Journal of Scientific & Industrial Research Vol. 3469, January 00, pp. 34-38 J SCI IN RES VOL 69 JANUARY 00 Image denoising using curvelet transform: an approach for edge preservation Anil A Patil * and Jyoti Singhai epartment of Electronics and Telecommunication, COE, Malegaon, Pune, India. epartment of Electronics and Communication Engineering, MANIT, Bhopal, India Received 03 March 009; revised 4 November 009; accepted 5 November 009 This paper suggests a soft thresholding multiresolution technique based on local variance estimation for image denoising. This adaptive thresholding with local variance estimation effectively reduce image noise and preserves edges. In proposed algorithm, fast discrete curvelet transform ( FCT) out performed wavelet based image denoising. PSNR using FCT is approximately doubled and it also preserves features at boundary of an image. Keywords: Curvelet transforms, Image denoising, Local variance, Sparse representation Introduction Images acquired through sensors [charge-coupled device (CC)] cameras may be contaminated by noise sources. Image processing technique also corrupts image with noise, leading to significant reduction in quality. Traditionally, linear filters (mean, median and wiener filters) are used for removing noise from images, but it blurs data. Edge preserving smoothing algorithm [symmetric nearest neighbor (SNN) filter, maximum homogeneity neighbor (MHN) filter and morphology-based filter] smooth noise in homogeneous regions and sharpens boundaries between regions. In non-linear techniques, waveletbased image denoising methods have attracted due to multi-resolution nature and ability to produce high level of noise reduction. Wavelet fails to give sparse representation along c curve 3. Wavelet effectively represents discontinuities for one dimensional signal. However, tensor-product construction is not flexible enough to reproduce this behavior in two-dimensions 4. Curvelet transform (CT) overcomes limitations of wavelet transform (WT). CT is a multiscale pyramid with many direction and position at each length scale and needle shaped element at fine scale. *Author for correspondence E-mail: panil@gmail.com In this paper, fast discrete curvelet transform ( FCT) is implemented with soft thresholding for image denoising. Proposed algorithm is evaluated by mean squared error (MSE) and peak signal-to-noise ratio (PSNR) as a measure of quality of denoised image. Experimental Curvelet Transform (CT) igital CT can be implemented in two ways (FCT via USFFT and FCT via wrapping), which differ by spatial grid used to translate curvelets at each scale and angle. In this paper, FCT via wrapping is implemented as it is simpler, faster and less redundant. This transform is constructed using parabolic scaling, anisotropic law, tight framing and wrapping. igital CT is expressed as c (, l, k ) f [ t, t] ϕ, l, k [ t, t ] = () 0> t, t < n where f [ t, t ] is an input of Cartesian arrays with t 0 and t < n. c (, l, k) are curvelet coefficients and ϕ, l, k are Riesz representers.
PATIL & JYOTI SINGHAI: IMAGE ENOISING USING CURVELET TRANSFORM 35 Fig. Thresholding: a) Hard; and b) Soft Parameter Estimation for Threshold Value FCT is applied for removing noise from image data. Let signal be{ f i, ; i, = L N}, where N is some integer power of. It is corrupted by additive noise and signal g, can be expressed as where { } i = + ; i, = L N () g i, fi, ε i, ε i, are independent and identically distributed ( iid ) as normal N ( 0, σ ) and independent of{ f i, }. Obective of proposed algorithm is to remove noise from{ g i, } to obtain an estimate{ f ˆ i, } of{ f } i,, which minimizes MSE and maximizes PSNR. Image denoising can be done either by hard or soft thresholding techniques 5. Hard thresholding operator with threshold T is defined as (Fig. a): ( g T i ) g i,, = for all g i, > T (3), = 0, otherwise Soft thresholding operator with threshold T is defined as (Fig. b): ( g i,, T ) = ( g ) max( g,0) sgn i i,, T (4) Soft thresholding is more efficient as it shrinks towards zero. It is used for entire algorithm because it achieves near minmax rate over a large number and yield visually more pleasing images 6. BayesShrink approach is used for calculating threshold for soft-thresholding. Adaptive threshold value is determined for different subband, depending upon noise variance and standard deviation of signal. For proposed algorithm, noise variance σ is estimated by robust median estimator 7-9. Robust median estimator for g i, in WT is expressed as ( g ) Median i, σ ˆ = g i, subband 0.6745 (5) WT gives only three orientations, where as CT has more orientations with tight frame. CT has scale, translation and orientations so curvelet coefficients are expressed in cells. In case of CT, robust median estimator is expressed as Median σˆ = 0.6745 ( g { }{ l} ) g{ }{ l} subband (6) Signal standard deviation is estimated by ( σˆ σˆ,0) ˆ x = max w σ (7)
36 J SCI IN RES VOL 69 JANUARY 00 Fig. Zoomed restored image with: a) Wavelet transform; and b) Curvelet transform where σˆ w is estimate of variance of observations. BayesShrink threshold T is given as ratio of noise variance and signal standard deviation as MSE = MN M i= N = ( x ( i, ) p( i, ) ) [ ] (9) T ˆ σ = (8) σˆ x BayesShrink soft thresholding is applied for both transforms for image denoising. Curvelet, which gives sparse representation 0, tracks curve better and hence denoise image better than WT. Experimental Setup and Results BayesShrink soft-thresholding algorithm can be implemented by following procedure: i) Perform FCT for noisy image and obtain curvelet coefficients; ii) Estimate subband dependent BayesShrink threshold valuet ; iii) Threshold curvelet coefficients; and iv) Perform inverse FCT and reconstruct denoised image. Three different grayscale images [Lena and Barbara (size, 5 x 5) and Cameraman (size, 56 x 56)] are used as test images with Gaussian noise level of σ =0, to demonstrate algorithm. Wavelets are implemented with blocks (3 x 3) and without blocks of noisy image. Variance measure is locally computed at each block and subband. Quality of denoising can be evaluated using MSE and PSNR. PSNR = 0 log ( 55) MSE db where x ( i, ) is original image and ( i ) (0) p, is denoised image. Experimental results show that PSNR and MSE are improved with CT as compared to WT with and without blocks. b4 wavelet gives better results than Haar Wavelet due to four point filter. Also, CT denoised images do not contain artifacts along edges. Quality of local reconstruction on zoomed restored images obtained via CT is especially promising (Fig. ). Line patterns on scarf of Barbara are prominent and with sharp edges in curvelet restored image as compared to that of wavelet restored image. Experimental results (Table ) shows PSNR value for denoised images using Haar, db4 WT with blocks and without blocks and CT for three standard test images (Fig. 3a). Fig. 3b show same images corrupted with Gaussian noise withσ = 0. Fig. 3c shows denoised images with db4 WT as it is showing best performance for wavelet. Fig. 3d shows denoised images with CT. Experimental work has
PATIL & JYOTI SINGHAI: IMAGE ENOISING USING CURVELET TRANSFORM 37 (a) Original Images (b) Noisy images with Gaussian noise σ =0 (c) enoised images with aubechies-4 wavelet (d) enoised images with Curvelet transform Fig. 3 Test results for three images (Barbara, Cameraman and Lena) Table PSNR and MSE of noisy and denoised image (σ =0) PSNR images MSE images Lena Barbara Cameraman Lena Barbara Cameraman Noisy 0.08 0. 0.3 638. 63.364 603.803 Haar 3.696 3.35 3.803 77.63 300.54 70.903 db4 4.806 3.594 4.678 5.03 84.39.463 Haar(Block) 7.0 4.4 5.605 9.3 44.89 78.903 db4(block) 7.904 5.95 5.703 05.37 96.587 74.98 CT 40.088 40.33 40.409 6.373 6.306 5.98
38 J SCI IN RES VOL 69 JANUARY 00 demonstrated that curvelet denoising is more effective than wavelet as evaluated in terms of MSE (Table ). Proposed method performs well both visually and in terms of MSE and PSNR for images contaminated by Gaussian noise. Conclusions A subband dependent threshold is implemented with CT for removing noise. It is based on generalized Gaussian distribution modeling of subband coefficients. Image denoising algorithm uses BayesShrink soft thresholding to improve smoothness and for better edge preservation. Curvelet outperforms wavelet results. Its images get denoised well and edges are preserved. CT reconstruction does not contain quality of disturbing artifacts along edges that is observed in wavelet reconstruction. References Rao R M & Bopardikar A S, Wavelet Transform: Introduction to Theory and Application (Addison Wesley Longman Inc.)998. Niu Y F & Shen L C, Wavelet denoising using the Pareto optimal threshold, Int J Comput Sci & Network Security, 7 (007) 30-34. 3 Candes E & onoho, Curvelets, in A Surprisingly Effective Non-adaptive Representation for Obects with Edges, Curve and Surface Fitting:Saint-Malo999, edited by A Cohen et al (Vanderbilt University Press, Nashville, TN) (999). 4 Candes E J, onoho L & Ying L, Fast discrete curvelet transform, J Multiscale Modeling & Simulation,5 (006) 86-899. 5 Starck J L, Candes E J & onoho L, The curvelet transform for image denoising, IEEE Trans Image Proc, (00)670-684. 6 onoho L, enoising by soft-thresholding, IEEE Trans Inform Theory, 4 (998) 63-67. 7 Chang S G, Yu B & Martin V, Adaptive wavelet thresholding for image denoising & compression, IEEE Trans Image Proc, 9 (000)53-546. 8 Chen Y & Han C, Adaptive wavelet thresholding for image denoising, Electron Lett, 4 (005) 586-587. 9 Achim A, Tsakalides P & Bererianos A, SAR image denoising via. Bayesian wavelet shrinkage based on heavy-tailed modeling, IEEE Trans Geosci, Remote Sens, (003) 773-784. 0 Candes E & onoho, New tight frames for curvelets and optimal representations of obects with C singularities, Cmmun Pure Appl Math, 57 (004) 9-66.