An Adaptive Fuzzy Image Smoothing Filter For Gaussian Noise

An Adaptive Fuzzy Image Smoothing Filter for Gaussian NoiseH. S. KAMFaculty of EngineeringMultimedia UniversityMALAYSIAS. N. CHEONGFaculty of EngineeringMultimedia UniversityMALAYSIAW. H. TANFaculty of EngineeringMultimedia UniversityMALAYSIAAbstract: - This paper presents a Gaussian noise filter that uses a sigmoid shaped membership function to modelimage information in the spatial domain. This function acts as a tunable smoothing intensification operator. With aproper choice of two sigmoid parameters ‘t’ and ‘a’, the filter strength can be tuned for removal of Gaussian noise inintensity images. An image information measure, Total Compatibility is used to adaptively select these sigmoidparameters. A visible improvement in the smoothness of images is observed, and the output of filter is also comparedwith those of other standard smoothing methods.Key-words: - image, noise, parameter, fuzzy logic, Gaussian, filter1 IntroductionIn the presence of noise, image smoothing is animportant pre-processing step followed by other taskssuch as edge detection, feature extraction and objectrecognition. Smoothing removes noise, but typicallyblurs edges as well. The conflicting need for bothsmoothing and preserving edges has given rise to thedevelopment of various filtering methods. Most of theeffective approaches are nonlinear and adaptive innature [3-4]. It is also desirable for the user to be able tocontrol the amount of smoothing according to theapplication.In a previous paper, we described the developmentof a tunable fuzzy filter for image smoothing [7]. Weproposed a parameter tuned filter based on the sigmoidfunction, with good performances in Gaussian, Impulseas well as a combination of both noise environments.The level of smoothing required was controlled bysetting the parameter value. In this paper, we presentthe same filter but with some modifications. Toimprove the ease of use, instead of the user having todetermine the parameter value (or use the default), animage information measure, Total Compatibility [6] isused to adaptively select the parameter value. Theeffectiveness of this modified filter in removingGaussian noise is highlighted through someexperiments.The paper is organized as follows. Section 2describes the sigmoid function and the parameters thatcontrol its shape. Section 3 describes the filter designand how the sigmoid function is to be used to controlthe removal of noise. Section 4 explains how the TotalCompatibility is used to adaptively tune the sigmoidparameters. Section 5 reports results of the filter on testimages and the comparison with other filters. Section 6concludes.2 The Sigmoid FunctionThe sigmoid shaped function is no stranger to fuzzyprocessing. Zadeh operator, which is a non-parametricsigmoid function, has been used as the fuzzyintensification operator, INT [5]. A parametric form ofthe sigmoid function was proposed for graylevel imagecontrast intensification by Madasu [2]. Later, it wasapplied to color image contrast intensification as well[1]. This function termed as the new intensificationoperator, NINT has more flexibility in determining theexact shape of the sigmoid than INT.We adopt the sigmoid function in [2] with 2parameters, t and a for the present study, given byµ(k)1 1 et ( d a )(1)where t controls the direction (and steepness) and acontrols the position of the curve on the horizontal axis.d is the value calculated from image information, e.g.,luminance differences. The shape of the sigmoid isdetermined by the choices for parameters a and t.Using the above function, we can design a tunablefuzzy filter, which smoothes depending on theparticular choice of parameters values.3 Filter DesignGenerally, an image I of size IxJ and intensity levels inthe range (0, L-1) can be considered as a collection offuzzy singletons in the fuzzy set notation,

I {µX(xi,j)} {µi,j /xi,j}; i 1,2 ,I; j 1,2 ,J(2)where µX(xi,j) represents the membership of someproperty µi,j of xi,j , where xi,j 0,1, ,L-1 is theintensity at (i,j)th pixel.For the transformation of the intensity xi,j in therange (0, L-1) to the fuzzy property plane in the interval(0,1), a membership function is used. The techniqueoperates on a window. For example, let us consider thewindow of size 3x3.As shown in Fig. 1, pixel luminance at location(i,j), is xi,j.jixi, jFig. 1: A 3x3 pixel windowThe window defines the central pixel, xi,j and itsneighbours. The membership of the pixel µX(xi,j) will beused to calculate a noise estimate at each xi,jNoise can be removed from the central pixel bymeans of subtracting a noise estimate. So, the windowmoves over every pixel, where we calculate a noiseestimate n. This n is to be subtracted from its originalintensity, xi,j to get the output intensity, yi,j.yi,j xi,j - n(3)The formula for the noise estimate at location (i,j)is obtained byn 1N µ (xkm, n km, n 03i m, j n)(xi, j xi m, j n )(4)where n is the weighted average of the differencesbetween the pixel of interest and its neighbours.µ3 is the membership function for thetransformation of the intensity xi,j in the range (0, L-1)to the fuzzy property plane in the interval (0,1).µ3(xi m,j n) 1 e(1t xm , n xm i , n j aL 1)(5)where m, n 0 , -k m,n k, k 1,2,3 (depends onwindow size, e.g. if window size is 3x3, k 1)Membership function µ3 is used for assigning theweight of the contribution from a particularneighbouring pixel towards the output. If the µ3 for aparticular neighbour pixel is higher, it contributes moreto the output intensity. In this membership function are2 tunable parameters, t and a and one variable d (seesection 2) derived from the image. To filter Gaussiannoise, t is set to be positive. d is the absolute intensitydifference between a particular neighbour and thecentral pixel. Parameter a sets the cutoff point for thisdifference, to result either in membership more or lessthan 0.5.If the central pixel xi , j is a Gaussian tail-end noisepixel, then a should be set larger, so that the noisycentral pixel can be neutralized by the neighbours’contribution, but not too large to cause detail blurring.Conversely, if the central pixel xi , j is part of a uniformarea or Gaussian noise, then a should be small to givehigh membership to only neighboring pixels similar inintensity to the central.In the membership function, the value of adetermines the selectivity. It should be observed that byvarying the value of a (0 a L 1), differentnonlinear behavior could be obtained. For removingGaussian noise, a bigger value of a will smooth more,but also result in more blurring. Therefore a needs to bechosen carefully.4 Parameter Tuning4.1 Parameter aWe propose to improve the performance of the filter bytuning the a parameter according to anothermembership function, which reflects the local imagecharacteristics.4.1.1 CompatibilityThe fuzzy membership function of total compatibilitywas defined by Choi and Krishnapuram [6] as a meansto quantify the local area characteristics in an image.If a given central pixel is an impulse noise pixel orGaussian tail-end noise pixel, then the gray level of thispixel will be significantly different from its neighbors.This means the degree that the neighboring pixels iscompatible with this noisy central pixel will be small.However, before we can start measuring compatibility,we also need to make sure that the neighboring pixelsare not also impulse noise pixels themselves. If theywere, then the compatibility measured would not beaccurate. So first, we have to establish the reliability ofthe neighboring pixels.To gauge the reliability of a pixel, we need to takeinto account the gray level differences between it andits neighbors.Reliability can be measured using variable β xi , j , whereβ xi , j 1N (xx m ,n Ai, j xm , n ) 2(6)

reflects the variance of the intensity differencesbetween the central pixel and its neighboring pixels. Ifβ xi , j is small, then it is likely to be reliable. Conversely,if β xi , j is large, then it could possibly be a Gaussiantail-end noise pixel.Let µ xi , j represents the degree of compatibility of aneighboring pixel xm ,n with respect to central pixelxi , j . The fuzzy membership function is defined by:µ xi , j (xi , j x m ,n )2 exp β xi , j (7)The variable β xi , j is large when the neighbouringpixel is unreliable, making the compatibility correctlylow even though the difference in intensity levels maybe small.In order to find out whether a particular central pixelis an Gaussian tail-end noise pixel, we have to considerthe compatibilities of all the neighboring pixels xm ,nwith respect to the central pixel xi , j . To evaluate thisproperty, we can simply take the mean of µ xi , j , asdescribedby1compatibility, µ C Nthefunction:k µm,n km. n 0xi , j(0 µ CTotal 1) (8)If the central pixel xi , j is part of an edge or Gaussiantail-end noise pixel, the compatibilities µ xi , j will below, resulting in a low µ C . Conversely, if the centralpixel xi , j is part of a uniform area or Gaussian noise,µ C would be high.Therefore it is useful to let parameter a varyaccording to µ C . When µ C is small it is desirable toset a higher a so that higher differences due to tail-endnoise can be corrected. However we do not want to blurthe edges, so there will be a upper limit to the value ofa.If µ C is high, we want to assign parameter a a smallvalue so that the Gaussian noise pixel will be replacedby the correction term contributed only by thecompatible neighbours. It is analogous to averagingbetween similar pixels. By experiment, it has beenfound that the filter usually performs well when a iswithin the range 40-80 for intensity images of 256levels.Thus, we set a -40 * µ C 80(9)4.2 Parameter tParameter t determines the direction of the sigmoidcurve. For Gaussian denoising purposes, t must be apositive number. Through experiments, it has beenfound that values of t above 20 work quite well and canbe fixed for the whole image as there was no significantimprovement found by varying the value of t from pixelto pixel.5 ResultsIn our experiments, we used the 256x256 Lena (Fig.2(a)) and 512x512 Baboon (Fig. 3(a)) images. Theimages were digitized into 256 gray levels. The windowsize used in our proposed filters was of size 7x7.Both images were corrupted by Gaussian noise withmean 0 and variance 0.005 as shown in Fig.s 2(b)and 3(b). The noise matrices were generated using aMATLAB subroutine. In addition to our proposedfilters, the noisy images were filtered with the 5x5Wiener filter and also with an image enhancementtechnique combining sharpening and noise reduction[8]. The reason for using the 5x5 window in the Wienerfilter is, because a 3x3 window tends not to smooth theflat areas enough whereas the 7x7 window tends to oversmooth with most details washed out. The results of allthe filters are presented in Fig. 3 and 4 below. Theresult of the application of the 5x5 Wiener filter onLena is shown in Fig. 3(c). The filter manages tosmooth the noise, but the edges are quite soft and theflat areas have a slight mottled appearance.The result of the combined sharpening and noisereduction filter [8] is shown in Fig. 3(d). It was appliedwith parameter alpha 50, as the parameter is userdefined and at this value it appears to balance sharpnessand reduced noise by visual inspection. This filteryields a sharper result but with increased unevenness inthe flat areas.The results of the proposed filter with parametersfixed [7] is show in Fig. 3(e). Values were fixed at a 50 and t 20. These values were chosen by experiment.Most areas are smoothed correctly and edges anddetails are preserved. However, some noise remains.Finally, the result of the application of our proposedfilter with tuned parameters is shown in Fig. 3(f). Thehomogenous areas are smooth and the edges are stillsharp.The next set of pictures involves a slice of the512x512 baboon image. The result of the application ofthe 5x5 Wiener filter on Lena is shown in Fig. 4(c). The