Adaptive Alpha-Trimmed Mean Filters Under Deviations ... - IEEE Xplore
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 5, MAY 2004
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Adaptive Alpha-Trimmed Mean Filters Under Deviations 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 for the restoration of signals and images corrupted by additive non-Gaussian noise. They are especially preferred if the underlying noise deviates from Gaussian with the impulsive noise components. The key design issue of these filters is to select its only parameter, , optimally for a given noise type. In image restoration, adaptive filters utilize the flexibility of selecting according to some local noise statistics. In the present paper, we first review the existing adaptive alpha-trimmed mean filter schemes. We then analyze the performance of these filters when the underlying noise distribution deviates from the Gaussian and does 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 implementation that detects the nonsymmetry points locally and applies alpha-trimmed mean filter that trims out the outlier pixels such as edges or impulsive noise according to this local decision. Comparisons of the speed and filtering performances under deviations from symmetry and Gaussian assumptions show that the proposed filter is a very good alternative to the existing schemes.
I. INTRODUCTION
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N many signal and image processing applications, nonlinear filters have been employed very effectively in removing nonGaussian noise present in the data. In particular, filters based on Order Statistics [1] have been extensively used in noise removal applications [2], [3], such as in the presence of impulsive noise. The most common and the simplest type of these filters is the median filter. It shows very good performance for the removal of long-tailed noise types (e.g., Laplacian) and preserving the edges. Another filter of this type is the alpha-trimmed mean filter [4], [11], [12]. It is a good compromise between median and moving average filter, which is known to be best for shorttailed noise types (e.g., Gaussian). For all three filters, the histograms of the restored signals are shown in Fig. 3(c)–(e) when the 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, . If these values are arranged in ascending order of their amplitude, the order statistics result is (1) Manuscript received November 9, 1999; revised December 11, 2002. This work was supported by the NSF under Grant CCR-9704262. The associate editor coordinating the review of this manuscript and approving it for publication was 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
is the minimum, is the maximum, and is the median of the above set of signal values. The is output of the alpha-trimmed mean filter,
where
(2) . As it where [.] denotes the greatest integer part and is easily seen from (2), indicates the percentage of the trimmed samples. Therefore, the alpha-trimmed mean filter performs like a median filter when is close to 0.5, and moving average filter when is close to 0. If we drop the time index and denote the , then the moving average filter is trimmed-mean filter by . Although never gets equal to 0.5, for simplicity we will since is very close to 0.5 for represent median filter as this filter. The conventional type of trimmed-mean filters that are mentioned in this paper are also called ’inner trimmed-mean filters’ since the outputs of these filters are produced by averaging the ’inner’ sample values (the ones close to the median). For the noise pdf’s, which have shorter tail lengths than that of a Gaussian pdf, outer trimmed-means are preferred. The output is equal to the average of an outer trimmed-mean filter, of the ’outer’ sample values(the ones close to the extremes). An example of this is the mid-point filter, which is the maximum likelihood estimate of location for uniform distribution. Midpoint filter trims out all the values except the extremes and can . The moving average filter output can be represented as . Here, also be represented as an outer trimmed-mean,i.e., for simplicity we will consider only inner trimmed-means, extensions to outer trimmed-means being trivial. Alpha-trimmed mean estimator is intended to be used for location estimation when the Gaussian data distribution contains some outliers. For these cases, they outperform other nonlinear filters in simplicity and noise reduction. In this paper, we focus on the generalization of this case and evaluate the robustness of the adaptive schemes that utilize alpha-trimmed mean filters when the noise pdf deviates from assumed Gaussian model. First, we review the existing adaptive trimmed-mean filters in Section II. Then, in Section III, compare their filtering performances when the underlying noise distribution deviates from the Gaussian and symmetry assumptions due to the clipping effects and impulsive noise. In Section IV, we introduce a new adaptive trimmed filter scheme by utilizing the symmetry conditions of the asymptotic distribution. This filter detects the nonsymmetry points locally and applies alpha-trimmed mean filter that trims out the outlier pixels such as edges or impulsive noise according to this local decision. This type of adaptation make this filter an “open-loop”
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adaptive filter as opposed to “closed-loop” adaptive filters that utilizes feedback [14]. All the filters considered in this paper use open-loop adaptation. Open-loop adaptive filters do not use previous estimates as an input, therefore, do not have convergence problems. However, in case of bad estimation schemes, they will perform very poorly. In Section V, the proposed filter is compared with the wellperforming adaptive trimmed mean filters of Section II and is shown to be a good alternative to these filters since it can achieve comparable results with considerably less complexity. II. ADAPTIVE FILTERS The design problem of alpha-trimmed mean filters reduces to selecting the “best” value for a given noise pdf. However, the selection of the best value may not be possible when the noise pdf is not known or varies with time. For these cases, one has to design an adaptive filter that changes its value according to some characteristics of the observed sample data. In still image procesing, adaptation can be done according to local noise statistics in a window. There are two types of filtering algorithms that have been proposed previously [5], [7]. These two and Jaeckel’s adaptive estimation scheme [8] are briefly reviewed below [8] in the context of image restoration filters. A. Filter I In [5], an adaptive scheme is proposed which selects the best from a previously determined collection of values according to some selector statistics whose threshold values are determined by Monte-Carlo analysis. The output of this filter is, at window if where if where
B. Filter II In [7], [13], another adaptive alpha-trimmed mean filter structure is proposed. Unlike [5] this filter utilizes the flexibility of alpha-trimmed means completely by allowing the use of whole ). The output of this filter at window range (i.e., is: (4) Here is the activity of the original signal in window defined as:
and (5)
where : if : otherwise is the sample variance of the signal values in , where is the variance of the noise in . and The main disadvantage of the above filter is that the noise variance is assumed to be known for all . This assumption is actually against the nature of adaptive filtering that we consider. It may be possible to use an estimate of the noise variance, but the performance will be highly affected by the error of the estimate, therefore a robust estimation scheme is critical for this filter. C. Filter III
where the selector statistic, , is a robust tail length estimate of the underlying noise pdf. In other words, adaptive filter performs one of the s outer mean filters, or one of the inner-mean filters according to the result of tail-length esti. The description and the mation of the local pdf in window can be found in [5] and [6]. properties of In [5], only five simple filters are employed: : : : : :
rigorous relation between the estimators and the threshold values. Since these thresholds are selected empirically from generalized exponential pdfs, this filter will not give acceptable results if the noise pdf is not in symmetric unimodal shape. Its performance is heavily dependent on the edge-detection scheme proposed in [5].
(3)
where the threshold values are selected via Monte Carlo simulation with the assumption that the underlying noise pdf can be modeled by the family of generalized exponential pdfs. The adaptive filter in (3) is easy to implement once the threshold values are selected. However, since it has to select from a set of few estimators, it cannot completely utilize the flexibility of alpha-trimmed mean filters. Although it is possible to increase the number of estimators, this will not improve the filter performance since there is no precise
Neither of the above two schemes uses a rigorous approach that is based on an optimization criteria. In this paper, we also consider Jaeckel’s estimator [8] which offers a more rigorous solution but has never been analyzed in a signal and image processing scenario. Jaeckel’s proposal is based on the minimization of the asymptotic variance estimate of the alpha-trimmed mean estimator. In the present paper, it is implemented as an adaptive alpha-trimmed mean filter. In Section III, we compare Jaeckel’s filter’s performance with Filter I and Filter II when the underlying noise pdf deviates from Gaussian. In particular, we examine the cases where clipping modifies the shape of the pdf and where the Gaussian noise is mixed with impulsive type noise. Their behavior both at the smooth regions and the edge regions is analyzed. 1) Asymptotic Properties of Alpha-Trimmed Means: Let be a sample of independent, identically distributed random variables with a common symmetric disdenote the order tribution , and let statistics. The alpha-trimmed mean is given by (6)
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With the assumption that and are unique, it is shown in [9] that (2) is an asymptotically normal estimator,i.e., (7) where (8) Fig. 1. Block diagram of Jaeckel’s adaptive alpha-trimmed mean filter. fx (i)g denotes the order statistics variables, fx (i); x (i); ...;x ( i) g .
and
(9) Asymptotic alpha-trimmed mean estimator, such that, by selecting an
, is optimized (10)
Now, we will present the utilization of this approach in Jaeckel’s adaptive alpha-trimmed mean filter for moderate number of samples inside a given window, which is usually the case in signal and image restoration applications. 2) Adaptive Filter Based on Jaeckel’s Estimator: Let be the observed data from a local neighborhood of a corrupted signal. It is assumed that , is locally constant and the noise, , the original signal, . The order is additive and symmetric, i.e., statistics of these sample points are as in (1). Therefore, the of alpha-trimmed mean, , asymptotic variance can be estimated by the Winsorized variance:
Fig. 2. The clipping effect on Filter I, Filter II, and Filter III is shown with average signal-to-noise ratios versus variance of the underlying Gaussian noise.
Minimization can be done by searching the best value in a , such that fixed interval, i.e., is an integer and computing for these values of , i.e., for , , and , will be selected among . In [8], it is shown that selected this of (10). way is asymptotically equivalent to value can In this procedure, starting from the largest reduce the computation time since for that value the number of terms in the sums of (2) and (6) is the smallest. These sums are in this form:
(11) (13) This means that the asymptotic variance estimate of the alphatrimmed mean can be computed from the observed samples. Therefore one can find an optimum alpha-trimmed mean by selecting an that minimizes the variance of the filter output. An adaptive alpha-trimmed mean filter that selects the best value, , which minimizes the sample asymptotic variance at a local window can be written as: estimate (12) where
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 in Section III. In this section, behaviors of these filters are examined on constant and edge regions separately.
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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) output of 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 filter is its robustness against outliers. In images, outliers can be defined as impulses in a local window generated by an impulsive noise source. One can simply choose a median filter to reject up to 50% of the outliers. However, this is not the best choice if the additive noise is the mixture of a short-tailed noise and an impulsive-type noise. In these cases, alpha-trimmed mean filters with the right values will generate better results.
The noise present in many applications can be modeled as an additive white Gaussian noise. In images, the regions between edges can be filtered successfully by local averaging if the noise is this type. However, in many cases the noise pdf deviates from its Gaussian shape due to different distortions such as clipping and/or impulsive noise. In Sections III-A and B, we will consider these cases. In the following simulations, for Filter I is taken 0.05 as is estimated presented in [5]. For Filter II, noise variance, before filtering by, (15)
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where is the total number of windows considered, and MAD is the median of absolute deviations from median, which is a very robust scale estimate with breakdown point equal to 0.5. For details on MAD see ([10]). It is scaled with 1.483 to produce unbiased estimates for Gaussian distribution. In other words, by (15) computes the local variance estimates at window for and selects the median of the M estimates as the noise variance, . This is a costly but very robust estimate. As you will see, it is very important for this filter to have a close estimate of the noise variance by discarding the outliers. is taken as 0, and is taken as 0.45. Taking For Filter III, and ) usually reduces the the full range (i.e., performance of this filter. A. Clipping Clipping occurs when the pixel intensity level exceeds the predetermined boundary values because of the limitation of the bit precision. For example, 8-bit pixel representation allows maximum pixel value of 255. This problem is more general but we will consider the cases where the added noise causes the pixel values to get higher (lower) values than the maximum(minimum) intensity level. An example is shown in Fig. 3(a) and (b) where the minimum intensity level is 0 and the maximum intensity level is 1. While Fig. 3(a) shows the histogram of the constant signal with pixel values equal to 0.75., Fig. 3(b) shows the histogram of the same signal corrupted by a Gaussian noise with variance 0.065. The pixel values greater than 1 are clipped and set to 1. As it can be easily seen from this figure the shape of the noise pdf significantly deviates from that of Gaussian. The performances of the above mentioned adaptive alphatrimmed mean filters are measured with their signal-to-noise ratios with respect to different variances of the Gaussian noise. The signal is the constant signal with pixel intensities equal to 0.75. The noise variance takes values from 0.005 to 0.065. The window sizes of the filters are 5 5. In this and the other experiments, the signal-to-noise ratios are defined as: (16)
is the pixel value of the nonnoisy original signal where is the pixel value of at the -th row and -th column, and the restored signal at that location. The results are shown in Fig. 2. The signal-to-noise ratios displayed are the average SNRs computed from the filtered outputs of 30 noisy rectangular image regions. Although, the Filter I shows very good performance at lower noise variances, it drops suddenly and decreases dramatically after the variance gets values higher than 0.03. This can be easily explained by the clipping effect. The thresholds of Filter I are static and selected with the assumption of the unimodal symmetric noise. The filter is very sensitive to the deviations from this assumption, therefore when the clipping ratio increased the performance of this filter decreased rapidly.
Fig. 4. Average SNRs of filter outputs vs the density of the impulses of mixed noise, when applied to corrupted constant 2-D regions.
On the other hand, Filter II and Filter III show more stable behavior against the increase of the clipping ratio. These filters are shown to be more robust in this case. Filter II has a slightly better performance over Filter III for all values of the noise variance. Although, there are slight differences among their average SNR values, all three filters show similiar performances when there is no clipping effect. For all three filters, the histograms of the restored signals are shown in Fig. 3(c), (d), and (e) when the noise variance is equal to 0.065. B. Mixed Noise Secondly, we measure the performance of these filters under mixed noise. The noise considered here consists of additive white Gaussian noise and impulsive noise. Particularly, we examined the case when the impulsive noise is salt & pepper noise. As it is clear from its name, this noise converts some of the pixels into very dark (low intensity) and very light (high intensity) pixels. In other words, the impulses appear as minimum and maximum intensity values. Performance on Constant Regions: In images, the pixel values of the regions between edges vary slowly. In many cases these regions can be assumed to have locally constant signal values if the window size is taken small enough. In this part of the paper, we examine the performance of the above filters on constant regions. and While Fig. 5(a) shows a constant 2-D signal its histogram, Fig. 5(b) shows the same signal corrupted with a mixed Gaussian and salt & pepper noise along with its histogram. Here, the Gaussian noise variance is 0.005 and impulse density is equal to 0.5. The Gaussian shape centered around 0.75 can be easily seen. The impulses of the salt & pepper noise are shown at the two ends of the histogram. In this case the mixed noise pdf can be written as:
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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 & pepper noise pdf. To be able to examine the effect of impulsive content, we computed the average signal-to-noise ratios of the three filter outputs while increasing the impulse density of the salt & pepper noise from 0.05 to 0.5. The signal is the same 2-D signal shown in Fig. 5(a) and the variance of the Gaussian noise is 0.005 for all cases. The results plotted in Fig. 4 show that the Filter I is affected by the impulsive content much more than the other two. As explained in the previous case, this filter is very sensitive to deviations from unimodal symmetric noise pdf assumption. As the impulse density increases, the two peaks at the two ends of the pdf dominate in the overall shape. Therefore, the selector
statistic of Filter I takes values as if the overall noise pdf is very short tailed and symmetric due to the effect of these peaks. Since this filter selects the midpoint filter for very short-tailed noise pdfs, the output will be the midpoint between the highest intensity value and the lowest intensity value. The output of Filter I and 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 argument. Although this filter produces very smooth output, it alters the signal level. Filter II, on the other hand, produces very good results compared to Filter I. However, this filter’s performance is also very sensitive to the estimation of the noise variance. To be able to
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Fig. 5. (Continued.)
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(d) Output of Filter II, and (e) output of the Jaeckel’s filter.
remove impulses, variance estimation should not be affected by is a these impulses. With a breakpoint of 0.5, very robust variance estimator, since it can reject upto 50% of the outliers. However, even MADs performance decays when the impulse density increases substantially. The effect of this on the filter’s average output SNR can be seen in Fig. 4 when the impulse density is around 0.5, which is the breakpoint of this estimator. Unlike Filter III, the performance of Filter II drops rapidly at this point. This indicates that Filter II is very sensitive against false estimation of the variance. The outputs and their histograms in Fig. 5(d) and (e) also gives the idea about their success at smoothing during the presence of mixed Gaussian and high impulsive noise. It can be concluded that, overall, Filter III is more robust than the other two when the mixed noise consists of impulses. Performance on the Edges: In image processing, the preservation of edges is as important as smoothing the slowly varying regions. Therefore, the rest of this section evaluates the performance of Filter I, Filter II, and Filter III on a step edge corrupted with mixed Gaussian and impulsive noise. The tools that we use to measure and present the results are same as the constant signal case 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 interested only in the performance of the bare filters. These components can be added any time to any of these filters.
Fig. 6. Average SNRs of Filter I, Filter II, and Filter III outputs versus impulse density of the mixed noise on a 2-D step edge.
The step edge we consider and its histogram representation is shown in Fig. 7(a). This is an edge with a height of 0.25 and a width of 10 pixels. The peak on the left of the histogram represents the lower pixel values, and the one on the right represents the higher pixel values. The corrupted signal and its histogram is presented in Fig. 7(b). There are still two peaks at the same location as
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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 the Gaussian component of the noise. As expected, the peaks resulting from the impulsive component are also shown in the far left and far right of the histogram. For this case the average signal-to-noise ratios show distinctive features (Fig. 6) for all three filters. The behavior of Filter I is similar to the constant signal case. Even for the very low impulse densities, this filter output has a lower SNR than the other two. The reason for this can be explained by the bi-modal distribution of the signal itself at the edges. The output of this filter at impulse density 0.5 and Gaussian noise variance 0.005 is shown Fig. 7(c). Again, mostly by the contribution of light and dark impulses the midpoint is selected as the output of the filter by destroying the edge. Filter II’s performance drops significantly when the impulse density gets closer to 0.5, which is the breakdown point of the noise variance estimator. The output of this filter and its histogram when the impulse density is equal to 0.5 is shown in Fig. 7(d). In this figure, it can be easily seen that two level structure of the edge is disappeared. Although this filter shows very good performance for lower impulse densities, it destroys the edges for higher impulse densities. Finally, on a step edge Filter III shows very robust performance against the increase in the impulse density. The two level structure of the filter output and its histogram is shown in Fig. 7(e). Filter III preserves the edges better than Filter I and Filter II under the high impulsive noise content.
IV. A NEW ADAPTIVE FILTER IMPLEMENTATION From the comparisons done in Sections II and III, we can say that Filter II and Filter III show comparable performances. However, actual processing times for these filters are long. The complexity of Filter II comes from the fact that it requires a good estimation of the underlying noise variance. Filter III is complex because it requires the computation of asymptotic variance estimate several times per pixel. Here, we propose another adaptive alpha-trimmed mean filter which has a strightforward implementation with less complexity and shows comparable performance with Filter II and Filter III. A. Theory The idea behind our implementation can be seen from an asymptotic analysis of noise samples from a symmetric distribution. As presented before, the asymptotic representation of alpha-trimmed mean is: (17) The meaning of this can be easily seen from Fig. 8. It corresponds to the ratio of the area between the -axis and the curve, between points and to the distance between these points.
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Fig. 7. (Continued.)
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(c) Output of Filter I, (d) output of Filter II, and (e) output of the Filter III.
If the distribution is symmetric between and and corresponds to the median of the distribution, then shifts the curve horizontally to the line. and Therefore, if a distribution is symmetric between and corresponds to the median of the distribution then
Therefore, the condition for symmetry between ,
and
(20) in asymptotic case can be represented as
(18)
(21)
will be equal to zero. One can easily verify that the above formula corresponds to , i.e., the derivative of
for small and moderate sample sizes. In our filter, we put a condition on to satisfy:
(19)
(22)
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TABLE IV FILTERING RESULTS FOR ZELDA IMAGE
TABLE V FILTERING RESULTS FOR PEPPERS IMAGE
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 0 to the distance between these
points.
TABLE I FILTERING RESULTS OF CONSTANT IMAGE REGIONS CORRUPTED BY GAUSSIAN NOISE
TABLE II FILTERING RESULTS OF THE STEP EDGE CORRUPTED WITH GAUSSIAN NOISE
TABLE III FILTERING RESULTS OF A CONSTANT IMAGE REGIONS CORRUPTED WITH MIXED NOISE
where is a constant threshold. Based on this, the output of the , where is computed adaptive filter we present is locally from the data samples:
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 the minimum value that will only utilize samples from the symmetric middle portion of the distribution while trimming the rest. The value of the threshold, , determines how much we will tolerate against deviations from the symmetry condition. from the sorted samples The algorithm to compute thru is explained in Section IV-B. B. Algorithm
(23)
To find the minimum value that satisifies (23), algorithm starts with the maximum value and decreases it until it fails to satisfy (23).
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(a)
(b)
(d)
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(c)
(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) Filter I, (c) Filter II, (d) Filter III, and (e) the proposed filter.
Let the sample size, . In this case, represents the median of the samples. Also, let us represent the sums of trimmmed samples as:
.. . (24)
The corresponding averages are represented as , . Note that the value decreases as increases. is nothing but the -trimmed mean with discrete values. The consequent algorithm is: Step Step Step Step Step
0. 1. While 1.1 Compute 1.2 2.
and
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(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 computation s. An efficient way is to utilize the fact that can of be derived from : (25) and . For some software and hardware architectures it may be better s of all the pixels that fit into to compute memory and process them like a vector processing.
V. PERFORMANCE OF THE PROPOSED FILTER In this section, we compare the performance of the proposed filter with Taguchi’s filter (Filter II) and the Jaeckel’s filter(Filter III). Here, the parameters of these filters are taken as previous simulations and threshold, of the proposed filter is 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
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performance against the deviations from the assumed noise model in Section III. The first comparison is done on constant image regions corrupted with Gaussian noise that results clipping. Table I presents the results of this comparison, where first row shows the different noise variances. The next comparison is done on step edges of 20 pixel width, which are corrupted by Gaussian noise. The SNRs are computed against different noise variances. The results are shown in Table II. In the next comparison, constant image regions corrupted by ) and salt and pepper noise with mixed Gaussian ( different impulse density are filtered with these three filters. the results are shown in Table III. Finally, we applied these three filters to the “Zelda” image and “Peppers” image that are corrupted with mixed noise described as in the previous simulation. The SNRs are shown in Tables IV and V and the results are shown in Figs. 10 and 11. These results show that the proposed filter shows comparable performance to the other filters. The speed comparisons of these filters are done on a PC with Pentium III 500 Mhz Intel processor. The filters are implemented on a MATLAB 6.0 platform. The processing time of these 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 adaptive alpha-trimmed mean filters and examined their performance when the underlying noise deviates from Gaussian noise model. It is shown that the adaptive filter suggested by Restrepo and Bovik [5] (Filter I) is very sensitive to deviations from the symmetric unimodal noise assumption. Although this filter is fast and easy to implement and produces very good results for unimodal symmetric noise distribution, it does not perform well for certain 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 requires the variance of the noise distribution as an input. This can be estimated but the estimator should be very robust against outliers (impulses and edges). The MAD estimator we suggested is very robust but it makes this filter computationally very complex. Taguchi’s filter (Filter II) is to be preferred over the other filters when the noise variance is known. We employed the Jaeckel’s estimator as an adaptive alphatrimmed mean filter for image restoration and observed its performance under certain conditions. This filter seems to be very robust against the noise model deviations such as clipping and high impulsive content as well as jumps (edges) of the signal itself. It performs better if the possible range of values can be taken smaller depending on the possible shapes of the underlying noise class. Jaeckel’s filter (Filter III) is to be preferred over the others when there is highly impulsive noise along with a short-tailed noise. However, this filter also suffers from complexity since it requires computation of the asymptotic variance several times per pixel.
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We also presented a new adaptive filter implementation which can be a good compromise between output quality and speed. This filter’s noise removal performance under the deviations from the assumed noise models is comparable to Filter II and Filter III, while its complexity is low as in the case of Filter I. REFERENCES [1] H. A. David, Order Statistics. New York: Wiley, 1980. [2] H. D. Tagare and R. J. P. deFigueiredo, “Order filters,” Proc. of the IEEE, vol. 73, pp. 163–165, 1985. [3] I. Pitas and A. N. Venetsanopoulos, Nonlinear Digital Filters: Kluwer Academic Publishers, 1990. [4] J. B. Bednar and T. L. Watt, “Alpha-trimmed means and their relationship to median filters,” IEEE Trans. on Acous.,Speech and Signal Processing, vol. 32, pp. 145–153, 1987. [5] A. Restrepo and A. C. Bovik, “Adaptive trimmed-mean filters for image restoration,” IEEE Tran. on Acous.,Speech and Signal Processing, vol. 36, no. 8, pp. 1326–1337, 1988. [6] R. V. Hogg, “Some observations on robust estimation,” Journal of American Statistical Association, vol. 62, pp. 1179–1186, 1967. [7] A. Taguchi, “Adaptive -trimmed mean filters with excellent detail-preserving,” in Proceedings of ICASSP ’94. IEEE International Conference on Acoustics, Speech and Signal Processing, Adelaide, SA, Australia, April 1994, pp. 19–22. [8] L. A. Jaeckel, “Some flexible estimates of location,” Annals of Mathematical Statistics, vol. 42, no. 5, pp. 1540–1552, 1971. [9] S. M. Stigler, “The asymptotic distribution of the trimmed mean,” Annals of Statistics, vol. 1, pp. 472–477, 1973. [10] F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw, and W. A. Stahel, Robust Statistics: The Approach Based on Influence Functions: John Wiley & Sons, 1985. [11] Y. B. Rytsar and I. B. Ivanesko, “Application of (alpha, beta)-trimmed mean filtering for removal of additive noise from images,” Proc. of SPIE, vol. 3238, 1997. [12] S. R. Peterson, Y. H. Lee, and S. A. Kassam, “Some statistical properties of alpha-trimmed mean and standard type M filters,” IEEE Trans. on Signal Proc., vol. 36, no. 5, pp. 707–713, 1988. [13] A. Taguchi, “An adaptive -trimmed mean filter with excellent detailpreservation and evaluation of its performance,” Elec. Comm. in Japan Part III, vol. 78, no. 10, pp. 46–56, 1995. [14] B. Widrow and S. D. Stearns, Adaptive Signal Processing. New Jersey: Prentice Hall, 1985.
Remzi Öten (M’94) received the B.S. degree in electrical and electronics engineering from Bilkent University, Ankara, Turkey, in 1992. He received the M.S. and Ph.D. degrees in electrical and computer engineering from University of California, Irvine, in 1993 and 1999, respectively. Since 1999, he has been working as a Systems Engineer and Consultant in the area of signal processing with emphasis to multimedia systems. He has written 13 journal and conference papers in this area. Dr. Oten is the recipient of the University of California Regents’ Fellowship and the 2003 IEEE Circuits and Systems Transactions Guillemin-Cauer Best Paper Award.
Rui J. P. de Figueiredo (LF’94) received the B.S. and M.S. degrees in electrical engineering from MIT, and the Ph.D. degree in applied mathematics from Harvard University. After being several years on the faculty of Rice University, Houston, TX, he joined the University of California, Irvine, in 1990, where he holds the position of Professor of electrical engineering and computer science, of bio-medical engineering, and of mathematics. He has published more than 370 papers, and chaired or co-chaired nine national or international conferences. Dr. de Figueiredo was the President of the IEEE Circuits and Systems Society in 1998, and served on several national and international committees and panels. For all these contributions, he received a number of awards, including the IEEE Fellow award, the 1994 IEEE Circuits and Systems Society Technical Achievement 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 IEEE Neural Networks Transactions Outstanding Paper Award, and the 2003 IEEE Circuits and Systems Transactions Guillemin-Cauer Best Paper Award.
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Adaptive Alpha-Trimmed Mean Filters Under Deviations 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 for the restoration of signals and images corrupted by additive non-Gaussian noise. They are especially preferred if the underlying noise deviates from Gaussian with the impulsive noise components. The key design issue of these filters is to select its only parameter, , optimally for a given noise type. In image restoration, adaptive filters utilize the flexibility of selecting according to some local noise statistics. In the present paper, we first review the existing adaptive alpha-trimmed mean filter schemes. We then analyze the performance of these filters when the underlying noise distribution deviates from the Gaussian and does 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 implementation that detects the nonsymmetry points locally and applies alpha-trimmed mean filter that trims out the outlier pixels such as edges or impulsive noise according to this local decision. Comparisons of the speed and filtering performances under deviations from symmetry and Gaussian assumptions show that the proposed filter is a very good alternative to the existing schemes.
I. INTRODUCTION
I
N many signal and image processing applications, nonlinear filters have been employed very effectively in removing nonGaussian noise present in the data. In particular, filters based on Order Statistics [1] have been extensively used in noise removal applications [2], [3], such as in the presence of impulsive noise. The most common and the simplest type of these filters is the median filter. It shows very good performance for the removal of long-tailed noise types (e.g., Laplacian) and preserving the edges. Another filter of this type is the alpha-trimmed mean filter [4], [11], [12]. It is a good compromise between median and moving average filter, which is known to be best for shorttailed noise types (e.g., Gaussian). For all three filters, the histograms of the restored signals are shown in Fig. 3(c)–(e) when the 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, . If these values are arranged in ascending order of their amplitude, the order statistics result is (1) Manuscript received November 9, 1999; revised December 11, 2002. This work was supported by the NSF under Grant CCR-9704262. The associate editor coordinating the review of this manuscript and approving it for publication was 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
is the minimum, is the maximum, and is the median of the above set of signal values. The is output of the alpha-trimmed mean filter,
where
(2) . As it where [.] denotes the greatest integer part and is easily seen from (2), indicates the percentage of the trimmed samples. Therefore, the alpha-trimmed mean filter performs like a median filter when is close to 0.5, and moving average filter when is close to 0. If we drop the time index and denote the , then the moving average filter is trimmed-mean filter by . Although never gets equal to 0.5, for simplicity we will since is very close to 0.5 for represent median filter as this filter. The conventional type of trimmed-mean filters that are mentioned in this paper are also called ’inner trimmed-mean filters’ since the outputs of these filters are produced by averaging the ’inner’ sample values (the ones close to the median). For the noise pdf’s, which have shorter tail lengths than that of a Gaussian pdf, outer trimmed-means are preferred. The output is equal to the average of an outer trimmed-mean filter, of the ’outer’ sample values(the ones close to the extremes). An example of this is the mid-point filter, which is the maximum likelihood estimate of location for uniform distribution. Midpoint filter trims out all the values except the extremes and can . The moving average filter output can be represented as . Here, also be represented as an outer trimmed-mean,i.e., for simplicity we will consider only inner trimmed-means, extensions to outer trimmed-means being trivial. Alpha-trimmed mean estimator is intended to be used for location estimation when the Gaussian data distribution contains some outliers. For these cases, they outperform other nonlinear filters in simplicity and noise reduction. In this paper, we focus on the generalization of this case and evaluate the robustness of the adaptive schemes that utilize alpha-trimmed mean filters when the noise pdf deviates from assumed Gaussian model. First, we review the existing adaptive trimmed-mean filters in Section II. Then, in Section III, compare their filtering performances when the underlying noise distribution deviates from the Gaussian and symmetry assumptions due to the clipping effects and impulsive noise. In Section IV, we introduce a new adaptive trimmed filter scheme by utilizing the symmetry conditions of the asymptotic distribution. This filter detects the nonsymmetry points locally and applies alpha-trimmed mean filter that trims out the outlier pixels such as edges or impulsive noise according to this local decision. This type of adaptation make this filter an “open-loop”
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adaptive filter as opposed to “closed-loop” adaptive filters that utilizes feedback [14]. All the filters considered in this paper use open-loop adaptation. Open-loop adaptive filters do not use previous estimates as an input, therefore, do not have convergence problems. However, in case of bad estimation schemes, they will perform very poorly. In Section V, the proposed filter is compared with the wellperforming adaptive trimmed mean filters of Section II and is shown to be a good alternative to these filters since it can achieve comparable results with considerably less complexity. II. ADAPTIVE FILTERS The design problem of alpha-trimmed mean filters reduces to selecting the “best” value for a given noise pdf. However, the selection of the best value may not be possible when the noise pdf is not known or varies with time. For these cases, one has to design an adaptive filter that changes its value according to some characteristics of the observed sample data. In still image procesing, adaptation can be done according to local noise statistics in a window. There are two types of filtering algorithms that have been proposed previously [5], [7]. These two and Jaeckel’s adaptive estimation scheme [8] are briefly reviewed below [8] in the context of image restoration filters. A. Filter I In [5], an adaptive scheme is proposed which selects the best from a previously determined collection of values according to some selector statistics whose threshold values are determined by Monte-Carlo analysis. The output of this filter is, at window if where if where
B. Filter II In [7], [13], another adaptive alpha-trimmed mean filter structure is proposed. Unlike [5] this filter utilizes the flexibility of alpha-trimmed means completely by allowing the use of whole ). The output of this filter at window range (i.e., is: (4) Here is the activity of the original signal in window defined as:
and (5)
where : if : otherwise is the sample variance of the signal values in , where is the variance of the noise in . and The main disadvantage of the above filter is that the noise variance is assumed to be known for all . This assumption is actually against the nature of adaptive filtering that we consider. It may be possible to use an estimate of the noise variance, but the performance will be highly affected by the error of the estimate, therefore a robust estimation scheme is critical for this filter. C. Filter III
where the selector statistic, , is a robust tail length estimate of the underlying noise pdf. In other words, adaptive filter performs one of the s outer mean filters, or one of the inner-mean filters according to the result of tail-length esti. The description and the mation of the local pdf in window can be found in [5] and [6]. properties of In [5], only five simple filters are employed: : : : : :
rigorous relation between the estimators and the threshold values. Since these thresholds are selected empirically from generalized exponential pdfs, this filter will not give acceptable results if the noise pdf is not in symmetric unimodal shape. Its performance is heavily dependent on the edge-detection scheme proposed in [5].
(3)
where the threshold values are selected via Monte Carlo simulation with the assumption that the underlying noise pdf can be modeled by the family of generalized exponential pdfs. The adaptive filter in (3) is easy to implement once the threshold values are selected. However, since it has to select from a set of few estimators, it cannot completely utilize the flexibility of alpha-trimmed mean filters. Although it is possible to increase the number of estimators, this will not improve the filter performance since there is no precise
Neither of the above two schemes uses a rigorous approach that is based on an optimization criteria. In this paper, we also consider Jaeckel’s estimator [8] which offers a more rigorous solution but has never been analyzed in a signal and image processing scenario. Jaeckel’s proposal is based on the minimization of the asymptotic variance estimate of the alpha-trimmed mean estimator. In the present paper, it is implemented as an adaptive alpha-trimmed mean filter. In Section III, we compare Jaeckel’s filter’s performance with Filter I and Filter II when the underlying noise pdf deviates from Gaussian. In particular, we examine the cases where clipping modifies the shape of the pdf and where the Gaussian noise is mixed with impulsive type noise. Their behavior both at the smooth regions and the edge regions is analyzed. 1) Asymptotic Properties of Alpha-Trimmed Means: Let be a sample of independent, identically distributed random variables with a common symmetric disdenote the order tribution , and let statistics. The alpha-trimmed mean is given by (6)
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With the assumption that and are unique, it is shown in [9] that (2) is an asymptotically normal estimator,i.e., (7) where (8) Fig. 1. Block diagram of Jaeckel’s adaptive alpha-trimmed mean filter. fx (i)g denotes the order statistics variables, fx (i); x (i); ...;x ( i) g .
and
(9) Asymptotic alpha-trimmed mean estimator, such that, by selecting an
, is optimized (10)
Now, we will present the utilization of this approach in Jaeckel’s adaptive alpha-trimmed mean filter for moderate number of samples inside a given window, which is usually the case in signal and image restoration applications. 2) Adaptive Filter Based on Jaeckel’s Estimator: Let be the observed data from a local neighborhood of a corrupted signal. It is assumed that , is locally constant and the noise, , the original signal, . The order is additive and symmetric, i.e., statistics of these sample points are as in (1). Therefore, the of alpha-trimmed mean, , asymptotic variance can be estimated by the Winsorized variance:
Fig. 2. The clipping effect on Filter I, Filter II, and Filter III is shown with average signal-to-noise ratios versus variance of the underlying Gaussian noise.
Minimization can be done by searching the best value in a , such that fixed interval, i.e., is an integer and computing for these values of , i.e., for , , and , will be selected among . In [8], it is shown that selected this of (10). way is asymptotically equivalent to value can In this procedure, starting from the largest reduce the computation time since for that value the number of terms in the sums of (2) and (6) is the smallest. These sums are in this form:
(11) (13) This means that the asymptotic variance estimate of the alphatrimmed mean can be computed from the observed samples. Therefore one can find an optimum alpha-trimmed mean by selecting an that minimizes the variance of the filter output. An adaptive alpha-trimmed mean filter that selects the best value, , which minimizes the sample asymptotic variance at a local window can be written as: estimate (12) where
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 in Section III. In this section, behaviors of these filters are examined on constant and edge regions separately.
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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) output of 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 filter is its robustness against outliers. In images, outliers can be defined as impulses in a local window generated by an impulsive noise source. One can simply choose a median filter to reject up to 50% of the outliers. However, this is not the best choice if the additive noise is the mixture of a short-tailed noise and an impulsive-type noise. In these cases, alpha-trimmed mean filters with the right values will generate better results.
The noise present in many applications can be modeled as an additive white Gaussian noise. In images, the regions between edges can be filtered successfully by local averaging if the noise is this type. However, in many cases the noise pdf deviates from its Gaussian shape due to different distortions such as clipping and/or impulsive noise. In Sections III-A and B, we will consider these cases. In the following simulations, for Filter I is taken 0.05 as is estimated presented in [5]. For Filter II, noise variance, before filtering by, (15)
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where is the total number of windows considered, and MAD is the median of absolute deviations from median, which is a very robust scale estimate with breakdown point equal to 0.5. For details on MAD see ([10]). It is scaled with 1.483 to produce unbiased estimates for Gaussian distribution. In other words, by (15) computes the local variance estimates at window for and selects the median of the M estimates as the noise variance, . This is a costly but very robust estimate. As you will see, it is very important for this filter to have a close estimate of the noise variance by discarding the outliers. is taken as 0, and is taken as 0.45. Taking For Filter III, and ) usually reduces the the full range (i.e., performance of this filter. A. Clipping Clipping occurs when the pixel intensity level exceeds the predetermined boundary values because of the limitation of the bit precision. For example, 8-bit pixel representation allows maximum pixel value of 255. This problem is more general but we will consider the cases where the added noise causes the pixel values to get higher (lower) values than the maximum(minimum) intensity level. An example is shown in Fig. 3(a) and (b) where the minimum intensity level is 0 and the maximum intensity level is 1. While Fig. 3(a) shows the histogram of the constant signal with pixel values equal to 0.75., Fig. 3(b) shows the histogram of the same signal corrupted by a Gaussian noise with variance 0.065. The pixel values greater than 1 are clipped and set to 1. As it can be easily seen from this figure the shape of the noise pdf significantly deviates from that of Gaussian. The performances of the above mentioned adaptive alphatrimmed mean filters are measured with their signal-to-noise ratios with respect to different variances of the Gaussian noise. The signal is the constant signal with pixel intensities equal to 0.75. The noise variance takes values from 0.005 to 0.065. The window sizes of the filters are 5 5. In this and the other experiments, the signal-to-noise ratios are defined as: (16)
is the pixel value of the nonnoisy original signal where is the pixel value of at the -th row and -th column, and the restored signal at that location. The results are shown in Fig. 2. The signal-to-noise ratios displayed are the average SNRs computed from the filtered outputs of 30 noisy rectangular image regions. Although, the Filter I shows very good performance at lower noise variances, it drops suddenly and decreases dramatically after the variance gets values higher than 0.03. This can be easily explained by the clipping effect. The thresholds of Filter I are static and selected with the assumption of the unimodal symmetric noise. The filter is very sensitive to the deviations from this assumption, therefore when the clipping ratio increased the performance of this filter decreased rapidly.
Fig. 4. Average SNRs of filter outputs vs the density of the impulses of mixed noise, when applied to corrupted constant 2-D regions.
On the other hand, Filter II and Filter III show more stable behavior against the increase of the clipping ratio. These filters are shown to be more robust in this case. Filter II has a slightly better performance over Filter III for all values of the noise variance. Although, there are slight differences among their average SNR values, all three filters show similiar performances when there is no clipping effect. For all three filters, the histograms of the restored signals are shown in Fig. 3(c), (d), and (e) when the noise variance is equal to 0.065. B. Mixed Noise Secondly, we measure the performance of these filters under mixed noise. The noise considered here consists of additive white Gaussian noise and impulsive noise. Particularly, we examined the case when the impulsive noise is salt & pepper noise. As it is clear from its name, this noise converts some of the pixels into very dark (low intensity) and very light (high intensity) pixels. In other words, the impulses appear as minimum and maximum intensity values. Performance on Constant Regions: In images, the pixel values of the regions between edges vary slowly. In many cases these regions can be assumed to have locally constant signal values if the window size is taken small enough. In this part of the paper, we examine the performance of the above filters on constant regions. and While Fig. 5(a) shows a constant 2-D signal its histogram, Fig. 5(b) shows the same signal corrupted with a mixed Gaussian and salt & pepper noise along with its histogram. Here, the Gaussian noise variance is 0.005 and impulse density is equal to 0.5. The Gaussian shape centered around 0.75 can be easily seen. The impulses of the salt & pepper noise are shown at the two ends of the histogram. In this case the mixed noise pdf can be written as:
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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 & pepper noise pdf. To be able to examine the effect of impulsive content, we computed the average signal-to-noise ratios of the three filter outputs while increasing the impulse density of the salt & pepper noise from 0.05 to 0.5. The signal is the same 2-D signal shown in Fig. 5(a) and the variance of the Gaussian noise is 0.005 for all cases. The results plotted in Fig. 4 show that the Filter I is affected by the impulsive content much more than the other two. As explained in the previous case, this filter is very sensitive to deviations from unimodal symmetric noise pdf assumption. As the impulse density increases, the two peaks at the two ends of the pdf dominate in the overall shape. Therefore, the selector
statistic of Filter I takes values as if the overall noise pdf is very short tailed and symmetric due to the effect of these peaks. Since this filter selects the midpoint filter for very short-tailed noise pdfs, the output will be the midpoint between the highest intensity value and the lowest intensity value. The output of Filter I and 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 argument. Although this filter produces very smooth output, it alters the signal level. Filter II, on the other hand, produces very good results compared to Filter I. However, this filter’s performance is also very sensitive to the estimation of the noise variance. To be able to
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Fig. 5. (Continued.)
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(d) Output of Filter II, and (e) output of the Jaeckel’s filter.
remove impulses, variance estimation should not be affected by is a these impulses. With a breakpoint of 0.5, very robust variance estimator, since it can reject upto 50% of the outliers. However, even MADs performance decays when the impulse density increases substantially. The effect of this on the filter’s average output SNR can be seen in Fig. 4 when the impulse density is around 0.5, which is the breakpoint of this estimator. Unlike Filter III, the performance of Filter II drops rapidly at this point. This indicates that Filter II is very sensitive against false estimation of the variance. The outputs and their histograms in Fig. 5(d) and (e) also gives the idea about their success at smoothing during the presence of mixed Gaussian and high impulsive noise. It can be concluded that, overall, Filter III is more robust than the other two when the mixed noise consists of impulses. Performance on the Edges: In image processing, the preservation of edges is as important as smoothing the slowly varying regions. Therefore, the rest of this section evaluates the performance of Filter I, Filter II, and Filter III on a step edge corrupted with mixed Gaussian and impulsive noise. The tools that we use to measure and present the results are same as the constant signal case 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 interested only in the performance of the bare filters. These components can be added any time to any of these filters.
Fig. 6. Average SNRs of Filter I, Filter II, and Filter III outputs versus impulse density of the mixed noise on a 2-D step edge.
The step edge we consider and its histogram representation is shown in Fig. 7(a). This is an edge with a height of 0.25 and a width of 10 pixels. The peak on the left of the histogram represents the lower pixel values, and the one on the right represents the higher pixel values. The corrupted signal and its histogram is presented in Fig. 7(b). There are still two peaks at the same location as
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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 the Gaussian component of the noise. As expected, the peaks resulting from the impulsive component are also shown in the far left and far right of the histogram. For this case the average signal-to-noise ratios show distinctive features (Fig. 6) for all three filters. The behavior of Filter I is similar to the constant signal case. Even for the very low impulse densities, this filter output has a lower SNR than the other two. The reason for this can be explained by the bi-modal distribution of the signal itself at the edges. The output of this filter at impulse density 0.5 and Gaussian noise variance 0.005 is shown Fig. 7(c). Again, mostly by the contribution of light and dark impulses the midpoint is selected as the output of the filter by destroying the edge. Filter II’s performance drops significantly when the impulse density gets closer to 0.5, which is the breakdown point of the noise variance estimator. The output of this filter and its histogram when the impulse density is equal to 0.5 is shown in Fig. 7(d). In this figure, it can be easily seen that two level structure of the edge is disappeared. Although this filter shows very good performance for lower impulse densities, it destroys the edges for higher impulse densities. Finally, on a step edge Filter III shows very robust performance against the increase in the impulse density. The two level structure of the filter output and its histogram is shown in Fig. 7(e). Filter III preserves the edges better than Filter I and Filter II under the high impulsive noise content.
IV. A NEW ADAPTIVE FILTER IMPLEMENTATION From the comparisons done in Sections II and III, we can say that Filter II and Filter III show comparable performances. However, actual processing times for these filters are long. The complexity of Filter II comes from the fact that it requires a good estimation of the underlying noise variance. Filter III is complex because it requires the computation of asymptotic variance estimate several times per pixel. Here, we propose another adaptive alpha-trimmed mean filter which has a strightforward implementation with less complexity and shows comparable performance with Filter II and Filter III. A. Theory The idea behind our implementation can be seen from an asymptotic analysis of noise samples from a symmetric distribution. As presented before, the asymptotic representation of alpha-trimmed mean is: (17) The meaning of this can be easily seen from Fig. 8. It corresponds to the ratio of the area between the -axis and the curve, between points and to the distance between these points.
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Fig. 7. (Continued.)
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(c) Output of Filter I, (d) output of Filter II, and (e) output of the Filter III.
If the distribution is symmetric between and and corresponds to the median of the distribution, then shifts the curve horizontally to the line. and Therefore, if a distribution is symmetric between and corresponds to the median of the distribution then
Therefore, the condition for symmetry between ,
and
(20) in asymptotic case can be represented as
(18)
(21)
will be equal to zero. One can easily verify that the above formula corresponds to , i.e., the derivative of
for small and moderate sample sizes. In our filter, we put a condition on to satisfy:
(19)
(22)
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TABLE IV FILTERING RESULTS FOR ZELDA IMAGE
TABLE V FILTERING RESULTS FOR PEPPERS IMAGE
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 0 to the distance between these
points.
TABLE I FILTERING RESULTS OF CONSTANT IMAGE REGIONS CORRUPTED BY GAUSSIAN NOISE
TABLE II FILTERING RESULTS OF THE STEP EDGE CORRUPTED WITH GAUSSIAN NOISE
TABLE III FILTERING RESULTS OF A CONSTANT IMAGE REGIONS CORRUPTED WITH MIXED NOISE
where is a constant threshold. Based on this, the output of the , where is computed adaptive filter we present is locally from the data samples:
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 the minimum value that will only utilize samples from the symmetric middle portion of the distribution while trimming the rest. The value of the threshold, , determines how much we will tolerate against deviations from the symmetry condition. from the sorted samples The algorithm to compute thru is explained in Section IV-B. B. Algorithm
(23)
To find the minimum value that satisifies (23), algorithm starts with the maximum value and decreases it until it fails to satisfy (23).
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(b)
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(c)
(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) Filter I, (c) Filter II, (d) Filter III, and (e) the proposed filter.
Let the sample size, . In this case, represents the median of the samples. Also, let us represent the sums of trimmmed samples as:
.. . (24)
The corresponding averages are represented as , . Note that the value decreases as increases. is nothing but the -trimmed mean with discrete values. The consequent algorithm is: Step Step Step Step Step
0. 1. While 1.1 Compute 1.2 2.
and
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(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 computation s. An efficient way is to utilize the fact that can of be derived from : (25) and . For some software and hardware architectures it may be better s of all the pixels that fit into to compute memory and process them like a vector processing.
V. PERFORMANCE OF THE PROPOSED FILTER In this section, we compare the performance of the proposed filter with Taguchi’s filter (Filter II) and the Jaeckel’s filter(Filter III). Here, the parameters of these filters are taken as previous simulations and threshold, of the proposed filter is 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
ÖTEN AND DE FIGUEIREDO: ADAPTIVE ALPHA-TRIMMED MEAN FILTERS UNDER DEVIATIONS FROM ASSUMED NOISE MODEL
performance against the deviations from the assumed noise model in Section III. The first comparison is done on constant image regions corrupted with Gaussian noise that results clipping. Table I presents the results of this comparison, where first row shows the different noise variances. The next comparison is done on step edges of 20 pixel width, which are corrupted by Gaussian noise. The SNRs are computed against different noise variances. The results are shown in Table II. In the next comparison, constant image regions corrupted by ) and salt and pepper noise with mixed Gaussian ( different impulse density are filtered with these three filters. the results are shown in Table III. Finally, we applied these three filters to the “Zelda” image and “Peppers” image that are corrupted with mixed noise described as in the previous simulation. The SNRs are shown in Tables IV and V and the results are shown in Figs. 10 and 11. These results show that the proposed filter shows comparable performance to the other filters. The speed comparisons of these filters are done on a PC with Pentium III 500 Mhz Intel processor. The filters are implemented on a MATLAB 6.0 platform. The processing time of these 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 adaptive alpha-trimmed mean filters and examined their performance when the underlying noise deviates from Gaussian noise model. It is shown that the adaptive filter suggested by Restrepo and Bovik [5] (Filter I) is very sensitive to deviations from the symmetric unimodal noise assumption. Although this filter is fast and easy to implement and produces very good results for unimodal symmetric noise distribution, it does not perform well for certain 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 requires the variance of the noise distribution as an input. This can be estimated but the estimator should be very robust against outliers (impulses and edges). The MAD estimator we suggested is very robust but it makes this filter computationally very complex. Taguchi’s filter (Filter II) is to be preferred over the other filters when the noise variance is known. We employed the Jaeckel’s estimator as an adaptive alphatrimmed mean filter for image restoration and observed its performance under certain conditions. This filter seems to be very robust against the noise model deviations such as clipping and high impulsive content as well as jumps (edges) of the signal itself. It performs better if the possible range of values can be taken smaller depending on the possible shapes of the underlying noise class. Jaeckel’s filter (Filter III) is to be preferred over the others when there is highly impulsive noise along with a short-tailed noise. However, this filter also suffers from complexity since it requires computation of the asymptotic variance several times per pixel.
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We also presented a new adaptive filter implementation which can be a good compromise between output quality and speed. This filter’s noise removal performance under the deviations from the assumed noise models is comparable to Filter II and Filter III, while its complexity is low as in the case of Filter I. REFERENCES [1] H. A. David, Order Statistics. New York: Wiley, 1980. [2] H. D. Tagare and R. J. P. deFigueiredo, “Order filters,” Proc. of the IEEE, vol. 73, pp. 163–165, 1985. [3] I. Pitas and A. N. Venetsanopoulos, Nonlinear Digital Filters: Kluwer Academic Publishers, 1990. [4] J. B. Bednar and T. L. Watt, “Alpha-trimmed means and their relationship to median filters,” IEEE Trans. on Acous.,Speech and Signal Processing, vol. 32, pp. 145–153, 1987. [5] A. Restrepo and A. C. Bovik, “Adaptive trimmed-mean filters for image restoration,” IEEE Tran. on Acous.,Speech and Signal Processing, vol. 36, no. 8, pp. 1326–1337, 1988. [6] R. V. Hogg, “Some observations on robust estimation,” Journal of American Statistical Association, vol. 62, pp. 1179–1186, 1967. [7] A. Taguchi, “Adaptive -trimmed mean filters with excellent detail-preserving,” in Proceedings of ICASSP ’94. IEEE International Conference on Acoustics, Speech and Signal Processing, Adelaide, SA, Australia, April 1994, pp. 19–22. [8] L. A. Jaeckel, “Some flexible estimates of location,” Annals of Mathematical Statistics, vol. 42, no. 5, pp. 1540–1552, 1971. [9] S. M. Stigler, “The asymptotic distribution of the trimmed mean,” Annals of Statistics, vol. 1, pp. 472–477, 1973. [10] F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw, and W. A. Stahel, Robust Statistics: The Approach Based on Influence Functions: John Wiley & Sons, 1985. [11] Y. B. Rytsar and I. B. Ivanesko, “Application of (alpha, beta)-trimmed mean filtering for removal of additive noise from images,” Proc. of SPIE, vol. 3238, 1997. [12] S. R. Peterson, Y. H. Lee, and S. A. Kassam, “Some statistical properties of alpha-trimmed mean and standard type M filters,” IEEE Trans. on Signal Proc., vol. 36, no. 5, pp. 707–713, 1988. [13] A. Taguchi, “An adaptive -trimmed mean filter with excellent detailpreservation and evaluation of its performance,” Elec. Comm. in Japan Part III, vol. 78, no. 10, pp. 46–56, 1995. [14] B. Widrow and S. D. Stearns, Adaptive Signal Processing. New Jersey: Prentice Hall, 1985.
Remzi Öten (M’94) received the B.S. degree in electrical and electronics engineering from Bilkent University, Ankara, Turkey, in 1992. He received the M.S. and Ph.D. degrees in electrical and computer engineering from University of California, Irvine, in 1993 and 1999, respectively. Since 1999, he has been working as a Systems Engineer and Consultant in the area of signal processing with emphasis to multimedia systems. He has written 13 journal and conference papers in this area. Dr. Oten is the recipient of the University of California Regents’ Fellowship and the 2003 IEEE Circuits and Systems Transactions Guillemin-Cauer Best Paper Award.
Rui J. P. de Figueiredo (LF’94) received the B.S. and M.S. degrees in electrical engineering from MIT, and the Ph.D. degree in applied mathematics from Harvard University. After being several years on the faculty of Rice University, Houston, TX, he joined the University of California, Irvine, in 1990, where he holds the position of Professor of electrical engineering and computer science, of bio-medical engineering, and of mathematics. He has published more than 370 papers, and chaired or co-chaired nine national or international conferences. Dr. de Figueiredo was the President of the IEEE Circuits and Systems Society in 1998, and served on several national and international committees and panels. For all these contributions, he received a number of awards, including the IEEE Fellow award, the 1994 IEEE Circuits and Systems Society Technical Achievement 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 IEEE Neural Networks Transactions Outstanding Paper Award, and the 2003 IEEE Circuits and Systems Transactions Guillemin-Cauer Best Paper Award.
