford University, 1999).

Feauveau, J.-C.

A. Cohen, I. Daubechies, and J.-C. Feauveau, "Biorthogonal bases of compactly supported wavelets," Commun. Pure Appl. Math. 45, 485-560 (1992).
[Crossref]

Field, D. J.

B. A. Olshausen and D. J. Field, "Sparse coding with an overcomplete basis set: a strategy employed by V1?" Vision Res. 37, 3311-3325 (1997).
[Crossref]

B. A. Olshausen and D. J. Field, "Emergence of simple-cell receptive field properties by learning a sparse code for natural images," Nature 381, 607-609 (1996).
[Crossref] [PubMed]

Flesia, A. G.

A. G. Flesia, H. Hel-Or, A. Averbuch, E. J. Candès, R. R. Coifman, and D. L. Donoho, "Digital implementation of ridgelet packets," in Beyond Wavelets, J.Stoeckler and G.V.Welland, eds. (Academic, 2003).
[Crossref]

Haskell, B. G.

A. N. Netravali and B. G. Haskell, Digital Pictures (Plenum, 1988).

Hel-Or, H.

A. G. Flesia, H. Hel-Or, A. Averbuch, E. J. Candès, R. R. Coifman, and D. L. Donoho, "Digital implementation of ridgelet packets," in Beyond Wavelets, J.Stoeckler and G.V.Welland, eds. (Academic, 2003).
[Crossref]

Israeli, M.

A. Averbuch, R. R. Coifman, D. L. Donoho, and M. Israeli, "Fast Slant Stack: A notion of Radon transform for data in a Cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible," Tech. Rep. (Stanford University, 2003).

Jiao, L.

S. Tan and L. Jiao, "Ridgelet bi-frame," Appl. Comput. Harmon. Anal. 20, 391-402 (2006).
[Crossref]

S. Tan and L. Jiao, "New evidences for sparse coding strategy employed in visual neurons: from the image processing and nonlinear approximation viewpoint," presented at the Thirteenth European Symposium on Artificial Neural Networks, Bruges, Belgium, April 27-29, 2005.

Joinstone, I. M.

I. M. Joinstone, "Wavelets and the theory of non-parametric function estimation," Philos. Trans. R. Soc. London, Ser. A 357, 2475-2493 (1999).
[Crossref]

Kolaczyk, E.

H. Chipman, E. Kolaczyk, and R. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Am. Stat. Assoc. 92, 1413-1421 (1997).
[Crossref]

Liao, J.

J. Villasenor, B. Belzer, and J. Liao, "Wavelet filter evaluation for image compression," IEEE Trans. Image Process. 2, 1053-1060 (1995).
[Crossref]

Mallat, S.

S. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Trans. Pattern Anal. Mach. Intell. 11, 674-693 (1989).
[Crossref]

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999).

McCulloch, R.

H. Chipman, E. Kolaczyk, and R. McCulloch, "Adaptive Bayesian wavelet shrinkage," J. Am. Stat. Assoc. 92, 1413-1421 (1997).
[Crossref]

Netravali, A. N.

A. N. Netravali and B. G. Haskell, Digital Pictures (Plenum, 1988).

Nowak, R. D.

M. S. Crouse, R. D. Nowak, and R. C. Baraniuk, "Wavelet-based statistical signal processing using hidden Markov models," IEEE Trans. Signal Process. 46, 886-902 (1998).
[Crossref]

Olshausen, B. A.

B. A. Olshausen and D. J. Field, "Sparse coding with an overcomplete basis set: a strategy employed by V1?" Vision Res. 37, 3311-3325 (1997).
[Crossref]

B. A. Olshausen and D. J. Field, "Emergence of simple-cell receptive field properties by learning a sparse code for natural images," Nature 381, 607-609 (1996).
[Crossref] [PubMed]

Po, D. D.-Y.

D. D.-Y. Po and M. N. Do, "Directional multiscale modeling of images using the contourlet transform," IEEE Trans. Image Process. (to be published).

Portilla, J.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Process. 12, 1338-1351 (2003).
[Crossref]

Shensa, M. J.

M. J. Shensa, U. Center, and C. S. Diego, "The discrete wavelet transform: wedding the àtrous and Mallat algorithms," IEEE Trans. Signal Process. 40, 2464-2482 (1992).
[Crossref]

Simoncelli, E. P.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Process. 12, 1338-1351 (2003).
[Crossref]

E. P. Simoncelli and E. H. Adelson, "Noise removal via Bayesian wavelet coring," in Proceedings of IEEE Conference on Image Processing (IEEE Press, 1996).
[Crossref]

Starck, J. L.

J. L. Starck, E. J. Candès, and D. L. Donoho, "The curvelet transform for image denoising," IEEE Trans. Image Process. 11, 670-684 (2002).
[Crossref]

J. L. Starck, D. L. Donoho, and E. Candès, "Very high quality image restoration by combining wavelets and curvelets," in Proc. SPIE 4478, 9-19 (2001).
[Crossref]

Strela, V.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Process. 12, 1338-1351 (2003).
[Crossref]

Tan, S.

S. Tan and L. Jiao, "Ridgelet bi-frame," Appl. Comput. Harmon. Anal. 20, 391-402 (2006).
[Crossref]

S. Tan and L. Jiao, "New evidences for sparse coding strategy employed in visual neurons: from the image processing and nonlinear approximation viewpoint," presented at the Thirteenth European Symposium on Artificial Neural Networks, Bruges, Belgium, April 27-29, 2005.

Vetterli, M.

M. N. Do and M. Vetterli, "The contourlet transform: an efficient directional multiresolution image representation," IEEE Trans. Image Process. 14, 2091-2106 (2005).
[Crossref] [PubMed]

M. N. Do and M. Vetterli, "The finite ridgelet transform for image representation," IEEE Trans. Image Process. 12, 16-28 (2003).
[Crossref]

Villasenor, J.

J. Villasenor, B. Belzer, and J. Liao, "Wavelet filter evaluation for image compression," IEEE Trans. Image Process. 2, 1053-1060 (1995).
[Crossref]

Wainwright, M. J.

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Process. 12, 1338-1351 (2003).
[Crossref]

Ying, L.

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, "Fast discrete curvelet transforms," (2005), http://www.curvelet.org/papers/FDCT.pdf.

Zhou, J.

A. L. Cunha, J. Zhou, and M. N. Do, "The nonsubsampled contourlet transform: theory, design, and applications," IEEE Trans. Image Process. (to be published).

Appl. Comput. Harmon. Anal. (2)

E. J. Candès, "Harmonic analysis of neural networks," Appl. Comput. Harmon. Anal. 6, 197-218 (1999).
[Crossref]

S. Tan and L. Jiao, "Ridgelet bi-frame," Appl. Comput. Harmon. Anal. 20, 391-402 (2006).
[Crossref]

Commun. Pure Appl. Math. (2)

E. J. Candès and D. L. Donoho, "New tight frames of curvelets and optimal representations of objects with piecewise-C2 singularities," Commun. Pure Appl. Math. 57, 219-266 (2004).
[Crossref]

A. Cohen, I. Daubechies, and J.-C. Feauveau, "Biorthogonal bases of compactly supported wavelets," Commun. Pure Appl. Math. 45, 485-560 (1992).
[Crossref]

IEEE Trans. Image Process. (5)

J. Villasenor, B. Belzer, and J. Liao, "Wavelet filter evaluation for image compression," IEEE Trans. Image Process. 2, 1053-1060 (1995).
[Crossref]

J. L. Starck, E. J. Candès, and D. L. Donoho, "The curvelet transform for image denoising," IEEE Trans. Image Process. 11, 670-684 (2002).
[Crossref]

M. N. Do and M. Vetterli, "The contourlet transform: an efficient directional multiresolution image representation," IEEE Trans. Image Process. 14, 2091-2106 (2005).
[Crossref] [PubMed]

M. N. Do and M. Vetterli, "The finite ridgelet transform for image representation," IEEE Trans. Image Process. 12, 16-28 (2003).
[Crossref]

J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Trans. Image Process. 12, 1338-1351 (2003).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

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Figures (9)

Fig. 1
Fig. 1

Comparison between biorthogonal wavelet transform and ridgelet bi-frame decomposition for a 256 × 256 image. (a) Original image, (b) three-level biorthogonal 7 9 wavelet transform, (c) Fast Slant Stack of original image, (d) ridgelet bi-frame decomposition (employing three-level biorthogonal 7 9 wavelet).

Fig. 2
Fig. 2

(a) Histogram for the HH wavelet subband built on the image in Fig. 1a, kurtosis = 26.22 , mean = 0.0019 . (b) Histogram of the same wavelet subband in the ridgelet bi-frame domain built on the same image, kurtosis = 121.86 , mean = 1.03 × 10 4 .

Fig. 3
Fig. 3

(a) Histogram of the HH wavelet subband in the modified version of the ridgelet bi-frame built on the image in Fig. 1a, kurtosis = 91.81 , mean = 0.045 . (b) Histogram of the same wavelet subband in the modified version of the ridgelet bi-frame built on a 256 × 256 Gaussian white-noise image with standard deviation 1, kurtosis = 3.11 , mean = 0.0028 . (c) Histogram of the same wavelet subband in the ridgelet bi-frame built on a 256 × 256 Gaussian white-noise image with standard deviation 1, kurtosis = 4.72 , mean = 8.3 × 10 5 . (d) Histogram of the same subband in the wavelet domain built on a 256 × 256 Gaussian white-noise image with standard deviation 1, kurtosis = 2.91 , mean = 0.0043 .

Fig. 4
Fig. 4

Example for curvelet bi-frame decomposition on the pepper image. (a) Original image. (b) Low-frequency subband P 0 f corresponds to standard wavelet subband j = 0 , 1 , 2 , 3 . (c) Subband Δ 1 f corresponds to standard wavelet subband j = 4 , 5 . (d) Subband Δ 2 f corresponds to standard wavelet subband j = 6 , 7 . (e) Curvelet bi-frame coefficients, obtained by assigning a monoscale ridgelet bi-frame with scale parameter s = 1 to subband Δ 1 f . (f) Curvelet bi-frame coefficients, obtained by assigning a monoscale ridgelet bi-frame with scale parameter s = 2 to subband Δ 2 f .

Fig. 5
Fig. 5

Example basis functions of the curvelet bi-frame.

Fig. 6
Fig. 6

Restoration images for visual comparison by different transforms (using hard-threshold procedure). (a) Decimated biorthogonal wavelet transform ( 7 9 filter), PSNR = 31.62 dB . (b) Ridgelet bi-frame ( 7 9 filter), PSNR = 34.12 dB . (c) Undecimated biorthogonal wavelet transform ( 7 9 filter), PSNR = 34.28 dB . (d) Undecimated ridgelet bi-frame ( 7 9 filter), PSNR = 36.99 dB .

Fig. 7
Fig. 7

Restoration images for visual comparison by different transforms (using the Bayesian coring algorithm). (a) Decimated biorthogonal wavelet transform ( 7 9 filter), PSNR = 33.88 dB . (b) Ridgelet bi-frame ( 7 9 filter), PSNR = 35.13 dB .

Fig. 8
Fig. 8

Restoration images for visual comparison using different methods on the Lena image (with additive Gaussian white noise of standard deviation 20, PSNR = 22.08 dB ). (a) UDWT, PSNR = 30.81 dB . (b) Contourlet, PSNR = 28.57 dB . (c) Second-generation curvelet (FDCṮWARP), PSNR = 30.92 dB . (d) DCBF, PSNR = 31.42 dB . (e) SMG, PSNR = 32.66 dB . (f) UCBF, PSNR = 32.41 dB .

Fig. 9
Fig. 9

Crop of restoration images for visual comparison using different methods on the Lena image (with additive Gaussian white noise of standard deviation 20, PSNR = 22.08 dB ). (a) DWT, PSNR = 28.68 dB . (b) UDWT, PSNR = 30.81 dB . (c) Contourlet, PSNR = 28.57 dB . (d) Second-generation curvelet (FDCṮWARP), PSNR = 30.92 dB . (e) SMG, PSNR = 32.66 dB . (f) UCBF, PSNR = 32.41 dB .

Tables (6)

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Table 1 Comparison of Computational Cost a of Orthonormal Ridgelet and Ridgelet Bi-frame

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Table 2 Comparison of Performance Using Biorthogonal Wavelet and Ridgelet Bi-frame: the Spline Case

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Table 3 Comparison of Performance Using Biorthogonal Wavelet and Ridgelet Bi-frame: the Villasenor Case

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Table 4 Comparison of Performance for Image Denoising on Pepper Image in Terms of PSNR a

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Table 5 Comparison of Performance for Image Denoising on Lena Image in Terms of PSNR

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Table 6 Comparison of Performance for Image Denoising on House Image in Terms of PSNR

Equations (25)

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A F R 2 λ Λ [ F , w λ ] R 2 B F R 2 ,
B 1 F R 2 λ Λ [ F , w ̃ λ ] R 2 A 1 F R 2 ,
F = λ Λ [ F , w ̃ λ ] R w λ = λ Λ [ F , w λ ] R w ̃ λ ,
A f L 2 ( R 2 ) 2 λ Λ f , ρ λ L 2 ( R 2 ) 2 B f L 2 ( R 2 ) 2 ,
B 1 f L 2 ( R 2 ) 2 λ Λ f , ρ ̃ λ L 2 ( R 2 ) 2 A 1 f L 2 ( R 2 ) 2 ,
f = λ Λ f , ρ λ L 2 ( R 2 ) ρ ̃ λ = λ Λ f , ρ ̃ λ L 2 ( R 2 ) ρ λ ,
ρ λ ( x ) = 1 4 π 0 2 π ψ j , k + ( x 1 cos θ + x 2 sin θ ) ω i , l ( θ ) d θ ,
ρ ̃ λ ( x ) = 1 4 π 0 2 π ψ ̃ j , k + ( x 1 cos θ + x 2 sin θ ) ω ̃ i , l ( θ ) d θ ,
f = λ Λ [ ( Δ + I ) ( R a f ) , w ̃ λ ] R ρ λ = λ Λ [ ( Δ + I ) ( R a f ) , w λ ] R ρ ̃ λ ,
[ F , w λ ] R = 1 2 π F , w λ L 2 ( R × [ 0 , 2 π ) ) ,
[ F , w ̃ λ ] R = 1 2 π F , w ̃ λ L 2 ( R × [ 0 , 2 π ) ) .
P x ( x ; a , p ) = e x a p Z ( a , p ) ,
σ 2 = a 2 Γ ( 3 p ) Γ ( 1 p ) , κ = Γ ( 1 p ) Γ ( 5 p ) Γ 2 ( 3 p ) .
F ̂ ( F ̂ i j ) i , j = 1 M = R F ( f ) = W R a ( f ) .
( y i j ) i , j = 1 N = ( f i j ) i , j = 1 N + σ ( ϵ i j ) i , j = 1 N ,
y = f + σ ε ,
Y ̂ ( Y ̂ k l ) k , l = 1 M = R F ( f + σ ε ) = W R a ( f + σ ε ) = W ( F + σ Υ ) = W ( ( F k l ) k , l = 1 M + σ ( Υ k l ) k , l = 1 M ) ,
Y ̂ ( Y ̂ k l ) k , l = 1 M = F ̂ + σ Υ ̂ = W ( F + σ Υ ) = W ( ( F k l ) k , l = 1 M + σ ( Υ k l ) k , l = 1 M ) = W ( ( F k l δ k l ) k , l = 1 M + σ ( Υ k l δ k l ) k , l = 1 M ) .
F σ = ( F k l , σ ) k , l = 1 M = W ̃ ( F ̂ σ ) = W ̃ D σ W ( F + σ Υ ) = W ̃ D σ W ( ( F k l δ k , l ) k , l = 1 M + σ ( Υ k l δ k , l ) k , l = 1 M ) .
f σ = ( F i j , σ ) i , j = 1 N = R a T ( ( δ k l F k l , σ ) k , l = 1 M ) ,
L = U + V ,
U ̂ ( L ) = d U P U L ( U L ) U = d U P L U ( L U ) P U ( U ) U d U P L U ( L U ) P U ( U ) = d U P v ( L U ) P U ( U ) U d U P v ( L U ) P U ( U ) ,
f 2 2 = P 0 f 2 2 + s Δ s f 2 2 ,
f ( x , y ) = C J ( x , y ) + j = 1 J D j ( x , y ) ,
Q = [ 2 s k 1 , 2 s ( k 1 + 1 ) ] × [ 2 s k 2 , 2 s ( k 2 + 1 ) ] ,

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