438 IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 12, DECEMBER 2002

Bivariate Shrinkage With Local Variance EstimationLevent Sendur and Ivan W. Selesnick, Member, IEEE

Abstract—The performance of image-denoising algorithmsusing wavelet transforms can be improved significantly by takinginto account the statistical dependencies among wavelet coeffi-cients as demonstrated by several algorithms presented in theliterature. In two earlier papers by the authors, a simple bivariateshrinkage rule is described using a coefficient and its parent.The performance can also be improved using simple models byestimating model parameters in a local neighborhood. This letterpresents a locally adaptive denoising algorithm using the bivariateshrinkage function. The algorithm is illustrated using both theorthogonal and dual tree complex wavelet transforms. Somecomparisons with the best available results will be given in orderto illustrate the effectiveness of the proposed algorithm.

Index Terms—Bivariate shrinkage, image denoising, statisticalmodeling, wavelet transforms.

I. INTRODUCTION

SOME RECENT research has addressed the development ofstatistical models of wavelet coefficients of natural images

and application of these models to image denoising [5], [11],[14], [15]. Recently, highly effective yet simple schemes mostlybased on soft thresholding have been developed [1], [2], [10].In [10], the wavelet coefficients are modeled with a Gaussianapriori density, and locally adaptive estimation is done for coef-ficient variances. Also, prior knowledge is taken into account toestimate coefficient variances more accurately. In [1], the inter-scale dependencies are used to improve the performance. In [2],the simple soft-thresholding idea is used for each of the waveletsubbands, and the threshold value is estimated to minimize themean-square error.

The models that exploit the dependency between coefficientsgive better results compared to the ones using an independenceassumption [5], [11], [14], [15]. However, some of these modelsare complicated and result in high computational cost. In [12], abivariate probability density function (pdf) is proposed to modelthe statistical dependence between a coefficient and its parent,and the corresponding bivariate shrinkage function is obtained.This new rule maintains the simplicity, efficiency, and intuitionof soft thresholding. An explicit multivariate shrinkage functionfor wavelet denoising is also presented in [16].

In this letter, the local adaptive estimation of necessary pa-rameters for the bivariate shrinkage function will be described.Also, the performance of this system will be demonstrated on

Manuscript received May 3, 2002; revised August 6, 2002. This researchwas supported by the National Science Foundation under CAREER grant CCR-9875452. The associate editor coordinating the review of this manuscript andapproving it for publication was Prof. Yu-Hen Hu.

The authors are with the Electrical and Computer Engineering, PolytechnicUniversity, Brooklyn, NY 11201 USA (e-mail: levent@taco.poly.edu; se-lesi@taco.poly.edu).

Digital Object Identifier 10.1109/LSP.2002.806054

both the orthogonal wavelet transform and the dual-tree com-plex wavelet transform (CWT) [7], [8], and some comparisonswith the best available wavelet-based image-denoising resultswill be given in order to illustrate the effectiveness of thesystem.

II. L OCAL ADAPTIVE ALGORITHM

In this letter, the denoising of an image corrupted by additiveindependent white Gaussian noise with variancewill be con-sidered. Let represent the parent of ( is the waveletcoefficient at the same position as theth wavelet coefficient

, but at the next coarser scale.) We formulate the problem inwavelet domain as and totake into account the statistical dependencies between a coeffi-cient and its parent. and are noisy observations ofand ; and , and are noise samples. We can write

no. of wavelet coefs (1)

where , , and. From this point, the coefficient index

will be omitted from the equations in order to improve thereadability of the equations.

The standard MAP estimator for given the corrupted ob-servation is

(2)

After some manipulations, this equation can be written as

(3)

In [12], we proposed a non-Gaussian bivariate pdf for thecoefficient and its parent as

(4)

The marginal variance is also dependent on the coefficientindex . Using (4) with (3), the MAP estimator of is derivedto be

(5)

1070-9908/02$17.00 © 2002 IEEE

SENDUR AND SELESNICK: BIVARIATE SHRINKAGE WITH LOCAL VARIANCE ESTIMATION 439

Fig. 1. (a) Original image. (b) Noisy image with PSNR= 20.18 dB (� = 25). (c) Denoised image using critically sampled transform PSNR= 27.16 dB.(d) Denoised image using dual-tree CWT PSNR= 28.61 dB.

which can be interpreted as a bivariate shrinkage function. Hereis defined as

ifotherwise.

(6)

This estimator requires the prior knowledge of the noisevariance and the marginal variance for each waveletcoefficient. In our algorithm, the marginal variance for theth

coefficient will be estimated using neighboring coefficients inthe region . Here is defined as all coefficients withina square-shaped window that is centered at theth coefficientas illustrated in Fig. 2.

To estimate the noise variance from the noisy wavelet co-efficients, a robust median estimator is used from the finest scalewavelet coefficients [6]

subband (7)

440 IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 12, DECEMBER 2002

Fig. 2. Illustration of neighborhoodN(k).

Let us assume we are trying to estimate the marginal variancefor the th wavelet coefficient. From our observation model,

one gets where is the marginal variance ofnoisy observations and . Since and are modeled aszero mean, can be found empirically by

(8)

where is the size of the neighborhood . Then, can beestimated as

(9)

The algorithm is summarized as follows.

1) Calculate the noise variance using (7).2) For each each wavelet coefficient ( number of

wavelet coefficients):

a) Calculate using (8).b) Calculate using (9).c) Estimate each coefficient usingand in (5).

III. RESULTS

We compared the proposed algorithm using the orthogonalwavelet transform to other effective systems in the literature,namely BayesShrink [2], AdaptShrink [3], locally adaptivewindow-based denoising using MAP (LAWMAP) estimator[10], and the hidden Markov tree (HMT) model [5]. We alsomade a comparison with our subband adaptive algorithmdescribed in [13]. (As seen from the results in Table I, thelocally adaptive estimation may improve the performance overthe subband adaptive estimation by 1 dB for some images.i.e., Barbara image.) The Daubechies length-eight filter and a7 7 window size [ ] are used. (We have also investigateddifferent window sizes. A 5 5 window size can also be agood choice. However, using a 33 window size resulted in aslight performance loss. In this letter, we have not considered

different shapes for .) The peak signal-to-noise ratio(PSNR) values of these systems are tabulated in Table I. Theperformance of this system is tested using the PSNR measure.Let denote the original and the denoised image. The rmserror is given by

(10)

where is the number of pixels. The PSNR in decibels is givenby

PSNR (11)

The PSNR values for AdaptShrink are taken from the paper [3].The Lena, Boat, and Barbara images are used for this purposewith different noise levels .

In our examples, in addition to the orthogonal wavelet trans-form, the dual-tree CWT will be used in order to take advan-tage of this transform (near shift-invariance and directional se-lectivity). The new bivariate shrinkage function will be appliedto the magnitude of the complex coefficients since the real andimaginary parts are not shift invariant individually but the mag-nitudes are. (We have also applied the same algorithm to thereal and imaginary parts separately but we have observed sig-nificant performance loss.) We assume the magnitudes of thecoefficients are corrupted by additive Gaussian noise althoughthis is an approximation. The performance of our algorithm sug-gests it is an acceptable assumption. The PSNR values, when theCWT is used, are listed in the last column of Table I.

We also compared the proposed algorithm with otherpublished results [3], [4], [9], [11] that use various redundantwavelet transforms, i.e., the undecimated wavelet transformin [3] and [9], the steerable pyramid in [11], and the dual-treeCWT in [4]. The PSNR values are listed in Table II (the valuesare taken from the corresponding papers). In [11], it is notedthat the Matlab implementation for that algorithm takes 12.8min for a 512 512 image on 900-MHz Pentium III, althoughthe Matlab program for the proposed algorithm takes 25 s for a512 512 image on 450-MHz Pentium II.

One example using a 512512 Barbara image is givenin Fig. 1. The original and the noisy images are illustratedin Fig. 1(a) and (b). The denoised images obtained usingthe data-driven denoising algorithm described above withthe orthogonal wavelet transform and the dual-tree CWT areillustrated in Fig. 1(c) and (d), respectively, and have PSNRvalues of 27.23 and 28.61 dB, respectively.

In [17] and [18], very high quality image denoising algo-rithms are presented using newly developed multiscale repre-sentation systems, namely the ridgelet and curvelet transform.In their experiments, the Lena image is used with the Gaussiannoise standard deviation 20. They reported the denoised PSNRvalue is 31.95 in [17] and and 32.72 in [18]. The result here liesbetween these two values.

SENDUR AND SELESNICK: BIVARIATE SHRINKAGE WITH LOCAL VARIANCE ESTIMATION 441

TABLE IPSNR VALUES OFDENOISEDIMAGES FORDIFFERENTTEST IMAGES AND NOISELEVELS (� ) OF NOISY, BAYESSHRINK [2], ADAPTSHRINK [3], HMT SYSTEM

[5], LAWMAP [10], PROPOSEDALGORITHM IN [13], PROPOSEDSYSTEM, AND PROPOSEDSYSTEM FORDUAL-TREECOMPLEX DISCRETEWAVELET TRANSFORM

TABLE IIPSNR VALUES OF DENOISEDIMAGES FORDIFFERENTTESTIMAGES AND NOISELEVELS (� ) OF NOISY, THE SYSTEM IN [9], SI-ADAPTSHR [3], CHMT [4], THE

SYSTEM IN [11], AND PROPOSEDSYSTEM FORDUAL-TREE COMPLEX DISCRETEWAVELET TRANSFORM

IV. CONCLUSION

This letter presents an effective and low-complexity image-denoising algorithm using the joint statistics of the waveletcoefficients of natural images. We presented our result for bothorthogonal and dual-tree CWTs and compared it with the otherpublished results in order to illustrate the effectiveness of the pro-posed algorithm. The comparison suggests the new denoisingresults are competitive with the best wavelet-based resultsreported in the literature.

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