a)

b)

Fig. 4. Result of non-linear approximation. (a) Original image, (b) Results: full line shows the dependence of PSNR of the re- constructed image on the number of coefficients kept for the new method. The dotted line shows the same dependence for the stan- dard method.

a)

b)

7.

ACKNOWLEDGMENTS

The third author would like to thank Prof. Minh Do for fruitful interactions on directional image analysis.

8. REFERENCES

[1] R. H. Bamberger and M. J. T. Smith. A filter bank for the di- rectional decomposition of images: Theory and design. IEEE Signal Proc., pages 882–893, April 1992.

c)

d)

[2] E. D. Bolker. The finite Radon transform. In S. Helgason R. L. Bryant, V. Guillemin and R. O. Wells Jr., editors, Integral Geometry (Contemporary Mathematics, Vol. 63), pages 27– 50. 1987.

Fig. 5. The image ’Cameraman’. (a) Original image, (b) noised

version (

), (c) denoised by the standard method

(

= 20.54dB), (d) denoised by the new method (

=

25.04dB).

[3] J. E. Bresenham. Algorithm for computer control of a digital plotter. IBM Systems Journal, 4(1):25–30, 1965.

[4] E. J. Cand`es and D. L. Donoho. Curvelets, multiresolution representation, and scaling laws. In A. Aldroubi, A. F. Laine, and M. A. Unser, editors, SPIE Wavelet Applications in Signal and Image Processing VIII, volume 4119, 2000.

[5] M. N. Do. Directional Multiresolution Image Representations. PhD thesis, Audio-Visual Laboratory, EPFL, October 2001.

[6] D. L. Donoho and I. M. Johnstone. Ideal spatial adaption via wavelet shrinkage. Biometrika, pages 425–455, Decem- ber 1994.

[7] V. K. Goyal, J. Kovacˇevic´, and J. A. Kelner. Quantized frame expansions with erasures. Applied & Computational Har- monic Analysis, 10(3):203–233, May 2001.

[8] M. Vetterli and J. Kovacˇevic´. Wavelets and Subband Coding. Prentice Hall PTR, New Jersey, 1995.

a) b)

[9] R. A. Zuidwijk. Directional and time-scale wavelet analy- sis. SIAM Journal on Mathematical Analysis, 31(2):416–430, 2000.

# Fig. 6. Result of denoising of the image ’Cameraman’. (a) The

dependence of

of the denoised image on the number of

the directions involved in the analysis, (b) The full line shows the dependence of of the denoised image on the input for the new method. The dotted line shows the same dependence

for the standard method.