Image de-noising by integer wavelet transforms and generalized cross validationMaarten Jansen, Geert Uytterhoeven, and Adhemar Bultheel Citation: Medical Physics 26, 622 (1999); doi: 10.1118/1.598562 View online: http://dx.doi.org/10.1118/1.598562 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/26/4?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Musculoskeletal ultrasound image denoising using Daubechies wavelets AIP Conf. Proc. 1499, 263 (2012); 10.1063/1.4768998 A time-domain wavelet-based approach for fluorescence diffuse optical tomography Med. Phys. 37, 2890 (2010); 10.1118/1.3431571 Accelerating Monte Carlo simulations of radiation therapy dose distributions using wavelet threshold de-noising Med. Phys. 29, 2366 (2002); 10.1118/1.1508112 The What, How, and Why of Wavelet Shrinkage Denoising Comput. Sci. Eng. 2, 12 (2000); 10.1109/5992.841791 An introduction to wavelet theory and application for the radiological physicist Med. Phys. 25, 1985 (1998); 10.1118/1.598387

Image de-noising by integer wavelet transforms and generalizedcross validation

Maarten Jansen,a) Geert Uytterhoeven, and Adhemar BultheelKatholieke Universiteit Leuven, Department of Computer Science

~Received 20 October 1998; accepted for publication 25 January 1999!

De-noising algorithms based on wavelet thresholding replace small wavelet coefficients by zero andkeep or shrink the coefficients with absolute value above the threshold. The optimal thresholdminimizes the error of the result as compared to the unknown, exact data. To estimate this optimalthreshold, we use Generalized Cross Validation. This procedure has linear complexity and is fullyautomatic, i.e., it does not require an estimate for the noise energy. This paper uses the method forwavelet transforms that map integer gray-scale pixel values to integer wavelet coefficients. Animage with artificial noise is used to illustrate the optimality properties of the estimator. Not alltheoretical requirements for a successful application of the method are strictly fulfilled in the integertransform case. However, this has little influence on practical results. ©1999 American Associa-tion of Physicists in Medicine.@S0094-2405~99!00404-6#

Key words: noise reduction, wavelets, integer transforms, thresholding, cross validation

I. INTRODUCTION

In spite of continuous improvements in image acquisitiontechniques and hardware, image enhancement remains a use-ful and often necessary step. In the last several years, waveletthresholding has shown remarkable results in digital imagede-noising.

A wavelet threshold procedure1 starts with a discretewavelet transform of the pixel matrix. In a second step, co-efficients beneath a certain threshold are replaced by zero.Inverse transform yields the result. The main issue in thisprocedure is the selection of the threshold. This parametershould be chosen so that the eventual result is as close aspossible to the unknown noise-free image. This paper usesthe method of generalized cross validation, which does notneed an estimate of the amount of noise.

A classical wavelet transform maps floating point num-bers to floating point numbers. However, most images con-sist of integer values only, so an algorithm with integer arith-metic could be interesting to avoid floating point calculus.Recently, an invertible transform has been proposed to con-vert integers to integers.2 This procedure is based on theso-called lifting scheme,3,4 which is in the first instance analternative and faster algorithm for a classical wavelet trans-form. The structure of this lifting scheme allows extension ofthe classical algorithm to cases with nonregular grids and toadapt the floating point algorithm to an integer version.

This paper describes experiments with a wavelet basedde-noising algorithm that uses integer transforms and athreshold selection procedure based on generalized crossvalidation.

II. THE DISCRETE WAVELET TRANSFORM

A one-dimensional discrete wavelet transform is a re-peated filter bank algorithm. The input is a vector, repre-sented by the row of circles on a shaded background in

Fig. 1. Typically neighboring points in this vector showstrong correlations. The objective of the transform is to usethese correlations to obtain a sparse representation of theinput. Therefore, in the first step, the input is convolved with

a high-pass filterg̃ and a low-pass filterh̃. The result of thelatter convolution is a smoothed version of the input. Thehigh frequency part is captured by the first convolution, andcontains a lot of very small numbers, due to the high corre-lations among neighboring input points. Since these convo-lutions both give a result with a size equal to that of theinput, this procedure doubles the total number of data. There-fore, theFast Wavelet Transform~FWT! omits half of thesedata bysubsampling. The resulting high frequency coeffi-cients are wavelet coefficients at thefinest level. The lowfrequency output are scaling coefficients. In the second andfollowing steps, the algorithm repeats the same procedure onthis smoothed version of the input. In each step, the resultingwavelet coefficients contain information about a certain de-gree of detail. All together, these coefficients constitute amultiresolution analysis of the input. These output coeffi-cients have a gray background in Fig. 1.

The basic building block of a wavelet transform is a filterbank with filtersg̃ and h̃. This is represented in Fig. 2. Thisfigure also shows the reconstruction, corresponding to onestep of the filter bank decomposition. This reconstructionstarts byupsamplingthe rows of coefficients: a zero is put inbetween two elements. The next step is a convolution withfilters g andh. The results of these convolutions are added.For an invertible wavelet transform, it is of course necessarythat the result from the filterbank with reconstruction equalsthe input. This is theperfect reconstruction property: it guar-antees a proper inverse transform into the original pixel rep-resentation of the image. An analysis~e.g., in frequency do-main! of this output yields conditions ong̃, h̃, g, andh. Wedo not go into detail on this problem.

In two dimensions, we first apply one step on the row

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vectors and then on the column vectors. Figure 3 shows howthis results in four classes of coefficients. Coefficients thatresult from a convolution withg̃ in both directions~HH!represent diagonal features of the image, whereas a convolu-tion with h̃ in one direction and withg̃ in the other, reflectsvertical and horizontal information~HG and GH!. In the nextstep we proceed with the low-pass~LL ! coefficients, whichare results of a convolution withh̃ in both directions. We endup with a transformed image in which each resolution levelconsists of coefficients for threecomponents: LH, HL, andHH. At the coarsest level, we also keep low-pass coefficientsLL.

Most noise reduction procedures based on this wavelettransform manipulate the coefficients, thereby often intro-ducing artifacts. To reduce these spurious effects, severalauthors5–10propose aRedundant Wavelet Transform~RWT!.The scheme of this alternative contains no subsamplingsteps, as illustrated by Fig. 4. This RWT is consistent withthe FWT, if all coefficients from the decimated transformre-appear in the output of the RWT. In Fig. 4, these coeffi-cients have a black center. To this end, the computation ofsuch an ‘‘original’’ coefficient may only use ‘‘original’’ co-efficients at the previous scale. Intermediate coefficients haveto be skipped. Therefore, the algorithm upsamples at eachscale the filter coefficients of the previous step. This redun-dant transform can also be seen as an interleaved collectionof the wavelet transforms of all translations of the input sig-nal. In two dimensions, each component~HH, HL, and LH!at each scale is now represented by a coefficient matrix with

the same size as the input image. A redundant wavelet trans-form is an overcomplete representation of the input data.Several independent reconstructions are possible. If the co-efficients were manipulated to reduce noise, the resulting co-efficients may not be an exact redundant wavelet transformof any image at all and all possible reconstructions may yielda different output. Averaging these outputs in some way willcause extra smoothing. More precisely, each application ofthe one-dimensional redundant filter operation throughoutthe algorithm doubles the data size. Therefore, for each for-ward step, there exist two independent reconstructions. Wecompute at each step the average of these two reconstruc-tions before proceeding to the next step. For more details werefer to Refs. 6, 7, and 10. For small images, the smoothingeffect of the redundancy largely compensates the overhead incomputation and storage@O (N logN) instead ofO (N), withN the total number of pixels#.

III. LIFTING AND INTEGER TRANSFORMS

The lifting scheme decomposes a filter bank operation ina number of consecutive lifting steps.11 This series starts bysplitting the input vector into points with odd and points witheven index~Fig. 5!. There exist two kinds of lifting opera-tions. The first, called dual lifting, subtracts a filtered versionof the ‘‘even’’ input from the ‘‘odd’’ input. The second,called primal lifting, adds a filtered version of the dual liftingoutput to the so far untouched ‘‘even’’ input. One way tointerpret the dual lifting step is the following: We assumethat the ‘‘even’’ input and the ‘‘odd’’ input are highly cor-related. This is certainly true in the first lifting step, wherethe ‘‘even’’ and the ‘‘odd’’ input are directly originatingfrom one input signal. We now try topredict the oddsamples by a prediction operator~a filter! on the even ones.

FIG. 3. A two-dimensional wavelet transform. First we apply one step of theone-dimensional transform to all rows~left!. Then, we repeat the same forall columns~middle!. In the next step, we proceed with the coefficients that

result from a convolution withh̃ in both directions~right!.

FIG. 4. Successive steps of a redundant, nondecimated wavelet transform. Inorder to maintain a multiresolution decomposition, the absence of subsam-pling is compensated by upsamling the filter coefficients.

FIG. 1. Successive steps of a fast decimated wavelet transform. After con-

volving the input with two filtersh̃ and g̃, the algorithm drops half of theresult ~downsampling, indicated by↓2!. Under certain conditions, perfectreconstruction of the original signal is possible from this decimated trans-form.

FIG. 2. One step of a wavelet decomposition and its reconstruction. This isa filter bank: The input is filtered and downsampled to get a low-pass signalLP and a high-pass signal HP. Reconstruction starts with upsampling byintroducing zeroes between every pair of points in LP and HP.

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By subtracting this prediction from the odd samples, we re-duce redundancy. Several alternating dual and primal liftingsteps have the same effect as a classical wavelet filter bankoperation. It can be proved that all traditional wavelet filterpairs can be decomposed into a sequence of lifting steps,with the appropriate choice of primal and dual operations.

The inverse transform is now very easy to construct: fol-low the arrows in opposite directions and replace all plussigns by minus signs and vice versa. Note that this is notpossible in the classical filter bank algorithm. It is possible inthe lifting scheme, because at all places where a filter opera-tion is performed, we also keep the input of this filter for thenext step.

To obtain an invertible transform, the filter operation inthe dual and primal lifting step need not even be linear. Forinstance, we could round the result of a given operation tothe nearest integer. This is the integer transform,2 illustratedin Fig. 6. If the input is a vector with integer values, allintermediate results consist of integers.

IV. THRESHOLDING AND GENERALIZED CROSSVALIDATION

A. Wavelet thresholding

We now suppose the input image is affected by additive,stationary, and zero-mean noise. This noise can be colored~correlated! or white. It can be proved12 that the~classical,noninteger! wavelet transform of stationary noise is station-ary at each resolution level and within each component~ver-tical, horizontal, diagonal!. Figure 7 illustrates this observa-tion: The noise is spread evenly over all coefficients withineach resolution level.

Second, we assume that the noise-free image can be wellrepresented by a limited number of large wavelet coeffi-cients. This decorrelating property also justifies the numer-ous wavelet based compression algorithms.

Third, the noise should not be ‘‘too large.’’ In that case,the noise has a relatively small influence on the importantlarge clean coefficients.

These three observations suggest replacing the small co-efficients by zero, because they are dominated by noise andcarry only a small amount of information. At thej -th reso-lution level, and for a given componentc, all wavelet coef-ficients with absolute value below a certain, level dependent,thresholdd j

c , are classified as ‘‘noisy.’’ The algorithm re-places them by zero. The coefficients above the threshold areshrunk with the valued j

c . Figure 8 compares this ‘‘soft-thresholding’’ operation with the ‘‘hard-thresholding’’ alter-native. The hard-thresholding procedure does not shrink thecoefficients with absolute value above the threshold. Al-though at first sight this may seem a more natural approach,soft-thresholding is a more continuous operation, and it ismathematically more tractable. The figure also shows a moresophisticatedshrinking function. It provides a continuous ap-proach, while keeping the largest coefficients untouched. Notall shrinking methods use a threshold parameter. However,these more sophisticated shrinkage policies are generallycomputationally more intensive. The soft-thresholding func-tion is thus a compromise between a fast and straightforwardmethod and a continuous approach.

B. Threshold selection

The main question in this procedure is how to choose thethreshold, at each level and for each component. If this

FIG. 5. Decomposition of a filter bank into lifting steps.

FIG. 6. Integer wavelet transform.

FIG. 7. Classical wavelet transform of stationary, colored noise. We useCohen–Daubechies–Feauveau~2,2!-wavelet filters. The noise was obtainedas a convolution of white gaussian noise with a FIR high-pass filter.

FIG. 8. Noise reduction by wavelet shrinking. On the left: Soft-thresholding:a wavelet coefficientw with an absolute value below the thresholdd isreplaced by 0. Coefficients with higher absolute values are shrunk. Middle:Hard-thresholding: Coefficients with an absolute value above the thresholdare kept. Right: a more sophisticated shrinking function.

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threshold is too small, the result is still noisy. On the otherhand, a large threshold also removes important image fea-tures and thus causes a bias.

The optimal thresholdd j ,optc for componentc at level j

minimizes the mean square error~MSE! of the result, ascompared with the unknown, noise-free coefficients:

Rjc~d!5

1

Njc iwj,d

c 2vjci2, ~1!

wherevjc is the vector of noise-free coefficients,wj,d

c is thevector of shrunk coefficients for a given thresholdd, andNj

c

represents the number of wavelet coefficients at levelj andcomponentc.

The ‘‘universal threshold’’ by Donoho and Johnstone:13

d jc5A2 log~Nj

c!s jc , ~2!

is widely known and states that the optimal threshold valued j

c at level j and for componentc is proportional to theamount of noise at that level and component, given by itsstandard deviations j

c . This universal threshold has a lot ofasymptotic optimality properties, even if the noise iscorrelated.12 In fact, for finite Nj

c this is an upper bound forthe optimal threshold.

Other choices include the ‘‘SURE’’ threshold14 and Baye-sian estimates.15–17 In this paper, we use the method of gen-eralized cross validation~GCV!.18–20

This method minimizes at each levelj and componentcthe following function:

GCVjc~d!5

1

Njc iwj

c2wj,dc i2

FNj 0c

Njc G2 , ~3!

wherewjc is the vector of wavelet coefficients at the resolu-

tion level j and for componentc; wj,dc is the vector of shrunk

coefficients for a given thresholdd, andNj 0c is the number of

zero elements in this vector. It is important to note that inthis formula no estimate for the noise energys j

c is needed.Cross validation is a widely used statistical technique andother applications in wavelet shrinkage procedures can befound in Refs. 21 and 22. In Refs. 20 and 10 the method ofgeneralized cross validation for threshold estimation isproved to beasymptotically optimal, i.e., for a large numberof coefficients, the minimizer of GCVj

c(d) also minimizesthe mean square error of the thresholded coefficients. Forfinite Nj

c it gives better estimates of the optimal thresholdthan the universal threshold. The mathematical description ofthe method in Ref. 20 assumes a continuous threshold opera-tion. This is why hard-thresholding is not allowed. In prin-ciple, it also requires Gaussian noise. However, other zero-mean, additive models mostly perform well, as far as theGCV threshold selection procedure is concerned. Note that a~linear! wavelet transform maps Gaussian noise onto Gauss-ian noise. For other models of input noise, it is harder tocompute the probability density of the wavelet coefficientsand the result depends on the filters used. In most cases, the

wavelet coefficients of the noise have a more normallike dis-tribution than the input. This, together with the generalityand well-known properties of the normal model, motivatesthe assumption of Gaussian noise. Thresholding by itselfmay not be the best de-noising procedure for all types ofnoise, especially for multiplicative noise.

Since a GCV-method approximates a minimum meansquare error~MSE! threshold, the procedure assumes thatMSE is a good image metric. In cases where the visual per-ception does not correspond so well to the MSE-measure, itwill be difficult to find a good threshold using GCV: In ouropinion, it would be hard to adapt the GCV formula andprocedure to other metrics. We also note that minimizingMSE in terms of wavelet coefficients corresponds exactly to

FIG. 9. An artificial test example.~a! A clean DSA test image.~b! The sameimage with artificial, additive and correlated noise. The noise is the result ofa convolution of white noise with a FIR high-pass filter. Signal-to-noiseratio is 10 dB.

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minimizing MSE on pixel level in the case of orthogonaltransforms. For biorthogonal wavelet transforms, the MSEnorm in pixel terms and the MSE norm in the wavelet do-main are equivalent, but not equal, i.e., the ratio of the twoerror norms is bounded by two positive constants:

c<SNR~pixels!

SNR~wavelet coefficients!<C,

and the minimization procedure in the wavelet domain findsan approximate minimal MSE in terms of pixels. We prefer aminimization in wavelet domain for three reasons: first thisspeeds up the computations since no forward or inversetransform is necessary during the minimization. Second,GCV only approximatesthe minimum MSE, both in waveletand pixel domain. Finally, MSE is only an approximation ofthe visual quality, both in the wavelet and the pixel domain.There seems to be no reason why MSE in the pixel domainyields better visual quality than MSE based on wavelet co-efficients.

V. INTEGER DE-NOISING

A. Generalized cross validation for integer waveletcoefficients

Since the input for the algorithm of this paper are inte-gers, the noise cannot take an arbitrary value and so its dis-tribution cannot be Gaussian. Moreover, the integer wavelettransform is nonlinear and so it does not preserve additivitynor stationarity. All of these conditions are necessary forcorrect use of a GCV-threshold estimation.

However, an artificial test example illustrates that, inpractice, these conditions do not pose serious problems. Fig-ure 9~a! shows a noise-free DSA test image. In Fig. 9~b!, weadd artificial, colored noise. This noise was the result of aconvolution of white noise with a FIR high-pass filter. Thesignal-to-noise ratio is 10 dB, where we define signal-to-noise ratio~SNR! as:

SNR520 log10

i f2 f̄iiei , ~4!

with f the noise-free image ande the noise.f̄ stands for themean value~dc-component! of f.

We compute the redundant, integer wavelet transform ofthe noisy and the noise-free image and estimate the optimalthreshold at each resolution level and for each component bythe GCV-procedure. Since we know the noise-free waveletcoefficients, we can compare the GCV function with themean square error as a function of the threshold. Figure 10~a!shows this comparison for the vertical component at the fin-est resolution level. Both GCVj

c(d) and Rjc(d) have about

the same minimum. Figure 10~b! compares both functions atthe one but finest resolution level. The optimal threshold atthis level is close to zero. This is not surprising: we haveadded high frequency noise which mainly manifests at finescales. As Fig. 11~b! shows, thresholding at this level causes

FIG. 10. An artificial test example: a comparison of GCVjc(d) andRj

c(d) forthe vertical component at the~a! finest resolution level and~b! one but finestresolution level. Both GCVj

c(d) andRjc(d) have about the same minimum.

At the one but finest level, the optimal threshold is close to zero, whichindicates that the noise at this level is neglectable.

FIG. 11. An artificial test example: reconstruction by inverse redundanttransform after removing small coefficients at~a! the finest resolution levelonly ~signal-to-noise ratio is 19.94 dB!, and ~b! the two finest resolutionlevels ~signal-to-noise ratio is 17.27 dB!. Thresholding at coarse levels in-troduces more visible artifacts.

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considerable blur and artifacts and loss of important details:small blood-vessels become very unclear or even completelydisappear. Moreover, computation of GCVj

c(d) for smallvalues ofd is unstable, since in Eq.~3! both numerator anddenominator approach zero. An optimal threshold close tozero is therefore hard to find. For these reasons, we get abetter result if we only apply the algorithm at the finest reso-lution level. Figure 11~a! shows this reconstruction: signal-to-noise ratio is now 19.94 dB, compared to 17.27 dB for athreshold at two levels. To illustrate the smoothing effect ofthe redundant transform, Fig. 12 shows the output of a noisereduction algorithm on only the first level of the fast wavelettransform. Signal-to-noise ratio for this case is 16.39 dB.

Table I compares the integer GCV procedure with other,classical threshold methods. It shows that, at least for thisexample, using the integer transform instead of the classical,linear transform poses no problem.~In this case, there is evena slight improvement.! The table also illustrates that theGCV method performs at least as well as other thresholdselection procedures, although GCV does not use informa-tion on the amount of noise~the deviations j

c!. The SUREand universal procedures do need the values ofs j

c . For thevalues in this table, we used the exacts j

c . In practical situ-ations, this value has to be estimated, which mostly deterio-rates the result. For a more detailed comparison of differentmethods, we refer to Ref. 10.

The quality of our result is of course limited by the pos-sibilities of thresholding as a de-noising method and—as

mentioned before—by the quality of MSE~expressed interms of wavelet coefficients! as an image metric.

B. The minimization procedure

If we assume that, for continuously varyingd, GCVjc(d)

is a convex function, we can use a Fibonacci minimizationprocedure.23 For a given relative accuracy, this procedureonly requires a limited and fixed number of function evalu-ations.

Since we are working with integers only, we only con-sider integer threshold values. Soft thresholding with a non-integer value would yield noninteger wavelet coefficients.We could try to find the integer threshold that minimizesGCVj

c(d) among all integerd. Since this is an absolute ac-curacy condition, the number of function evaluations woulddepend logarithmically on the largest possible threshold~thesmallest is assumed to be zero!, which is the largest waveletcoefficient at the given resolution level and for the givencomponent.

In practice, GCVjc(d) is not strictly convex. Figure 13

shows a detail of the GCV function, depicted in Fig. 10~a!.This nonsmooth character is due to the jumps of the denomi-nator: if, for a given wavelet coefficientw, the thresholdvalue increases fromw2e to w1e, the value ofN0 in thedenominator decreases at least by one. Note that the numera-tor is continuous since we use the continuous soft-thresholdoperation. Since most wavelet coefficients are small, most ofthe jumps of GCVj

c(d) appear at small threshold values. Forpractical use of the GCV procedure this lack of convexitymostly poses no problem. We also remark that GCVj

c(d) isjust anestimateof the mean square error as a function ofd.So, it is no use spending too much energy in finding theoptimal threshold, if a fast algorithm finds a good approxi-mation.

C. An example

We now illustrate the algorithm with an example. Figure14~a! shows an MRI image of 128 by 128 pixels. Figure14~b! contains the result of the de-noising algorithm for afast wavelet transform. Figure 14~c! is the result for a non-decimated transform. Figures 14~d! and 14~e! show the re-sults if one uses universal thresholds. This illustrates that theuniversal threshold is in fact not appropriate for image de-noising, since it acts as an probabilistic upper bound. Thismeans that the threshold is chosen so that asymptotically all

FIG. 12. An artificial test example: reconstruction by inverse fast wavelettransform after removing small coefficients at the finest resolution level.Signal-to-noise ratio is 16.39 dB.

TABLE I. Comparison of different threshold procedures, applied to the finestscale of the coefficients of Fig. 9. In all cases, we used a redundant trans-form with Cohen–Daubechies–Feauveau~2,2!-filters.

Integer GCV Classical GCV SURE Universal

19.94 19.88 19.26 18.72

FIG. 13. Detail of GCVjc(d), depicted in Fig. 10~a!. GCVj

c(d) is not strictlyconvex.

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pure noise coefficients are below the threshold with probabil-ity one. This procedure introduces a lot of visual artifacts,and lower thresholds are preferable.

We used biorthogonal Cohen-Daubechies-Feauveau~2,2!-wavelet filters.24 This is one of the popular filters for imageprocessing. Its decomposition into lifting steps is particularly

easy.25 In principle, the success of GCV in estimating theoptimal threshold does not depend on the choice of a waveletbasis. This choice determines the visual quality by the filterproperties, an issue which is beyond the scope of this paper.

Figure 15 shows the GCV functions of the first~finest!and second resolution level of the fast wavelet transform.

FIG. 16. GCV functions for a redun-dant wavelet transform of MRI imageof Fig. 14.~a! The three components atthe finest resolution level.~b! Thethree components at the second resolu-tion level.

FIG. 14. An example.~a! The input image, an MRI image (1283128 pixels) with noise.~b! Result after thresholding the fast wavelet coefficients at the firstand second resolution level, using GCV thresholds.~c! Result after thresholding the redundant wavelet coefficients at the first and second resolution level,using GCV thresholds.~d! Result after thresholding the fast wavelet coefficients at the first and second resolution level, using universal thresholds.~e! Resultafter thresholding the redundant wavelet coefficients at the first and second resolution level, using universal thresholds.

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Since the GCV-procedure is only asymptotically optimal, itis no use applying it for components with only a small num-ber of coefficients. Our experience indicates that 1000 coef-ficients is about the minimum to guarantee a successful useof GCV. Therefore, we only shrink coefficients at the finestlevel (64364 coefficients in each component! and at thesecond level (32332 coefficients in each component!. Thenumber of coefficients at coarser levels poses no problem inthe nondecimated case. If the amount of noise justifies noiseremoval at these levels, we could use GCV to estimate theoptimal threshold. However, to compare results with the fasttransform, we restrict ourselves to two levels. The GCV-functions for this case are shown in Fig. 16.

VI. CONCLUSIONS

We have presented an image de-noising algorithm, basedon wavelet thresholding, that combines a redundant integerwavelet transform with the fully automatical threshold selec-tion method by generalized cross validation.

The method selects a separate threshold for each compo-nent at each scale. This level-dependent approach is moreadaptive than a single threshold choice and moreover, it canhandle colored noise. Our experiments show that resolutionlevels with little noise should be kept untouched.

Although the theoretical conditions for application of gen-eralized cross validation are not strictly fulfilled in an integertransform framework, the minimizer of the generalized crossvalidation function is a good estimate for the optimal thresh-old. All computations use integers only, which could be use-ful, e.g., for DSP-implementation.

We use a redundant, nondecimated wavelet transform,which better preserves edges and introduces less artifacts andblur.

ACKNOWLEDGMENTS

This paper presents research results of the Belgian Pro-gramme on Interuniversity Poles of Attraction, initiated bythe Belgian State, Prime Minister’s Office for Science, Tech-nology and Culture. The scientific responsibility rests withits authors. The first author is financed by a grant from theFlemish Institute for the Promotion of Scientific and Tech-nological Research in the Industry~IWT!. Research of thesecond author is supported by the Flemish Information Tech-nology Action Program~’Vlaams Actieprogramma Informa-tietechnologie’!, project number ITA/950244. We used thesoftware library WAILI, developed at K. U. Leuven, Depart-ment of Computer Science~Ref. 26!.

a!Correspondence address: Maarten Jansen, K. U. Leuven, Department ofComputer Science, Celestijnenlaan 200A, B-3001 Heverlee, Belgium,Phone: 32 16 32 7080; Fax: 32 16 32 7996, Electronic mail:maarten.jansen@cs.kuleuven.ac.be, http://www.cs.kuleuven.ac.be/˜maarten/1D. L. Donoho, ‘‘De-noising by soft-thresholding,’’ IEEE Trans. Inf.Theory41, 613–627~1995!.

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FIG. 15. GCV functions for a fastwavelet transform of MRI image ofFig. 14. ~a! The three components atthe finest resolution level.~b! Thethree components at the second resolu-tion level.

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8M. Malfait and D. Roose, ‘‘Wavelet based image denoising using a mar-kov random field a priori model,’’ IEEE Trans. Image Process.6, 549–565 ~1997!.

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