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# Image denoising using curvelet transform

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### Image denoising using curvelet transform

1. 1. Seminar on “Image Denoising Method based on Curvelet Transform” Master of Engineering (Electronics and Communication ) Year 2011-12. Rajput Sandeep Kumar Jawaharlal (100370704036) Prepared By: Guided By: Rajput Sandeep J Prof. A.R. Yadav ME (EC-213) Professor , EC Dept. PIET, Limda. PIET, Limda.
2. 2. Introduction  Image acquired through sensors charge coupled device (CCD) cameras may be influenced by noise sources.  Image processing technique also corrupts image with noise, leading to significant reduction in quality. Traditionally,  Linear filters  Edge preserving smoothing algorithm New Methods,  Non-linear techniques : Wavelet Transform : Curvelet Transform
3. 3. Original Image Sub-band decomposition Smooth partitioning and Renormalization Each subzone of each block to carry out analysis of the Ridgelet Block image n x n Ridgelet TransformRadon Transform WT 1D Angle Inverse FFT 1 D FFT 2D Frequency WT 2D Process of Curvelet Transform Figure: 1 Curvelet transform flow block diagram
4. 4. Sub-band Decomposition fP0 f1∆ f2∆ f ( ) ,,, 210 fffPf ∆∆
5. 5. Smooth Partitioning
6. 6. Smooth Partitioning  The windowing function w is a nonnegative smooth function.  Partition of the intensity: The intensity of certain pixel (x1,x2) is divided between all sampling windows of the grid. ( ) 1, 21, 2211 2 ≡−−∑kk kxkxw
7. 7.  Ridgelet are an orthonormal set {ρλ} for L2 (R2 ). Ridgelet Analysis 2-s 2-2s 1 2-s 2s radius 2s 2s divisions Ridge in Square It’s Fourier TransformRidge in Square Ridgelet Tiling Fourier Transform within Tiling
8. 8. Ridgelet Analysis  The ridgelet element in the frequency domain: where, ωi,l are periodic wavelets for [-π, π ). i is the angular scale. ψj,k are wavelets for R. j is the ridgelet scale and k is the ridgelet location. ( ) ( ) ( ) ( ) ( )( )πθωψθωψρ likjlikjλ +⋅−+⋅= − ,,,,2 1 ξˆξˆξξˆ 2 1
9. 9. Curvelet Transform The four stages of the Curvelet Transform were:  Sub-band decomposition  Smooth partitioning  Renormalization  Ridgelet analysis ( ) ,,, 210 fffPf ∆∆ fwh sQQ ∆⋅= QQQ hTg 1− = ( ) λQQ,λ ρgα ,=
10. 10. Image Reconstruction The Inverse of the Curvelet Transform:  Ridgelet Synthesis  Renormalization  Smooth Integration  Sub-band Recomposition ( ) λ λ Q,λQ ραg ⋅= ∑ QQQ gTh = ∑∈ ⋅=∆ sQ QQs hwf Q ( ) ( )∑ ∆∆+= s ss ffPPf 00
11. 11. Thresholding methods Window Shrink Method  Set di, j is the parameter which is from curvelet transformed noise image; choose a di, j centered window of n×n as the processing subject. 3X 3 Window Shrink The curvelet coefficients to be thresholded
12. 12. Set Symbolic function:  σ is the variance of Gaussian white noise in the image , then shrinking processing parameter is Then the thresholded parameter can be calculated as: Thresholding methods The sum of all the parameter’s square in the n×n window is calculated.
13. 13. Bayes Shrink method Thresholding methods  In this method σ2 D is the variance of an image containing noise, σ2 is the variance of noise, and σ2 X is the original image’s variance. Now, noise variance is: The variance of original image is calculated by,  Setting Threshold is σ2 / σ2 X then begin the processing of removing noise.
14. 14. Combination of Window shrink and Bayes shrink  The variance σ2 X is estimated of the original picture using Bayes shrink theory, then η is calculated using σ2 X instead of the noise variance σ 2 ,such as  At last shrink factors αi, j are known and the noise coefficient is filtered out by taking advantage of αi, j . Thresholding methods x
15. 15. Thresholding methods
16. 16. Image denoising Algorithm Original image σ = 20 noise image 2-D Wavelet transform Traditional Curvelet transform
17. 17. Image denoising Algorithm Quad tree Decomposition algorithm Now, The Q(x,y) that define the matrix of mxm image and S(vi) denote the element of the Q(x,y) where vi denote the number of decomposition required for that element.
18. 18. Image denoising Algorithm Algorithm : Denote result image of improved algorithm as R, this pixel fusion based algorithm is described as follows.  Applying wavelet transform to obtain result image W.  Applying curvelet transform to obtain result image C.  Get quad tree matrix Q with applying quad tree decomposition to C.  R(x, y) is calculated as R(x, y) = cW(x, y) + dC(x, y) Where,
19. 19. Image denoising Algorithm Result of algorithm Original image σ = 20 noise image Improved Curvelet transform
20. 20. Image denoising Algorithm
21. 21. Image denoising Algorithm
22. 22. Conclusion  To overcome the disadvantages of the wavelet transform along the curves in the images the curvelet transform is used and it gives high PSNR.  A new method of combination of the Window Shrink and Bayes Shrink based on Curvelet transform is used to remove noise from image. It has better PSNR. So the image we get by this method is better and that of the traditional wavelet methods.
23. 23. References i. Introduction to Wavelet: Bhushan D Patil PhD Research Scholar Department of Electrical Engineering Indian Institute of Technology, Bombay. ii. Pixel Fusion Based Curvelets and Wavelets Denoise Algorithm, Liyong Ma, Member, IAENG, Jiachen Ma and Yi Shen Advance online publication: 16 May 2007 iii. The Curvelet Transform - Jean-Luc Starck, Emmanuel J. Candès, and David L. Donoho IEEE transactions on image processing, vol. 11, no. 6, june 2002. iv. Image denoising using wavelet transform: an approach for edge Preservation Received 03 March 2009; revised 24 November 2009; accepted 25 November 2009 v. Image Denoising Method Based on Curvelet Transform -University of Science and Technology, IEEE transactions on image processing, vol. 11, no. 6, june 2008. vi. New Method Based on Curvelet Transform for Image Denoising Donglei Li, Zhemin Duan, Meng Jia vii. Department of Electronics and Information Northwestern Polytechnical University, China, 2010 International Conference on Measuring Technology and Mechatronics Automation viii. Improved Image Denoising Method based on Curvelet Transform Proceedings of the 2010 IEEE International Conference on Information and Automation June 20 - 23, Harbin, China ix. Image Denoising Based on Curvelet Transform and Continuous Threshold YUAN Ruihong TANG Liwei WANG Ping YAO Jiajun Department of Artillery Engineering Ordnance Engineering College Shijiazhuang ,China, 2010 First International Conference on Pervasive Computing, Signal Processing and Applications.