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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 5, MAY 2004 627

Adaptive Alpha-Trimmed Mean Filters UnderDeviations From Assumed Noise Model

Remzi Öten, Member, IEEE, and Rui J. P. de Figueiredo, Life Fellow, IEEE

Abstract—Alpha-trimmed mean filters are widely used forthe restoration of signals and images corrupted by additivenon-Gaussian noise. They are especially preferred if the un-derlying noise deviates from Gaussian with the impulsive noisecomponents. The key design issue of these filters is to select itsonly parameter, , optimally for a given noise type. In imagerestoration, adaptive filters utilize the flexibility of selectingaccording to some local noise statistics. In the present paper,we first review the existing adaptive alpha-trimmed mean filterschemes. We then analyze the performance of these filters whenthe underlying noise distribution deviates from the Gaussian anddoes not satisfy the assumptions such as symmetry. Specifically,the clipping effect and the mixed noise cases are analyzed.

We also present a new adaptive alpha-trimmed filter implemen-tation that detects the nonsymmetry points locally and appliesalpha-trimmed mean filter that trims out the outlier pixels suchas edges or impulsive noise according to this local decision. Com-parisons of the speed and filtering performances under deviationsfrom symmetry and Gaussian assumptions show that the proposedfilter is a very good alternative to the existing schemes.

I. INTRODUCTION

I N many signal and image processing applications, nonlinearfilters have been employed very effectively in removing non-

Gaussian noise present in the data. In particular, filters basedon Order Statistics [1] have been extensively used in noise re-moval applications [2], [3], such as in the presence of impulsivenoise. The most common and the simplest type of these filtersis the median filter. It shows very good performance for the re-moval of long-tailed noise types (e.g., Laplacian) and preservingthe edges. Another filter of this type is the alpha-trimmed meanfilter [4], [11], [12]. It is a good compromise between medianand moving average filter, which is known to be best for short-tailed noise types (e.g., Gaussian). For all three filters, the his-tograms of the restored signals are shown in Fig. 3(c)–(e) whenthe noise variance is equal to 0.065.

Alpha-trimmed Mean Filters:Let , where

be a set of n sample signal values observed in a window, . Ifthese values are arranged in ascending order of their amplitude,the order statistics result is

(1)

Manuscript received November 9, 1999; revised December 11, 2002. Thiswork was supported by the NSF under Grant CCR-9704262. The associate ed-itor coordinating the review of this manuscript and approving it for publicationwas Prof. Scott T. Acton.

R. Öten is with IC-Media Corp., Santa Clara, CA 95050 USA.R. J. P. de Figueiredo is with the Department of Electrical and Computer

Engineering, University of California at Irvine, Irvine, CA 92697-2625 USA(e-mail: [email protected]).

Digital Object Identifier 10.1109/TIP.2003.821115

where is the minimum, is the maximum, andis the median of the above set of signal values. The

output of the alpha-trimmed mean filter, is

(2)

where [.] denotes the greatest integer part and . As itis easily seen from (2), indicates the percentage of the trimmedsamples. Therefore, the alpha-trimmed mean filter performs likea median filter when is close to 0.5, and moving average filterwhen is close to 0. If we drop the time index and denote thetrimmed-mean filter by , then the moving average filter is

. Although never gets equal to 0.5, for simplicity we willrepresent median filter as since is very close to 0.5 forthis filter.

The conventional type of trimmed-mean filters that are men-tioned in this paper are also called ’inner trimmed-mean fil-ters’ since the outputs of these filters are produced by averagingthe ’inner’ sample values (the ones close to the median). Forthe noise pdf’s, which have shorter tail lengths than that of aGaussian pdf, outer trimmed-means are preferred. The outputof an outer trimmed-mean filter, is equal to the averageof the ’outer’ sample values(the ones close to the extremes). Anexample of this is the mid-point filter, which is the maximumlikelihood estimate of location for uniform distribution. Mid-point filter trims out all the values except the extremes and canbe represented as . The moving average filter output canalso be represented as an outer trimmed-mean,i.e., . Here,for simplicity we will consider only inner trimmed-means, ex-tensions to outer trimmed-means being trivial.

Alpha-trimmed mean estimator is intended to be used for lo-cation estimation when the Gaussian data distribution containssome outliers. For these cases, they outperform other nonlinearfilters in simplicity and noise reduction. In this paper, we focuson the generalization of this case and evaluate the robustnessof the adaptive schemes that utilize alpha-trimmed mean fil-ters when the noise pdf deviates from assumed Gaussian model.First, we review the existing adaptive trimmed-mean filters inSection II. Then, in Section III, compare their filtering perfor-mances when the underlying noise distribution deviates from theGaussian and symmetry assumptions due to the clipping effectsand impulsive noise.

In Section IV, we introduce a new adaptive trimmed filterscheme by utilizing the symmetry conditions of the asymptoticdistribution. This filter detects the nonsymmetry points locallyand applies alpha-trimmed mean filter that trims out the outlierpixels such as edges or impulsive noise according to this localdecision. This type of adaptation make this filter an “open-loop”

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628 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 5, MAY 2004

adaptive filter as opposed to “closed-loop” adaptive filters thatutilizes feedback [14]. All the filters considered in this paperuse open-loop adaptation. Open-loop adaptive filters do not useprevious estimates as an input, therefore, do not have conver-gence problems. However, in case of bad estimation schemes,they will perform very poorly.

In Section V, the proposed filter is compared with the well-performing adaptive trimmed mean filters of Section II and isshown to be a good alternative to these filters since it can achievecomparable results with considerably less complexity.

II. ADAPTIVE FILTERS

The design problem of alpha-trimmed mean filters reducesto selecting the “best” value for a given noise pdf. However,the selection of the best value may not be possible when thenoise pdf is not known or varies with time. For these cases,one has to design an adaptive filter that changes its value ac-cording to some characteristics of the observed sample data. Instill image procesing, adaptation can be done according to localnoise statistics in a window. There are two types of filteringalgorithms that have been proposed previously [5], [7]. Thesetwo and Jaeckel’s adaptive estimation scheme [8] are briefly re-viewed below [8] in the context of image restoration filters.

A. Filter I

In [5], an adaptive scheme is proposed which selects thebest from a previously determined collection of valuesaccording to some selector statistics whose threshold values aredetermined by Monte-Carlo analysis. The output of this filterat window is,

ifwhereifwhere

where the selector statistic, , is a robust tail length es-timate of the underlying noise pdf. In other words, adaptivefilter performs one of the s outer mean filters, or one of the

inner-mean filters according to the result of tail-length esti-mation of the local pdf in window . The description and theproperties of can be found in [5] and [6].

In [5], only five simple filters are employed:

:::::

(3)

where the threshold values are selected via MonteCarlo simulation with the assumption that the underlying noisepdf can be modeled by the family of generalized exponentialpdfs.

The adaptive filter in (3) is easy to implement once thethreshold values are selected. However, since it has to selectfrom a set of few estimators, it cannot completely utilizethe flexibility of alpha-trimmed mean filters. Although itis possible to increase the number of estimators, this willnot improve the filter performance since there is no precise

rigorous relation between the estimators and the thresholdvalues. Since these thresholds are selected empirically fromgeneralized exponential pdfs, this filter will not give acceptableresults if the noise pdf is not in symmetric unimodal shape.Its performance is heavily dependent on the edge-detectionscheme proposed in [5].

B. Filter II

In [7], [13], another adaptive alpha-trimmed mean filter struc-ture is proposed. Unlike [5] this filter utilizes the flexibility ofalpha-trimmed means completely by allowing the use of wholerange (i.e., ). The output of this filter at window

is:

(4)

Here is the activity of the original signal in window anddefined as:

(5)

where

: if: otherwise

where is the sample variance of the signal values in ,and is the variance of the noise in .

The main disadvantage of the above filter is that the noisevariance is assumed to be known for all . This assumption isactually against the nature of adaptive filtering that we consider.It may be possible to use an estimate of the noise variance, butthe performance will be highly affected by the error of the es-timate, therefore a robust estimation scheme is critical for thisfilter.

C. Filter III

Neither of the above two schemes uses a rigorous approachthat is based on an optimization criteria. In this paper, we alsoconsider Jaeckel’s estimator [8] which offers a more rigoroussolution but has never been analyzed in a signal and image pro-cessing scenario. Jaeckel’s proposal is based on the minimiza-tion of the asymptotic variance estimate of the alpha-trimmedmean estimator. In the present paper, it is implemented as anadaptive alpha-trimmed mean filter. In Section III, we compareJaeckel’s filter’s performance with Filter I and Filter II whenthe underlying noise pdf deviates from Gaussian. In particular,we examine the cases where clipping modifies the shape of thepdf and where the Gaussian noise is mixed with impulsive typenoise. Their behavior both at the smooth regions and the edgeregions is analyzed.

1) Asymptotic Properties of Alpha-Trimmed Means: Letbe a sample of independent, identically

distributed random variables with a common symmetric dis-tribution , and let denote the orderstatistics. The alpha-trimmed mean is given by

(6)

ÖTEN AND DE FIGUEIREDO: ADAPTIVE ALPHA-TRIMMED MEAN FILTERS UNDER DEVIATIONS FROM ASSUMED NOISE MODEL 629

With the assumption that and are unique,it is shown in [9] that (2) is an asymptotically normal esti-mator,i.e.,

(7)

where

(8)

and

(9)

Asymptotic alpha-trimmed mean estimator, , is optimizedby selecting an such that,

(10)

Now, we will present the utilization of this approach inJaeckel’s adaptive alpha-trimmed mean filter for moderatenumber of samples inside a given window, which is usually thecase in signal and image restoration applications.

2) Adaptive Filter Based on Jaeckel’s Estimator: Letbe the observed data from

a local neighborhood of a corrupted signal. It is assumed thatthe original signal, , is locally constant and the noise, ,is additive and symmetric, i.e., . The orderstatistics of these sample points are as in (1). Therefore, theasymptotic variance of alpha-trimmed mean, ,can be estimated by the Winsorized variance:

(11)

This means that the asymptotic variance estimate of the alpha-trimmed mean can be computed from the observed samples.

Therefore one can find an optimum alpha-trimmed mean byselecting an that minimizes the variance of the filter output.An adaptive alpha-trimmed mean filter that selects the bestvalue, , which minimizes the sample asymptotic varianceestimate at a local window can be written as:

(12)

where

Fig. 1. Block diagram of Jaeckel’s adaptive alpha-trimmed meanfilter. fx (i)g denotes the order statistics variables, fx (i); x (i);. . . ; x (i)g.

Fig. 2. The clipping effect on Filter I, Filter II, and Filter III is shown withaverage signal-to-noise ratios versus variance of the underlying Gaussian noise.

Minimization can be done by searching the best value in afixed interval, i.e., , such thatis an integer and computing for these values of , i.e.,for , , and , will be selected

among . In [8], it is shown that selected thisway is asymptotically equivalent to of (10).

In this procedure, starting from the largest value canreduce the computation time since for that value the number ofterms in the sums of (2) and (6) is the smallest. These sums arein this form:

(13)

where and corresponds to sorted data samples.Therefore, successive values can be computed by

(14)

which needs only two additions at a time.In this filter, , is defined similar to (2) where

and the general structure is shown in Fig. 1.The performance comparisons of Filter I, Filter II, and the

Filter III under clipping effect and mixed noise are presented inSection III. In this section, behaviors of these filters are exam-ined on constant and edge regions separately.


Fig. 3. Histogram of (a) constant 2-D signal region and (b) constant signal in (a) corrupted with Gaussian noise (� = 0:065). (c) Output of Filter I, (d) outputof Filter II, and (e) output of the Filter III.

III. PERFORMANCE UNDER DEVIATIONS FROM ASSUMED

NOISE MODEL

The most significant feature of an alpha-trimmed mean filteris its robustness against outliers. In images, outliers can be de-fined as impulses in a local window generated by an impulsivenoise source. One can simply choose a median filter to reject upto 50% of the outliers. However, this is not the best choice if theadditive noise is the mixture of a short-tailed noise and an im-pulsive-type noise. In these cases, alpha-trimmed mean filterswith the right values will generate better results.

The noise present in many applications can be modeled as anadditive white Gaussian noise. In images, the regions betweenedges can be filtered successfully by local averaging if the noiseis this type. However, in many cases the noise pdf deviates fromits Gaussian shape due to different distortions such as clippingand/or impulsive noise. In Sections III-A and B, we will con-sider these cases.

In the following simulations, for Filter I is taken 0.05 aspresented in [5]. For Filter II, noise variance, is estimatedbefore filtering by,

(15)


where is the total number of windows considered, and MADis the median of absolute deviations from median, which is avery robust scale estimate with breakdown point equal to 0.5.For details on MAD see ([10]). It is scaled with 1.483 to produceunbiased estimates for Gaussian distribution. In other words,(15) computes the local variance estimates at window by

for and selects the median ofthe M estimates as the noise variance, . This is a costly butvery robust estimate. As you will see, it is very important for thisfilter to have a close estimate of the noise variance by discardingthe outliers.

For Filter III, is taken as 0, and is taken as 0.45. Takingthe full range (i.e., and ) usually reduces theperformance of this filter.

A. Clipping

Clipping occurs when the pixel intensity level exceedsthe predetermined boundary values because of the limitationof the bit precision. For example, 8-bit pixel representationallows maximum pixel value of 255. This problem is moregeneral but we will consider the cases where the added noisecauses the pixel values to get higher (lower) values than themaximum(minimum) intensity level. An example is shownin Fig. 3(a) and (b) where the minimum intensity level is 0and the maximum intensity level is 1. While Fig. 3(a) showsthe histogram of the constant signal with pixel values equalto 0.75., Fig. 3(b) shows the histogram of the same signalcorrupted by a Gaussian noise with variance 0.065. The pixelvalues greater than 1 are clipped and set to 1. As it can be easilyseen from this figure the shape of the noise pdf significantlydeviates from that of Gaussian.

The performances of the above mentioned adaptive alpha-trimmed mean filters are measured with their signal-to-noiseratios with respect to different variances of the Gaussian noise.The signal is the constant signal with pixel intensities equal to0.75. The noise variance takes values from 0.005 to 0.065. Thewindow sizes of the filters are 5 5.

In this and the other experiments, the signal-to-noise ratiosare defined as:

(16)

where is the pixel value of the nonnoisy original signalat the -th row and -th column, and is the pixel value ofthe restored signal at that location.

The results are shown in Fig. 2. The signal-to-noise ratiosdisplayed are the average SNRs computed from the filteredoutputs of 30 noisy rectangular image regions. Although, theFilter I shows very good performance at lower noise variances,it drops suddenly and decreases dramatically after the variancegets values higher than 0.03. This can be easily explained by theclipping effect. The thresholds of Filter I are static and selectedwith the assumption of the unimodal symmetric noise. Thefilter is very sensitive to the deviations from this assumption,therefore when the clipping ratio increased the performance ofthis filter decreased rapidly.

Fig. 4. Average SNRs of filter outputs vs the density of the impulses of mixednoise, when applied to corrupted constant 2-D regions.

On the other hand, Filter II and Filter III show more stablebehavior against the increase of the clipping ratio. These filtersare shown to be more robust in this case. Filter II has a slightlybetter performance over Filter III for all values of the noise vari-ance.

Although, there are slight differences among their averageSNR values, all three filters show similiar performances whenthere is no clipping effect.

For all three filters, the histograms of the restored signals areshown in Fig. 3(c), (d), and (e) when the noise variance is equalto 0.065.

B. Mixed Noise

Secondly, we measure the performance of these filters undermixed noise. The noise considered here consists of additivewhite Gaussian noise and impulsive noise. Particularly, weexamined the case when the impulsive noise is salt & peppernoise. As it is clear from its name, this noise converts someof the pixels into very dark (low intensity) and very light(high intensity) pixels. In other words, the impulses appear asminimum and maximum intensity values.

Performance on Constant Regions: In images, the pixel valuesof the regions between edges vary slowly. In many cases theseregions can be assumed to have locally constant signal values ifthe window size is taken small enough. In this part of the paper,we examine the performance of the above filters on constantregions.

While Fig. 5(a) shows a constant 2-D signal andits histogram, Fig. 5(b) shows the same signal corrupted witha mixed Gaussian and salt & pepper noise along with its his-togram. Here, the Gaussian noise variance is 0.005 and impulsedensity is equal to 0.5. The Gaussian shape centered around 0.75can be easily seen. The impulses of the salt & pepper noise areshown at the two ends of the histogram. In this case the mixednoise pdf can be written as:


Fig. 5. The 2-D signal and histogram of (a) constant signal (s = 0:75) and (b) constant signal in (a) corrupted with mixed Gaussian (� = 0:005) and salt &pepper (impulse density = 0.5) noise. (c) Output of Filter I.

where is the Gaussian noise pdf and is the salt & peppernoise pdf. To be able to examine the effect of impulsive content,we computed the average signal-to-noise ratios of the three filteroutputs while increasing the impulse density of the salt & peppernoise from 0.05 to 0.5. The signal is the same 2-D signal shownin Fig. 5(a) and the variance of the Gaussian noise is 0.005 forall cases. The results plotted in Fig. 4 show that the Filter I is af-fected by the impulsive content much more than the other two.As explained in the previous case, this filter is very sensitive todeviations from unimodal symmetric noise pdf assumption. Asthe impulse density increases, the two peaks at the two ends ofthe pdf dominate in the overall shape. Therefore, the selector

statistic of Filter I takes values as if the overall noise pdf is veryshort tailed and symmetric due to the effect of these peaks. Sincethis filter selects the midpoint filter for very short-tailed noisepdfs, the output will be the midpoint between the highest inten-sity value and the lowest intensity value. The output of Filter Iand its histogram for impulse density 0.5 is shown in Fig. 5(c).The peak at 0.5 of the this histogram justifies our above argu-ment. Although this filter produces very smooth output, it altersthe signal level.

Filter II, on the other hand, produces very good results com-pared to Filter I. However, this filter’s performance is also verysensitive to the estimation of the noise variance. To be able to


Fig. 5. (Continued.) (d) Output of Filter II, and (e) output of the Jaeckel’s filter.

remove impulses, variance estimation should not be affected bythese impulses. With a breakpoint of 0.5, is avery robust variance estimator, since it can reject upto 50% ofthe outliers. However, even MADs performance decays whenthe impulse density increases substantially. The effect of this onthe filter’s average output SNR can be seen in Fig. 4 when theimpulse density is around 0.5, which is the breakpoint of thisestimator. Unlike Filter III, the performance of Filter II dropsrapidly at this point. This indicates that Filter II is very sensitiveagainst false estimation of the variance.

The outputs and their histograms in Fig. 5(d) and (e) alsogives the idea about their success at smoothing during the pres-ence of mixed Gaussian and high impulsive noise.

It can be concluded that, overall, Filter III is more robust thanthe other two when the mixed noise consists of impulses.

Performance on the Edges: In image processing, the preser-vation of edges is as important as smoothing the slowly varyingregions. Therefore, the rest of this section evaluates the perfor-mance of Filter I, Filter II, and Filter III on a step edge corruptedwith mixed Gaussian and impulsive noise. The tools that we useto measure and present the results are same as the constant signalcase analyzed above.

In [5] and [7], special components such as edge detector [5]and double-window [7] is offered to operate on edges. However,we have not included these components since we are interestedonly in the performance of the bare filters. These componentscan be added any time to any of these filters.

Fig. 6. Average SNRs of Filter I, Filter II, and Filter III outputs versus impulsedensity of the mixed noise on a 2-D step edge.

The step edge we consider and its histogram representation isshown in Fig. 7(a). This is an edge with a height of 0.25 and awidth of 10 pixels. The peak on the left of the histogram repre-sents the lower pixel values, and the one on the right representsthe higher pixel values.

The corrupted signal and its histogram is presented inFig. 7(b). There are still two peaks at the same location as


Fig. 7. The 2-D signal and histogram of (a) step edge and (b) signal in (a) corrupted with mixed Gaussian (� = 0:005) and salt & pepper (impulse density =0.5) noise.

Fig. 7(a), however, they are Gaussian shaped because of theGaussian component of the noise. As expected, the peaksresulting from the impulsive component are also shown in thefar left and far right of the histogram.

For this case the average signal-to-noise ratios show distinc-tive features (Fig. 6) for all three filters. The behavior of FilterI is similar to the constant signal case. Even for the very lowimpulse densities, this filter output has a lower SNR than theother two. The reason for this can be explained by the bi-modaldistribution of the signal itself at the edges. The output of thisfilter at impulse density 0.5 and Gaussian noise variance 0.005is shown Fig. 7(c). Again, mostly by the contribution of lightand dark impulses the midpoint is selected as the output of thefilter by destroying the edge.

Filter II’s performance drops significantly when the impulsedensity gets closer to 0.5, which is the breakdown point of thenoise variance estimator. The output of this filter and its his-togram when the impulse density is equal to 0.5 is shown inFig. 7(d). In this figure, it can be easily seen that two level struc-ture of the edge is disappeared. Although this filter shows verygood performance for lower impulse densities, it destroys theedges for higher impulse densities.

Finally, on a step edge Filter III shows very robust perfor-mance against the increase in the impulse density. The twolevel structure of the filter output and its histogram is shown inFig. 7(e). Filter III preserves the edges better than Filter I andFilter II under the high impulsive noise content.

IV. A NEW ADAPTIVE FILTER IMPLEMENTATION

From the comparisons done in Sections II and III, we can saythat Filter II and Filter III show comparable performances. How-ever, actual processing times for these filters are long. The com-plexity of Filter II comes from the fact that it requires a goodestimation of the underlying noise variance. Filter III is com-plex because it requires the computation of asymptotic varianceestimate several times per pixel. Here, we propose another adap-tive alpha-trimmed mean filter which has a strightforward im-plementation with less complexity and shows comparable per-formance with Filter II and Filter III.

A. Theory

The idea behind our implementation can be seen from anasymptotic analysis of noise samples from a symmetric distri-bution. As presented before, the asymptotic representation ofalpha-trimmed mean is:

(17)

The meaning of this can be easily seen from Fig. 8. It cor-responds to the ratio of the area between the -axis and the

curve, between points and to thedistance between these points.


Fig. 7. (Continued.) (c) Output of Filter I, (d) output of Filter II, and (e) output of the Filter III.

If the distribution is symmetric between andand corresponds to the median of the distribution, then

shifts the curve horizontally to the line.Therefore, if a distribution is symmetric between and

and corresponds to the median of the distri-bution then

(18)

will be equal to zero.One can easily verify that the above formula corresponds to

the derivative of , i.e.,

(19)

Therefore, the condition for symmetry between and,

(20)

in asymptotic case can be represented as

(21)

for small and moderate sample sizes. In our filter, we put a con-dition on to satisfy:

(22)


Fig. 8. Plot of a cumulative distribution function of a symmetric distribution.M(�) corresponds to the ratio of the area between the y-axis and the F (�)curve, between points y = � and y = 1 � � to the distance between thesepoints.

TABLE IFILTERING RESULTS OF CONSTANT IMAGE

REGIONS CORRUPTED BY GAUSSIAN NOISE

TABLE IIFILTERING RESULTS OF THE STEP EDGE CORRUPTED WITH GAUSSIAN NOISE

TABLE IIIFILTERING RESULTS OF A CONSTANT IMAGE REGIONS CORRUPTED WITH

MIXED NOISE

where is a constant threshold. Based on this, the output of theadaptive filter we present is , where is computedlocally from the data samples:

(23)

TABLE IVFILTERING RESULTS FOR ZELDA IMAGE

TABLE VFILTERING RESULTS FOR PEPPERS IMAGE

Fig. 9. Processing times of the adaptive alpha-trimmed mean filters for“Zelda” and “Peppers” images.

In other words, with this scheme, we would like to find theminimum value that will only utilize samples from the sym-metric middle portion of the distribution while trimming therest.

The value of the threshold, , determines how much we willtolerate against deviations from the symmetry condition.

The algorithm to compute from the sorted samplesthru is explained in Section IV-B.

B. Algorithm

To find the minimum value that satisifies (23), algorithmstarts with the maximum value and decreases it until it failsto satisfy (23).


(a) (b) (c)

(d) (e)

Fig. 10. (a) A portion of the “Zelda” image corrupted with mixed Gaussian (� = 0:005) and salt & pepper noise (impulse density = 0.4). Outputs of (b) FilterI, (c) Filter II, (d) Filter III, and (e) the proposed filter.

Let the sample size, . In this case, repre-sents the median of the samples. Also, let us represent the sumsof trimmmed samples as:

...

(24)

The corresponding averages are represented as, . Note that the value decreases

as increases. is nothing but the -trimmed mean withdiscrete values.

The consequent algorithm is:

Step 0.Step 1. While andStep 1.1 ComputeStep 1.2Step 2.


(a) (b)

(c) (d)

(e)

Fig. 11. (a) “Peppers” image corrupted with mixed Gaussian (� = 0:005) and salt & pepper noise (impulse density = 0.4). Outputs of (b) Filter I, (c) Filter II,(d) Filter III, and (e) the proposed filter.

This algorithm does not say anything about the computationof s. An efficient way is to utilize the fact that canbe derived from :

(25)

and .For some software and hardware architectures it may be better

to compute s of all the pixels that fit intomemory and process them like a vector processing.

V. PERFORMANCE OF THE PROPOSED FILTER

In this section, we compare the performance of the pro-posed filter with Taguchi’s filter (Filter II) and the Jaeckel’sfilter(Filter III). Here, the parameters of these filters are takenas previous simulations and threshold, of the proposed filteris taken as 0.5. All the filters used a 5 by 5 sliding window.Filter I is omitted from the comparisons because of its poor


performance against the deviations from the assumed noisemodel in Section III.

The first comparison is done on constant image regions cor-rupted with Gaussian noise that results clipping. Table I presentsthe results of this comparison, where first row shows the dif-ferent noise variances.

The next comparison is done on step edges of 20 pixel width,which are corrupted by Gaussian noise. The SNRs are com-puted against different noise variances. The results are shownin Table II.

In the next comparison, constant image regions corrupted bymixed Gaussian ( ) and salt and pepper noise withdifferent impulse density are filtered with these three filters. theresults are shown in Table III.

Finally, we applied these three filters to the “Zelda” imageand “Peppers” image that are corrupted with mixed noise de-scribed as in the previous simulation. The SNRs are shown inTables IV and V and the results are shown in Figs. 10 and 11.

These results show that the proposed filter shows comparableperformance to the other filters.

The speed comparisons of these filters are done on a PCwith Pentium III 500 Mhz Intel processor. The filters are im-plemented on a MATLAB 6.0 platform. The processing time ofthese filters are measured for “Zelda” and “Peppers” images.The results are plotted in Fig. 9.

VI. CONCLUSION

In this paper, we have provided an overview of the adap-tive alpha-trimmed mean filters and examined their performancewhen the underlying noise deviates from Gaussian noise model.It is shown that the adaptive filter suggested by Restrepo andBovik [5] (Filter I) is very sensitive to deviations from the sym-metric unimodal noise assumption. Although this filter is fastand easy to implement and produces very good results for uni-modal symmetric noise distribution, it does not perform well forcertain cases such as the ones in Section IV.

The adaptive filter suggested by Taguchi [7], in general,shows a very good performance. However, this filter requiresthe variance of the noise distribution as an input. This can beestimated but the estimator should be very robust against out-liers (impulses and edges). The MAD estimator we suggestedis very robust but it makes this filter computationally verycomplex. Taguchi’s filter (Filter II) is to be preferred over theother filters when the noise variance is known.

We employed the Jaeckel’s estimator as an adaptive alpha-trimmed mean filter for image restoration and observed its per-formance under certain conditions. This filter seems to be veryrobust against the noise model deviations such as clipping andhigh impulsive content as well as jumps (edges) of the signalitself. It performs better if the possible range of values canbe taken smaller depending on the possible shapes of the under-lying noise class. Jaeckel’s filter (Filter III) is to be preferredover the others when there is highly impulsive noise along witha short-tailed noise. However, this filter also suffers from com-plexity since it requires computation of the asymptotic varianceseveral times per pixel.

We also presented a new adaptive filter implementation whichcan be a good compromise between output quality and speed.This filter’s noise removal performance under the deviationsfrom the assumed noise models is comparable to Filter II andFilter III, while its complexity is low as in the case of Filter I.

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Remzi Öten (M’94) received the B.S. degree in electrical and electronics engi-neering from Bilkent University, Ankara, Turkey, in 1992. He received the M.S.and Ph.D. degrees in electrical and computer engineering from University ofCalifornia, Irvine, in 1993 and 1999, respectively.

Since 1999, he has been working as a Systems Engineer and Consultant in thearea of signal processing with emphasis to multimedia systems. He has written13 journal and conference papers in this area.

Dr. Oten is the recipient of the University of California Regents’ Fellowshipand the 2003 IEEE Circuits and Systems Transactions Guillemin-Cauer BestPaper Award.

Rui J. P. de Figueiredo (LF’94) received the B.S. and M.S. degrees in elec-trical engineering from MIT, and the Ph.D. degree in applied mathematics fromHarvard University.

After being several years on the faculty of Rice University, Houston, TX, hejoined the University of California, Irvine, in 1990, where he holds the posi-tion of Professor of electrical engineering and computer science, of bio-medicalengineering, and of mathematics. He has published more than 370 papers, andchaired or co-chaired nine national or international conferences.

Dr. de Figueiredo was the President of the IEEE Circuits and Systems So-ciety in 1998, and served on several national and international committees andpanels. For all these contributions, he received a number of awards, includingthe IEEE Fellow award, the 1994 IEEE Circuits and Systems Society TechnicalAchievement Award, this society’s 2002 M. E. Van Valkenburg Society Award,and its 1999 Golden Jubilee Medal, the IEEE Third Millennium Medal, the Gh.Asachi Medal from the Technical University of Iasi, Romania, the 2000 IEEENeural Networks Transactions Outstanding Paper Award, and the 2003 IEEECircuits and Systems Transactions Guillemin-Cauer Best Paper Award.

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