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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 14, NO. 11, NOVEMBER 2005 1747

A Universal Noise Removal AlgorithmWith an Impulse Detector

Roman Garnett, Timothy Huegerich, Charles Chui, Fellow, IEEE, and Wenjie He, Member, IEEE

Abstract—We introduce a local image statistic for identifyingnoise pixels in images corrupted with impulse noise of randomvalues. The statistical values quantify how different in intensity theparticular pixels are from their most similar neighbors. We con-tinue to demonstrate how this statistic may be incorporated into afilter designed to remove additive Gaussian noise. The result is anew filter capable of reducing both Gaussian and impulse noisesfrom noisy images effectively, which performs remarkably well,both in terms of quantitative measures of signal restoration andqualitative judgements of image quality. Our approach is extendedto automatically remove any mix of Gaussian and impulse noise.

Index Terms—Bilateral filter, denoising, Gaussian noise, imagerestoration, impulse noise, mixed noise, nonlinear filters.

I. INTRODUCTION

NOISE can be systematically introduced into images duringacquisition and transmission. A fundamental problem of

image processing is to effectively remove noise from an imagewhile keeping its features intact. The nature of the problem de-pends on the type of noise added to the image. Fortunately, twonoise models can adequately represent most noise added to im-ages: additive Gaussian noise and impulse noise.

Additive Gaussian noise is characterized by adding to eachimage pixel a value from a zero-mean Gaussian distribution.Such noise is usually introduced during image acquisition. Thezero-mean property of the distribution allows such noise to beremoved by locally averaging pixel values. Ideally, removingGaussian noise would involve smoothing inside the distinctregions of an image without degrading the sharpness of theiredges. Classical linear filters, such as the Gaussian filter,smooth noise efficiently but blur edges significantly. To solvethis problem, nonlinear methods have to be used, most notablythe anisotropic diffusion technique of Perona and Malik [1].Another interesting method is the bilateral filter studied by

Manuscript received February 14, 2004; revised September 27, 2004. Thiswork was supported in part by the National Science Foundation under GrantsCCR-0098331 and CCR-9988289 and in part by the Army Research Officeunder Grant DAAD 19-00-1-0512. The associate editor coordinating the reviewof this manuscript and approving it for publication was Dr. Nicolas Rougon.

R. Garnett is with the Departments of Mathematics and Computer Science,Washington University in Saint Louis, Saint Louis, MO 63130 USA (e-mail:[email protected]).

T. Huegerich is with the Department of Physics, Rice University, Houston,TX 77005 USA (e-mail: [email protected]).

C. Chui is with the Department of Mathematics and Computer Science, Uni-versity of Missouri–Saint Louis, Saint Louis, MO 63121 USA, and also with theDepartment of Statistics, Stanford University, Stanford, CA 94305 USA (e-mail:[email protected]; [email protected]).

W. He is with the Department of Mathematics and Computer Science,University of Missouri–Saint Louis, Saint Louis, MO 63121 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TIP.2005.857261

Tomasi and Manducci [2] based on the original idea of Overtonand Weymouth [3]. The essence of these methods is to uselocal measures of an image to quantitatively detect edges andto smooth them less than the rest of the image.

Impulse noise is characterized by replacing a portion of animage’s pixel values with random values, leaving the remainderunchanged. Such noise can be introduced due to transmissionerrors. The most noticeable and least acceptable pixels in thenoisy image are then those whose intensities are much differentfrom their neighbors.

The Gaussian noise removal methods mentioned abovecannot adequately remove such noise because they interpret thenoise pixels as edges to be preserved. For this reason, a separateclass of nonlinear filters have been developed specifically forthe removal of impulse noise; many are extensions of the me-dian filter [4], [5], or otherwise use rank statistics [6]–[8]. Thecommon idea among these filters is to detect the impulse pixelsand replace them with estimated values, while leaving the re-maining pixels unchanged. When applied to images corruptedwith Gaussian noise; however, such filters are not effective, andin practice leave grainy, visually disappointing results.

There has not been much work carried out on building filtersthat can effectively remove both Gaussian and impulse noise,or any mixture thereof. Such “mixed noise” could occur, forinstance, when sending an already noisy image over faultycommunication lines. Peng and Lucke suggested a fuzzy filterdesigned specifically for mixed noise [9]. Additionally, in 1996Abreu et al., proposed the median-based SD-ROM filter toremove impulse noise, and their method proved quite effective[10]. They also gave quantitative measures demonstrating theSD-ROM filter’s ability to remove Gaussian noise as well asmixed Gaussian and impulse noise. Although their filter hasimpressive quantitative results, when applied to images withGaussian or mixed noise it often produces visually disap-pointing output similar to that of other median-based filters.

In this paper, we introduce a framework for creating a uni-versal noise removal filter that is based on a simple statisticto detect impulse noise pixels in an image. Instead of applyingthe “detect and replace” methodology of most impulse noise re-moval techniques, we show how to integrate such a statistic intoa filter designed to remove Gaussian noise. The behavior of thefilter can be adaptively changed to remove impulses while re-taining the ability to smooth Gaussian noise. Additionally, thefilter can be easily adapted to remove mixed noise.

This paper is arranged as follows. In Section II, we introducea local image statistic for detecting impulses. In Section III, webriefly explain the bilateral filter and describe how to incorpo-rate the statistic into the filter to create a universal filter. Finally,

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1748 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 14, NO. 11, NOVEMBER 2005

in Section IV, we provide visual examples and numerical resultsthat demonstrate our method’s soundness.

II. NEW STATISTIC FOR DETECTING IMPULSES

A. Noise Models

Throughout this paper, we will use standard matrix notationfor images. For example, when is an image, will representthe intensity value of at the pixel location in the imagedomain. For the case of additive Gaussian noise, the noisy image

is related to the original image by

where each noise value is drawn from a zero-mean Gaussiandistribution.

Impulse noise, again to be denoted by , is characterized byreplacing a portion of the original pixel values of the image withintensity values drawn from some distribution, usually eithera uniform or discrete distribution over the intensity range.Throughout this paper, we consider the uniform noise distri-bution model, although the methods we discuss could be usedwithout modification for the discrete model.1 Therefore, for im-ages corrupted with impulse noise, the noisy image is relatedto the original image by

with probabilitywith probability .

B. Motivation of the Impulse Detection Scheme

The problem of deciding which pixels in an image are im-pulses is clearly not well defined. In particular, Lichtenstein orSeurat might be dismayed if he saw one of his paintings “de-noised.” Therefore, we must be content with detecting pixelsthat are like impulses, that is, pixels that vary greatly in intensityfrom most surrounding pixels. Thankfully, uncorrupted naturalimages rarely contain details isolated to a single pixel and gen-erally have few impulse-like pixels.

Impulse noise removal methods use many different tech-niques to determine whether a given pixel is an impulse inthis sense. These approaches vary in complexity from beingrelatively simple to highly complex. The most basic impulsedetectors are based on two-state methods that attempt to defini-tively characterize each image pixel as either an impulse oran unaffected pixel. The underlying goal of these two-statemethods is to find pixels that are significant outliers whencompared to their neighbors. One of the simplest and mostintuitive method is to compare a pixel’s intensity with themedian intensity in its neighborhood, as in [5]. Other methods,such as the two-state SD-ROM filter of Abreu et al. [10] and therecent CSAM filter of Pok et al. [8], use more complex criteriato judge whether a pixel is an impulse. The advantage of thesetwo-state methods is their simplicity, which makes them easilycustomizable.

1In contrast, conditional signal-adaptive median (CSAM) filtering [8], is de-signed for salt-and-pepper noise, a discrete impulse noise model in which thenoisy pixels take only the values 0 and 255. It can remove salt-and-pepper typenoise pixels very well, but it cannot perform similarly for uniformly distributedimpulse noise and is outperformed by even the median filter.

More complex methods are naturally more successful for de-tecting impulses in general, but there is a tradeoff for this im-provement in detection. The most complicated methods requiretraining procedures to make an optimal classification based onmeasures of pixels and their neighbors. Methods that requiretraining are bound to be less easily controlled and more un-predictable than simpler methods. If a method is trained on animage with many impulse-like pixels (such as an image withmany fine details), its ability to detect and remove actual im-pulse noise from a different image will be inhibited. On the otherhand, if the method is trained on a smooth image with few im-pulse-like pixels, it will overly smooth an image with many finedetails that could otherwise be preserved. Even if care is takento select an appropriate training image—assuming one is avail-able—the performance of an automatically trained filter is es-sentially unpredictable.

An additional concern with existing methods arises from thefact that when impulse noise is introduced to an image, a por-tion of the pixels will be replaced with intensities only slightlydifferent from their original values. Two-state detectors, with orwithout training, will most likely fail to detect such small im-pulses since they look exclusively for large outliers. They re-move the most conspicuous noise, but the lesser impulses re-main, creating a grainy appearance. In response to this problem,we adopt a continuous function to represent how impulse-like aparticular pixel may be. By considering the magnitude of thisfunction, we can adapt the behavior of the filter in a straightfor-ward way according to how impulse-like each pixel is.

C. Definition of the Rank-Ordered Absolute Differences(ROAD) Statistic

Let be the location of the pixel under consider-ation, and let

(1)

be the set of points in a neighborhoodcentered at for some positive integer . In the following dis-cussion, let us only consider , though the same procedurecan be extended to any . Hence

represents the set of points in a 3 3 deleted neighborhood of. For each point , define as the absolute difference

in intensity of the pixels between and , i.e.,

Finally, sort the values in increasing order and define

(2)

where and

th smallest for

We call the statistic defined in (2) ROAD. In this paper, we willconsider only, and set .

GARNETT et al.: UNIVERSAL NOISE REMOVAL ALGORITHM WITH AN IMPULSE DETECTOR 1749

Fig. 1. Closeups of an artificially added impulse (upper left) and a typical edgepixel (lower right). ROAD of impulse: 525; ROAD of edge pixel: 88.

Fig. 2. Demonstrating how to calculate ROAD.

The ROAD statistic provides a measure of how close a pixelvalue is to its four most similar neighbors. The logic underlyingthe statistic is that unwanted impulses will vary greatly in inten-sity from most or all of their neighboring pixels, whereas mostpixels composing the actual image should have at least half oftheir neighboring pixels of similar intensity, even pixels on anedge. Fig. 1 shows examples from the Lena image comparing atypical impulse noise pixel to an edge pixel. Notice that the edgepixel has neighbors of similar intensity despite forming part ofan edge, and, thus, has a significantly lower ROAD value. Fig. 2demonstrates how the latter value was calculated.

We illustrate numerically that the ROAD statistic is a goodindicator of impulse noise. Ideally we want our statistic to bevery high for impulse noise pixels and much lower for uncor-rupted pixels. Fig. 3 displays quantitative results from the Lenaimage. The upper dashed line represents the mean ROAD valuefor noise pixels as a function of the amount of impulse noiseadded, and the lower dashed line represents the mean ROAD

Fig. 3. Mean ROAD values of noise pixels and uncorrupted pixels in the Lenaimage as a function of the impulse noise probability, with standard deviationerror bars demonstrating the significance of the difference.

value for uncorrupted pixels. The noise pixels consistently havemuch higher mean ROAD values than the uncorrupted pixels,whose mean ROAD values remain nearly constant even withvery large amounts of noise.

III. INTRODUCING THE ROAD AND TRILATERAL FILTERS

It would be relatively simple to introduce the ROAD statisticinto many existing filtering techniques, allowing them to detectand properly handle impulse-like pixels in a noisy image. Forexample, one could modify the popular anisotropic diffusionmethod to utilize the ROAD statistic. Below we describe howone might extend the bilateral filter to create a filter capable ofremoving both impulse and additive Gaussian noise from im-ages. We begin with a brief introduction to the bilateral filter.

A. Bilateral Filter

The bilateral filter, as described in [2], applies a nonlinearfilter to to remove Gaussian noise while retaining the sharp-ness of edges. Each pixel is replaced by a weighted average ofthe intensities in a neighborhood. Theweighting function is designed to smooth in regions of similarintensity while keeping edges intact, by heavily weighting thosepixels that are both near the central pixel spatially and similarto the central pixel radiometrically.

More precisely, let be the location of the pixel under con-sideration, and let

(3)

be the pixels in a neighborhood of . Theweight of each with respect to is the product of twocomponents, one spatial and one radiometric

(4)

where

(5)

and

(6)


The weights must be normalized, so the restored pixel isgiven by

(7)

The weighting function decreases as the spatial distancebetween and increases, and the weighting function de-creases as the radiometric “distance” between the intensities

and increases. The spatial component of the weight de-creases the influence of pixels far away from to generally re-duce blurring, while the radiometric component diminishes theinfluence of pixels with significantly different intensities to keepthe edges of distinct image regions sharp. Notice that theand weighting functions need not be Gaussians—any suit-able nonnegative functions that decrease to zero may be usedinstead.

In our particular weighting functions, the parameters andcontrol the behavior of the weights. They are the values at

which the respective Gaussian weighting functions take theirmaximum derivatives, so they serve as rough thresholds foridentifying pixels sufficiently close spatially or radiometrically.Note, in particular, that as and radiometric differencesare rendered irrelevant by this high threshold, the bilateral filterapproaches a Gaussian filter of standard deviation . As both

so that all neighboring pixels easily meet boththresholds, the bilateral filter approaches the mean filter.

The idea for bilateral filtering was originally introduced in1979 by Overton and Weymouth [3] as a simple image prepro-cessing tool to remove noise. In 1998, Tomasi and Manduchi [2]provided the name “bilateral filtering” and improved the methodin several aspects. Specifically, they used Gaussian functions inthe weighting functions (instead of the rational functions usedin [3]) to improve the filter’s performance, analyzed the inter-action between the weighting functions, and proposed metricsto be used for color images. Elad [11] established a connectionbetween bilateral filtering and several other methods in terms ofminimizing functionals.

B. New Weighting Function

We incorporate the ROAD statistic into the bilateral filteringframework by introducing a third weighting function influencedby how impulse-like each pixel of the image is. The “impulsive”weight at a point is given by

(8)

The parameter determines the approximate threshold abovewhich to penalize high ROAD values.

We would like to integrate this impulsive component into anonlinear filter designed to weight pixels based on their spa-tial, radiometric, and impulsive properties. Unfortunately, theimpulsive component is not directly compatible with the radio-metric component already present in the bilateral filter. To illus-trate this observation, let us consider black impulses on a whitebackground. If the point under consideration were such an im-pulse, the radiometric component would weight any other blackimpulses in the neighborhood of much more than the white

background pixels we desire. As a result, the black impulses re-main black impulses. The radiometric weight works contrary toour goal because it was not designed to remove impulse noise.However, if used selectively, the radiometric weight can still behelpful for removing impulse noise. It can help smooth impulsesthat are only slightly different from their surrounding pixelswithout blurring edges, while the impulsive weight works to re-move the larger outliers. If we can use the radiometric compo-nent for only small impulses, we can improve upon the commontwo-state methods for impulse noise removal by not only re-moving the larger outliers, but also smoothing away smaller im-pulses.

To add the impulsive weight while still retaining the radio-metric component of the bilateral filter, we introduce a switchto determine how much to use the radiometric component in thepresence of impulse noise. If is the central pixel under con-sideration, and is a pixel in the neighborhood of ,we define the “joint impulsivity” of with respect to as

(9)

The function assumes values in . The parametercontrols the shape of the function. Again, any suitably nonneg-ative function that decreases to zero may be used in place of theGaussian. If at least one of or is impulse-like and has a highROAD value with respect to , then . If neitherpixel is impulse like, and, thus, neither has a high ROAD value,then .

We would like to use the radiometric weight more heavilywhen to smooth regions without large impulsesand less heavily when , because if either pixel isan impulse, the radiometric weight fails to function correctly asillustrated above. Conversely, we would like to use the impulsiveweight less heavily when and more heavily when

, to suppress large impulses. With this in mind,we define the final, “trilateral” weight of with respect to thecentral point as

(10)

Referring to the Gaussian forms of and , we see thatraising these functions to the specified exponents has the ef-fect of modifying their effective standard deviations or “thresh-olds.” When so that , the ra-diometric threshold becomes very large so that radiometric dif-ferences become irrelevant, while the impulsive weight is un-affected. When , the opposite happens and onlythe radiometric weight is used to distinguish pixels because theeffective impulsive threshold is so high. In this way, the appro-priate weighting function is applied on a pixel-by-pixel basis.We will call the nonlinear filter of form (7) with the weightingfunction given in (10) the “trilateral filter,” since it com-bines three different measures of neighboring pixels in deter-mining its weights.

In general, the trilateral weighting function works well to re-move impulse noise without compromising the bilateral filter’sability to remove Gaussian noise. For images with no impulsenoise—and, thus, few points with high ROAD values—the


Fig. 4. Original Lena image, the image corrupted with impulse noise (p = 20%), and the result after trilateral filtering.

term in (10) effectively “shuts off” the impulsive com-ponent of the weight and only uses the radiometric and spatialweights. Essentially, the trilateral filter reverts to the bilateralfilter when processing images without impulse noise.

The function also allows for future work to furtherextend the filter. Specifically, we are exploring methods to au-tomatically choose the control parameters andlocally. This would allow us to remove different types of noisefrom the same image, as well as mixed Gaussian and impulsenoise in a single pass.

We have found that even for very high levels of impulse noise,one pass of the trilateral filter will remove almost all of thenoise. However, for %, a few spots of unremoved im-pulses often remain. This happens because impulses sometimes“clump” together in the original noisy image to form regionsof similar intensity so large that they are mistaken for mean-ingful features. To remove such residual spots, it is often helpfulto process the image with an iterative version of the filter. Wesimply run the image through the trilateral filter several times,using the output of the previous iteration as the input of the next.For low and moderate levels of noise % , one iteration issufficient and usually provides the best results. For high levelsof noise % , applying two to five iterations providesbetter results.

C. Selection of Parameters

To remove Gaussian noise, we take . For suppressingimpulse noise, however, smaller values of , on the order of0.5, work best. We set to be about twice the standard devia-tion of the added Gaussian noise for automatically selecting .

In [14], Immerkær provides a fast method of estimating thestandard deviation of additive Gaussian noise in an image. Wemodify his method slightly by using a weighted average to dis-count pixels with high ROAD values. Our estimator for the stan-dard deviation of “quasi-Gaussian noise” exhibited by small im-pulses, , is given by

where

is the Laplacian filter.We have also found through experimentation that a good ini-

tial value of is 40. In general, should never be outside theinterval [25, 55]. Experimental results have shown that the op-timal value of for most images is around 50, but any value inthe interval [30, 80] should work well to remove both impulseand Gaussian noise.

D. Removing Mixed Gaussian and Impulse Noise

The trilateral filter can be easily extended to remove any mix-ture of Gaussian and impulse noise. The ideal solution would beto locally vary parameters so that they are finely tuned to removethe precise amount and type of noise present in each section ofthe image. This solution, however, would require a deep statis-tical study of the ROAD statistic for the automatic selection ofparameters, and the best way to do this is not immediately clear.

A simpler, yet still quite effective, solution to restore an imagecorrupted by mixed noise is to apply the trilateral filter twicewith two different values of —once with a smaller value of

, to remove the impulse noise, and another time with a largervalue of , to smooth the remaining Gaussian noise. A myriadof other options are available for altering the parameters be-tween filtering, but our simple approach produces visually ap-pealing results and only requires changing one parameter.

IV. RESULTS

We have extensively tested the noise removal capabilities ofthe trilateral filter and compared the results with several ex-isting methods. Our method produced results superior to theother methods we tested in both visual image quality and quan-titative measures of signal restoration. Among the many im-ages we tested are the 512 512 8-bit grayscale Lena image,available online from Mike Wakin at Rice University [15], andthe 512 512 “bridge” image, available online from the UEASignal and Image Processing Group [16].

A. Implementation and Testing Procedure

Our implementation of the trilateral filter used a 5 5window size and performed multiple iterations when it pro-vided better results (for %). Image boundaries werehandled by assuming symmetric boundary conditions.

In each of our experiments, we strove to be impartial whilecollecting data. For each method tested, we varied its param-eters exhaustively [as suggested by its author(s)] to obtain thebest possible result. Furthermore, to eliminate the bias created


Fig. 5. Lena image corrupted with a high level of impulse noise and the results of applying several filters.

Fig. 6. Comparing the trilateral filter and SD-ROM filter on images with Gaussian noise and mixed Gaussian and impulse noise.

by different manifestations of noise, we created a standard setof noisy images. Five noisy images were created for each testimage and noise level and formed the common input to eachmethod. The numerical results shown are the average results forthese five images.

B. Image Quality

Our first goal was to ensure that our approach provides visu-ally pleasing output. The trilateral filter can restore images cor-rupted with low to moderate levels of impulse noise %with virtually unblemished results. Fig. 4 shows the Lena imagecorrupted with 20% impulse noise and the result after trilateralfiltering. Comparing with the original, it is clear that the trilat-eral filter can eliminate a fair amount of noise while preservingedge boundaries and fine details.

Although exceptional for lower impulse noise levels, our filtertruly excelled when treating highly corrupted % im-ages. Fig. 5 shows the Lena image with 50% impulse noise andcompares the outputs of the trilateral filter with the output of theSD-ROM filter of Abreu et al. The trilateral filter’s output hasfewer spots and other artifacts and is generally more pleasing.

We also verified that the trilateral filter retains the ability toremove Gaussian noise and that it can effectively remove mixednoise. Fig. 6 shows the Lena image corrupted with Gaussiannoise and mixed noise and the outputs of the SD-ROM and tri-lateral filters. The trilateral filter continues to adequately sup-press Gaussian noise, even after the introduction of the impul-sive weight and the joint impulsivity function.

For both Gaussian and mixed noise, the trilateral filter leavesless noise in the restored image than the SD-ROM filter, and


TABLE IRESULTS AFTER APPLYING VARIOUS FILTERS TO IMAGES CORRUPTED WITH IMPULSE NOISE

its output is generally more visually appealing. However, somefine details are lost in the process. For example, the individualstrands of hair in the image are better kept in the SD-ROMfilter’s output, at the expense of a less smooth and less pleasingoutput.

C. Signal Restoration

Once the visual quality of images restored by the trilateralfilter had been confirmed, we concentrated on directly compa-rable, quantitative measures of signal restoration. In particular,we measured the peak signal-to-noise ratio (PSNR). If is theoriginal image and is a restored image of , the PSNRof is given by

(11)

Larger PSNR values signify better signal restoration. Wetested each method on impulse noise levels from % (thatis, no noise) to % in steps of 5%. Table I comparesthe mean PSNR values of the five restored test images for

% % % % . Again, the trilateral filter pro-vided results with higher PSNR values than the results of theother methods tested, especially for very high levels of noise.In particular, for the Lena image with 50% noise, the trilateralfilter produces PSNR values almost a full three decibels higherthan the closest competing method.

TABLE IIPSNR VALUES (IN DECIBELS) OF NOISY IMAGES

RESTORED WITH VARIOUS FILTERS

We also compared the performance of the trilateral filterwith the performance of the previously tested filters on imagescorrupted with Gaussian noise and mixed noise

% . The trilateral filter consistently yielded thehighest PSNR for each image and noise level. Table II showsthe results for the 3 3 median filter, the trilateral filter, and thepreviously-tested methods with training procedures: the RCRSand SD-ROM filters. Notice that the other methods generallyperformed only slightly better than the median filter for mixednoise.


V. CONCLUSION

Many noise removal algorithms, such as the bilateral filtering,tend to treat impulse noise as edge pixels, and, hence, end withunsatisfactory results. In order to process impulse pixels andedge pixels differently, we introduce a new statistic based onROAD in some neighborhood of a pixel. This statistic representshow impulse-like a particular pixel is in the sense that the largerthe impulse, the greater the ROAD value.

We then incorporate the ROAD statistic into the bilateralfiltering by adding a third component to the weighting function.The new nonlinear filter is called the trilateral filter, whoseweighting function contains spatial, radiometric, and impulsivecomponents. The radiometric component combined with thespatial component smooths away Gaussian noise and smallerimpulse noise, while the impulsive component removes largerimpulses. A switch based on the ROAD statistic is adopted toadjust weight distribution between the radiometric and impul-sive components. The resulting trilateral filter performs wellin removing Gaussian and mixed noise as well as in removingimpulse noise.

REFERENCES

[1] P. Perona and J. Malik, “Scale-space and edge detection usinganisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 12,no. 5, pp. 629–639, May 1990.

[2] C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color im-ages,” in Proc. IEEE Int. Conf. Computer Vision, 1998, pp. 839–846.

[3] K. J. Overton and T. E. Weymouth, “A noise reducing preprocessingalgorithm,” in Proc. IEEE Computer Science Conf. Pattern Recognitionand Image Processing, Chicago, IL, 1979, pp. 498–507.

[4] H. Lin and A. N. Willson Jr., “Median filters with adaptive length,” IEEETrans. Circuits Syst., vol. 35, no. 6, pp. 675–690, Jun. 1988.

[5] T. Sun and Y. Neuvo, “Detail-preserving median based filters in imageprocessing,” Pattern Recognit. Lett., vol. 15, pp. 341–347, Apr. 1994.

[6] A. C. Bovik, T. S. Huang, and D. C. Munson, “A generalization of me-dian filtering using linear combinations of order statistics,” IEEE Trans.Acoust., Speech, Signal Process., vol. ASSP-31, no. 6, pp. 1342–1350,Dec. 1983.

[7] R. C. Hardie and K. E. Barner, “Rank conditioned rank selection filtersfor signal restoration,” IEEE Trans. Image Process., vol. 3, no. 3, pp.192–206, Mar. 1994.

[8] G. Pok, J. Liu, and A. S. Nair, “Selective removal of impulse noise basedon homogeneity level information,” IEEE Trans. Image Process., vol.12, no. 1, pp. 85–92, Jan. 2003.

[9] S. Peng and L. Lucke, “Multi-level adaptive fuzzy filter for mixed noiseremoval,” in Proc. IEEE Int. Symp. Circuits Systems, vol. 2, Seattle, WA,Apr. 1995, pp. 1524–1527.

[10] E. Abreu, M. Lightstone, S. Mitra, and K. Arakawa, “A new efficient ap-proach for the removal of impulse noise from highly corrupted images,”IEEE Trans. Image Process., vol. 5, no. 6, pp. 1012–1025, Jun. 1996.

[11] M. Elad, “On the origin of the bilateral filter and ways to improve it,”IEEE Trans. Image Process., vol. 11, no. 10, pp. 1141–1151, Oct. 2002.

[12] R. L. Lagendijk, J. Biemond, and D. E. Boekee, “Regularized iterativeimage restoration with ringing reduction,” IEEE Trans. Acoust., Speech,Signal Process., vol. 36, no. 6, pp. 1874–1887, Dec. 1988.

[13] M. J. Black and G. Sapiro, “Edges as outliers: Anisotropic smoothingusing local image statistics,” in Proc. Scale-Space Theories in Comput.Vision Second Int. Conf., Corfu, Greece, Sep. 1999, pp. 259–270. LNCS1682.

[14] J. Immerkær, “Fast noise variance estimation,” Comput. Vis. Image Un-derstand., vol. 64, pp. 300–302, Sep. 1996.

[15] Standard Test Images—Lena, M. Wakin. [Online]. Available:http://www-ece.rice.edu/~/wakin/images/

[16] Standard Image Page [Online]. Available: http://www.sys.uea.ac.uk/ Re-search/researchareas/imagevision/images_ftp/

Roman Garnett received the B.S. degree in mathematics and the M.Sc. degreein computer science from Washington University in Saint Louis, Saint Louis,MO, in May 2004.

The work for this paper was completed while he was a Research Aide at TheInstitute for Computational Harmonic Analysis, University of Missouri–SaintLouis. His research interests include image and video processing and computervision.

Timothy Huegerich received the B.S. degree in physics from Rice University,Houston, TX, in May 2004.

The work for this paper was completed while he was a Research Aide at TheInstitute for Computational Harmonic Analysis, University of Missouri–SaintLouis. His research interests include mathematical ecology and epidemiology.

Charles Chui (F’94) received the B.S., M.S., and Ph.D. degrees from the Uni-versity of Wisconsin, Madison.

He is currently the Curators’ Professor at the University of Missouri–SaintLouis, and a Consulting Professor of statistics at Stanford University, Stan-ford, CA. He is the Co-Editor-in-Chief of Applied and Computational HarmonicAnalysis and serves on the editorial boards of seven other journals. In addition,he is the Editor of a three-book series, as well as the Editorial Advisor of theElsevier series in electrical engineering. His research interests are in the areas ofapproximation theory, computational harmonic analysis, surface subdivisions,and mathematics of imaging.

Wenjie He (M’04) received the B.S. degree in mathematics from Peking Uni-versity, Beijing, China, in 1988, and the Ph.D. degree in mathematics from theUniversity of Georgia, Athens, in 1998.

He is currently an Associate Professor of computer science at the Universityof Missouri–Saint Louis. His research interests include wavelets, wavelet tightframes, computer graphics, and image processing.

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