IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008 139

A New Interscale and Intrascale Orthonormal WaveletThresholding for SURE-Based Image Denoising

Fengxia Yan, Lizhi Cheng, and Silong Peng

AbstractThe interscale Steins unbiased risk estimator(SURE)-based approach introduced by Luisier is a recent state ofthe art in orthonormal wavelet denoising, but it is not very effectivefor those images that have substantial high-frequency contents.To solve this problem, we introduce an effective integration ofthe intrascale correlations within the interscale SURE-based ap-proach. We show that the consideration of both the intrascale andinterscale dependencies of wavelet coefficients brings more de-noising gains than those obtained with the interscale SURE-basedapproach, especially for denoising of images that have substantialtextures such as the Barbara image.

Index TermsImage denoising, intrascale and interscale depen-dencies, Steins unbiased risk estimator (SURE) minimization.

I. INTRODUCTION

AN IMAGE is often corrupted by noise in its acquisitionor transmission and noise elimination is still one of themost fundamental, widely studied, and largely unsolved prob-lems in computer vision and image processing. In image de-noising, a compromise has to be found between noise suppres-sion and the preservation of the important image features. Toachieve a good performance in this respect, a denoising algo-rithm has to adapt to image discontinuities. The wavelet repre-sentation naturally facilitates the construction of such spatiallyadaptive algorithms. In particular, Donoho and Johnstone [1] de-veloped a very simple denoising procedure consisting in thresh-olding the noisy wavelet coefficients (shrinkage). Indeed, thewavelet transform is good at energy compaction, small coeffi-cients are more likely due to noise, and large coefficients dueto important signal features (such as edges). Wavelet shrinkageis, thus, effective for signals with sparse wavelet representa-tions. In this context, the key issue is the threshold value se-lection. From an asymptotically minimax analysis, Donoho andJohostone have proposed to apply the universal threshold inthe VisuShrink method. Subsequently, they suggested to choose

Manuscript received July 8, 2007; revised October 31, 2007. This work wassupported in part by the National Natural Science Foundation of China (NSFC)under Grant No. 60573027 and Grant No. 10601068. The associate editor co-ordinating the review of this manuscript and approving it for publication wasProf. Philippe Salembier.

F.-X. Yan and L.-Z. Cheng are with the Department of Mathe-matics and System, School of Sciences, National University of De-fense Technology, Changsha 410073, China (e-mail: xialang3@163.com;clzcheng@vip.sina.com).

S.-L. Peng is with the Institute of Automation, Chinese Academy of Sciences,Beijing 100080, China (e-mail: silong.peng@ia.ac.cn).

Digital Object Identifier 10.1109/LSP.2007.914790

the optimal threshold by minimizing Steins unbiased risk esti-mator (SURE) [2] and called the corresponding denosing ap-proach SureShrink [3]. The SURE principle was also used byPesquet et al. to develop sophisticated multivariate estimatesfor multicomponent image denoising [4][6]. Recently, by ex-ploiting the Steins principle, Luisier et al. introduced a newinterscale SURE-based approach to orthonormal wavelet imagedenoising that does not need any prior statistical modelizationof the wavelet coefficients [7]. Instead of postulating a statisticalmodel for the wavelet coefficients, they directly parameterizedthe denoising process as a sum of elementary nonlinear pro-cesses with unknown weights. The key point of their techniquesis to take advantage of Steins MSE estimate, which depends onthe noisy image alone, not on the clean one. Using this approach,they designed an image denoising algorithm that takes into ac-count interscale dependencies, but discards intrascale correla-tions. In order to compensate for feature misalignment, they de-veloped a rigorous procedure based on the relative group delaybetween the scaling and wavelet filters-group delay compen-sation. Their denoising results demonstrated that, for most ofthe images, the interscale SURE-based approach is competitivewith the best techniques available that consider nonredundantorthonormal transforms, but with the noteworthy exception forimages that have substantial high-frequency contents such as theBarbara image. The main reason for that is that the interscalecorrelations may be too weak for such images, which indicatesthat an efficient denoising process may require intrascale infor-mation as well.

In this letter, we will create an orthonormal wavelet thresh-olding which considering both the intrascale and interscaledependencies of the wavelet coefficients. Specifically, we definea local spatial predictor to capture the intrascale correlationsand propose the intrascale and interscale wavelet thresholdingwithin the SURE-based approach. Experiments results showthat our algorithm is superior to the interscale SURE-baseddenoising approach, and especially efficient for those imagesthat have substantial textures such as the Barbara image.

II. SURE-BASED INTRASCALE AND INTERSCALETHRESHOLDING

In this letter, the denoising of an image corrupted by additivewhite Gaussian noise with variance will be considered, i.e.,

(1)

where are independent and identically distributed(i.i.d.) as normal and independent of . The

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140 IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008

goal is to obtain an estimate of which mini-mizes the mean-squared error (MSE)

(2)

An orthogonal wavelet transformation of the noisy inputyields an equivalent additive white noise model in the waveletdomain. In each wavelet subband at a given scale (there arethree subbands in each scale) and orientation, we have

(3)

where are noisy wavelet coefficients, are noise-freewavelet coefficients, are i.i.d. normal random variables,which are statistically independent from , and is thenumber of coefficients in a subband. Here, we will only con-sider orthonormal wavelet transform, so the MSE in the spacedomain defined in (2) is a weighted sum of the MSE of eachindividual wavelet subband.

Different from Luisiers bivariate denoising function whichonly considers interscale dependencies, in this letter, we chooseto estimate each by the following pointwise denoising func-tion which takes into account both interscale and intrascale de-pendencies:

(4)

where is a (weakly) differentiable function from tois the interscale predictor of , and is the intrascale predictorof .

In order to compensate for feature misalignment, we also usethe group delay compensation procedure [7] which is based onthe relative group delay between the scaling and wavelet filtersto obtain the interscale predictor . Then, we introduce a localspatial activity indicator [9] (LSAI) as an intrascale predictoras follows:

(5)

Here, represents the locally averaged magnitude of the co-efficients in a relatively small square window of a fixed size

, and to be derived with respect to its center component (butis not included). Our goal is to find a function that mini-

mizes the MSE of each wavelet subband

(6)

In practice, we only have access to the noisy signaland not to the original signal . In (6),

we, thus, need to remove the explicit dependence on . Sincehas no influence in the minimization process,

we do not need to estimate it. The remaining problematic termis only . However, for Gaussian ad-ditive noise, Steins formula applied to our multivariate modelallows to express in terms of ob-served data only. Recall that the randomness of

only results from the Gaussian white noise , becauseno statistical model is assumed on the noise-free data ; andhere, we only consider an orthonormal transformwhich trans-forms Gaussian white noise into Gaussian white noise, that is,any wavelet coefficient other than is independent of . As de-fined in (5), the intrascale predictor is a weighted sum of themagnitude of the coefficients in the neighborhood of . Thus,the interscale predictor (which is built out of the lowpass sub-band at the same scale as ) and the intrascale predictor areboth independent of . Consequently, according to Steins the-orem [2], we have

(7)

is an unbiased estimator of the MSE defined in (6), i.e.,

(8)

where is expectation operator. Note that variables andare both independent of ; it is easy to proof (8), and the con-crete proof procedure is similar with [7, proof of Theorem 1].

To suitably integrate the parent and the LSAI into thepointwise denoising function, we use them as discriminators be-tween high SNR wavelet coefficients and low SNR wavelet co-efficients and choose to build a linearly parameterized denoisingfunction of the following form:

(9)

In (9), the function depends linearly on parameters, ,and degree of freedomwhich we will determine exactlyby minimizing . For the sake of convenience, we put the pa-rameters and together into a vector

, and then for; at the same time, we define the weight variables

YAN et al.: NEW INTERSCALE AND INTRASCALE ORTHONORMAL WAVELET THRESHOLDING 141

TABLE ICOMPARISON OF SEVERAL DENOISING ALGORITHMS

as for and forand for .

If we introduce (9) into the estimate of the MSE given in (8),according to (7), perform differentiations over the parameters

, we have for all

(10)

It is equivalent to

(11)

These equations can be summarized in matrix form as, where and are

vectors of size 6 1, and is a matrix ofsize 6 6. This linear system is solved for by

(12)Formulation (9) is a general form of denoising function. Now,

we should choose suitable basis functions and the decisionfunction . As in Luisiers work, we choose the basis functions

as derivatives of Gaussian (DOG) as follows:

(13)where , and these basis functions can guarantee thatthe denoising function in (9) satisfies differentiability, anti-symmetry, and linear behavior of large coefficients [7]. Then,we choose the decision function in (9) as

(14)

Substituting the formulations (14) and (13) into the formula-tion (9), we can obtain our interscale and intrascale-dependentdenoising function.

Then, the procedure of our proposed intrascale and interscaleSURE-based orthonormal wavelet denoising can be summed upas follows.

1) Perform a level discrete wavelet transform (DWT) to thenoisy data and obtain noisy waveletsubimages .

2) For each wavelet subimages, compute the interscale pre-dictor using group delay compensation andcompute the intrascale predictor using (5).

3) Solve the linear system to obtain the parameters , andaccording to (14) and (12).

4) Compute estimates of the noise-free high-pass subbands using (4) and (14).

5) Reconstruct the denoised image by applying the inversediscrete wavelet transform (IDWT) on the processedhighpass wavelet subimages to obtain an es-timate of the noise-free data

.

III. EXPERIMENTAL RESULTS

We tested our algorithm on Barbara, which representthose images that have substantial textures, and Boat, whichrepresent those images that have less textures, to make acomparison with some other related algorithms [7][9]. Tocompare with Luisiers interscale SURE-based method, we alsouse a critically sampled orthornormal wavelet basis with eightvanishing moments (sym8) over five decomposition stages.For the tested Barbara image, when integrating the intrascaledependencies, the window size 5 5 yielded maximum PSNR.For those tested imaged with less texture, the window size3 3 yielded maximum PSNR. Experimental results in Table Ishow that the resulting SURE-based method which considerthe interscale and intrascale dependencies always yields animproved PSNR as compared to the interscale version [7],and also, superior to some selected techniques reflecting thestate of the art in orthonormal wavelet denoising [8], [9].When compared to the interscale SURE-based approach [7],for textural images, the PSNR gains are up to db, andfor those images with less texture, the PSNR gains are about

db. Even though these PSNR gains may seem marginal,the differences can be seen visually. The visual improvement

142 IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008

Fig. 1. (a) Part of the noisy Barbara image, PSNR = 22:11 dB. (b) Denoised result using the interscale SURE-based approach, PSNR = 27:96dB. (c) Denoised result using our interscale and intrascale-dependent thresholding function (9): PSNR = 28:88 dB.

mainly consists of better suppressing noise in uniform areasas can be seen from Fig. 1.

IV. CONCLUSIONIn this letter, improvements to the original interscale SURE

orthonormal wavelet image denoising method introduced in [7]were proposed. In order to efficiently integrate the intrascalecorrelations within the SURE-based approach, we introduce aLSAI to describe the interscale dependencies. Experimental re-sults show that the proposed algorithm gives better denoisingperformance, especially for images that have substantial high-frequency contents. For the next step, we will work on a moreeffective integration of the intrascale and interscale correlationswithin the SURE-based approach.

ACKNOWLEDGMENTThe authors would like to thank the anonymous reviewers for

their constructive suggestions that improved the presentation ofthis letter. The authors also would like to thank F. Luisier, T.Blu, M. Unser, and their Biomedical Imaging Group (BIG) forgenerously sharing the source codes for the interscale SURE-based denoising.

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