Paper:

# Dyadic Curvelet Transform (DClet) for Image Noise Reduction

## Marjan Sedighi Anaraki^{*}, Fangyan Dong^{*},

Hajime Nobuhara^{**}, and Kaoru Hirota^{*}

^{*}Dept. of Computational Intelligence and Systems Science, Tokyo Institute of Technology, G3-49, 4259 Nagatsuta, Midori-ku, Yokohama 226-8502, Japan

^{**}Dept. of Intelligent Interaction Technologies, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba science city, Ibaraki 305-8573, Japan

Dyadic Curvelet transform (DClet) is proposed as a tool for image processing and computer vision. It is an extended curvelet transform that solves the problem of conventional curvelet, of decomposition into components at different scales. It provides simplicity, dyadic scales, and absence of redundancy for analysis and synthesis objects with discontinuities along curves, i.e., edges via directional basis functions. The performance of the proposed method is evaluated by removing Gaussian, Speckles, and Random noises from different noisy standard images. Average 26.71 dB Peak Signal to Noise Ratio (PSNR) compared to 25.87 dB via the wavelet transform is evidence that the DClet outperforms the wavelet transform for removing noise. The proposed method is robust, which makes it suitable for biomedical applications. It is a candidate for gray and color image enhancement and applicable for compression or efficient coding in which critical sampling might be relevant.

Hajime Nobuhara, and Kaoru Hirota, “Dyadic Curvelet Transform (DClet) for Image Noise Reduction,”

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.11, No.6, pp. 641-647, 2007.

- [1] M. N. Do and M. Vetterli, “The Contourlet Transform: an Efficient Directional Multiresolution Image Representation,” IEEE Trans. on Image Processing, 14-12, pp. 2091-2106, 2005.
- [2] D. D. -Y. Po and M. N. Do, “Directional Multiscale Modeling of Images using the Contourlet Transform,” IEEE Trans. on Image Processing, 15-6, pp. 1610-1620, 2006.
- [3] D. L. Donoho and M. R. Duncan, “Digital Curvelet Transform: Strategy, Implementation and Experiments,” Proc. SPIE, Vol.4056, pp. 12-29, 2000.
- [4] J. L. Starck, F. Murtagh, E. J. Candes, and D. L. Donoho, “Gray and Color Image Contrast Enhancement by the Curvelet Transform,” IEEE Trans. on Image Processing, 12-6, pp. 706-717, 2003.
- [5] F. J. Herrmann, “Curvelet Imaging and Processing: an Overview,” CSEG National Convention, 2004.
- [6] E. J. Candes, “What Is A Curvelet?,” Notice of the AMS, 50-11, pp. 1402-1403.
- [7] E. J. Candes and D. L. Donoho, “Curvelets Multiresolution Representation, and Scaling Laws,”

http:www.acm.caltech.edu/˜emmanuel/papers/SPIE_Curvelets.pdf - [8] E. Candes and D. Donoho, “New Tight Frams of Curvelets and Optimal Representations of Objects with Piecewise-C2 Singularities,” Commun. on Pure and Appl. Math., 57, pp. 219-266, 2004.
- [9] E. J. Candes and D. L. Donoho, “Ridgelets, A Key to Higher Dimensional Intermittency?,” Phil. Trans. Royal Society London, 1999.
- [10] E. J. Candes and D. L. Donoho, “Curvelet A Surprisingly Effective Nanadaptation for Objects with Edges,” Curve and Surface Filtering, Saint-Malo, 1999.
- [11] M. N. Do and M. Vetterli, “The Finite Ridgelet Transform for Image Representation,” IEEE Trans. on Image Processing, 12-1, pp. 16-28, 2003.
- [12] M. N. Do and M. Vetterli, “The Finite Ridgelet Transform for Image Representation,” IEEE Trans. on Image Processing, IP EDICS: 2-WAVP (Wavelets and Multiresolution Processing), 2001.
- [13] F. Matus and J. Flusser, “The Image Representations via a Finite Radon Transform,” IEEE Trans. on Pattern Analysis and Machine Intelligence, 15-10, pp. 996-1006, 1993.
- [14] E. D. Bolker, S. Helgason, R. L. Bryant, V. Guillemin, and R. O. Wells Jr. (Eds.), “The Finite Radon Transform,” Internal Geometry, Contemporary Mathematics, 63, pp. 27-50, 1987.
- [15] G. Beylkin, “Discrete Radon Transform,” IEEE Trans. on Acoustic and Speech Signal Processing, 35, pp. 162-172, 1987.
- [16] R. Rangarajan, R. Kataramanan, and S. Shah, “Image De-noising using Wavelet,” 2002.

http://www-personal.engin.umich.edu/˜sidshah/projects/denoise.pdf - [17] E. Bala and A. Ertuzun, “Application of Multiwavelet Techniques to Image Denoising,” IEEE ICIP 2002, III581-584.
- [18] J. L. Starck, E. J. Candes, and D. L. Donoho, “The Curvelet Transform for Image De-noising,” 2000.

http://www.acm.caltech.edu/˜emmanuel/papers/CurveDenoise.pdf - [19] J. L. Starck, E. J. Candes, and D. L. Donoho, “The Curvelet Transform for Image Denoising,” IEEE, Transaction on Image Processing, 11-6, pp. 670-684, 2002.
- [20] E. Candes, L. Demanet, D. Donoho, and L. Ying, “Fast Discrete Curvelet Transforms,” 2005.

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