Periodic Noise Recognition and Elimination UsingRFPCM Clustering

Souradeep Dutta∗, Arijit Mallick†, Sourya Roy‡, Utkarsh Kumar§Department of Instrumentation and Electronics Engineering

Jadavpur UniversitySalt Lake Campus, Kolkata, India

Email: duttasouradeep39@gmail.com∗ Email: aribryan@gmail.com† Email: souroy099@gmail.com‡ Email: utkarsh.iee.ju@gmail.com§

Abstract—Frequency domain filtering is a conventionalmethod in eliminating periodic noise in an image. Conventionalmethods include filtering via bandpass filter and notch filter fordetecting both periodic and quasi-periodic noise in an image.Inthis paper, RFPCM clustering has been used to detect periodicnoise spikes in Fourier amplitude domain. Qualitative and com-parative analysis has been presented in this paper for clusteredmask method and the conventional method as well. PSNR valuesof the resultant images are taken as filtering quality factor andcomparison analysis is taken with respect to this value.

Keywords—Frequency domain filtering, periodic noise, cluster-ing, rough fuzzy, fourier amplitude domain, PSNR

I. INTRODUCTION

In the fourier amplitude domain the periodic and quasiperiodic noise appear as a bright peak points around centralfrequency. Conventional methods of filtering include introduc-ing lowpass, bandpass and notch filters [1]–[3] in order tosuppress the effect of the noise spikes. Modified filters suchas Windowed Gaussian Notch Filter (WGNF) [4] ,adaptedmedianfilters [4], [5] have also been used. Previously, in thefollowing section we will apply rough fuzzy technique inorder to cluster the noise spikes, resulting into most efficientsuppression of noise frequency. Comparative analysis is alsogiven in order with respect to conventional filtering techniquesto that of our technique.

II. IMAGE REPRESENTATION IN FREQUENCY DOMAIN

Digital image is basically two dimensional discrete signal.In spatial domain, changes in image position correspond tothe change in pixel intensity. Whereas in case of frequencydomain representation this corresponds to change in spatialfrequency. To transform a digital image from spatial to itsfrequency representation, discrete Fourier transform is used.The 2-D discrete Fourier transform of an M×N image f(x, y)is denoted by F (u, v), is given by

F (u, v) =

M−1∑x=0

N−1∑y=0

f(x, y)e−j2π(uxM + vx

N ) (1)

for x = 0, 1, ,M − 1 and y = 0, 1, , N − 1

Inverse operation can be done by the following equation

f(x, y) =1

MN

M−1∑u=0

N−1∑v=0

F (u, v)ej2π(uxM + vx

N ) (2)

for u = 0, 1, ,M − 1 and v = 0, 1, , N − 1

Fig. 1. A grayscale image(Pepper)

Fig. 2. Fourier transform of image shown in figure 1

III. IMAGE FILTERING IN FREQUENCY DOMAIN

Image restoration from any noise corrupted image requiresimage filtering. Image filtering can be done in frequencydomain as well as in spatial domain. Convolution theoremplays the main role in both the cases. In spatial domain,to achieve filtering we need to convolve the image f(x, y)with the spatial filterh(x, y). As per convolution theorem sameresults can be achieved, if we just multiply Fourier transform of

the image, F (u, v)by H(u, v), Fourier transform of the spatialfilter [6]. So we can write this as,

f(x, y) ∗ h(x, y)⇐⇒ H(u, v)F (u, v) (3)

Inversion of the product H(u, v)F (u, v) gives the spatiallyfiltered image. On the basis of effect of filtering on thefrequency counter-part of the image, there are mainly threetypes of filter: low-pass, high-pass and band-pass filter. As thename suggests, low-pass filter retains the lower frequencies ofthe image. High-pass filter does the opposite. Band pass allowsa specific range of frequencies and diminishes others. Specificfilter is chosen, as per our requirement and also this choiceis very much dependent on the nature of noise that broughtcorruption. There are various type of noise that can corrupt animage. In our work, we only dealt with specifically periodicnoise which is a very common one and our main objective wasto remove periodic noise more efficiently.

Fig. 3. A periodic noise corrupted image

Fig. 4. Fourier transform of noisy image

IV. PROPOSED ALGORITHM

In the following stages an algorithm is proposed, in orderto efficiently identify the noise components in the Fourierspectrum of the image, and thus remove them using some well-known filtering techniques. The techniques used have beenexplained in detail in the following paragraphs.

A. RFPCM Clustering

Fig. 5. An RFPCM Cluster represented by crisp lower bound and fuzzyboundary

RFPCM [7] incorporates both fuzzy and rough sets, it addsthe concept of fuzzy membership (both probabilistic and pos-sibilistic) of fuzzy sets and, lower and upper approximationsof rough sets into a c-means algorithm. While the member-ship of fuzzy sets enables efficient handling of overlappingpartitions, the rough sets deal with uncertainty, vagueness andincompleteness in class definition. In the RFPCM, each clusteris represented by a centroid, a crisp lower approximation,and a fuzzy boundary, see figure 5. The image obtained fromFourier transform of the noisy image was clustered using theRFPCM algorithm, with two randomly chosen centroids. Thepixels belonging to the higher gray valued cluster were set tobackground and all other clusters were set to foreground theresultant image has been shown in figure 6.

Fig. 6. Resultant image after RFPCM clustering

B. Separation of the Central frequency

The image obtained in the above step was spatially seg-regated, i.e. to remove the overlapping between the centralfrequency and the noise components by building an imaginaryboundary of white pixels between them. This was accom-plished by dilating [8] the image obtained from Step-I, usinga 33 structuring element. The resultant image had the centralfrequency and the noise components separated spatially, asshown in figure 7.

Fig. 7. Resultant image after dilation of clustered image

Thus the image now, has a clear demarcation betweenthe central frequency and the noise components, as shown infigure 7.

C. Elimination of the Central frequency

In order to identify the central frequency, a connectedcomponent analysis [8] was initiated from each black pixelsin the central zone of the image in figure 7. The central zonewas approximated to an optimum limit on both axes, confinedprimarily to the spatial central of the image obtained fromSTEP-III. Since the central frequency corresponds fundamen-tally to image frequencies, hence, the regions obtained from theconnected component analysis were turned white, or in otherwords, removed. The resultant image is shown in figure 8.

Fig. 8. Resultant image after central frequency elimination

D. Identification of the Noise components

The process in STEP-III provides a good approximation ofthe noise components. For a further refinement of the process,a connected component analysis was once again initiated,and the regions obtained were either retained or eliminatedbased on the number of pixels it had. The pixel count wascompared to a particular threshold, any region having pixelcount less than the particular threshold was eliminated .It wasobserved that, for a threshold of 50 pixels/region, the resultswere satisfactory. The result obtained from the above step wasfurther dilated, and the image obtained has been shown infigure 9.

Fig. 9. Cluster filter mask Fig. 10. CMF filtered image

E. Detection of noise components and filtering

Next, to obtain the coordinates of the noise components,from each of the regions identified in the above step, furtheranalysis was performed on the image as shown in figure 9,and a cluster filter mask was formed. Figure 10 shows thefinal output image of clustered mask filter (CMF).

V. RESULTS

We have described our method gradually in the previoussection with one grayscale image, named Pepper. For furthervalidation, we have applied these algorithm also on other3 grayscale images with sinusoidal noise corruption. Thesethree images are Gatlin, Mandrill and Street. The outputimages are shown in this section with their original and noisecorrupted versions. As a proof of better periodic noise filtrationa corresponding PSNR table (Table 1) is also provided withthe output images. To draw conclusions about the superiorityof our proposed we have also compared the results withconventional manual Bandpass and notch( BP + notch filter)filtering. ( figure 13, figure 17, figure 21)

Fig. 11. Original image Fig. 12. Corrupted image

Fig. 13. BP+ notch filtered image Fig. 14. CMF filtered image

Fig. 15. Original image Fig. 16. Corrupted image

Fig. 17. BP+ notch filtered image Fig. 18. CMF filtered image

Fig. 19. Original image Fig. 20. Corrupted image

Fig. 21. BP+ notch filtered image Fig. 22. CMF filtered image

Conventionally manual combination of high pass and lowpass filters are combined manually covering the noise peaks,forming a bandpass filter along with selective notch filter forminor suppressing action of additional noise peaks, if any.Fourier transform of noisy image reveals that the noise peaks

are not exactly uniform, rather they extended surrounding thecentral noise peaks in a non uniform clustered pattern. Hencefrequency domain should also be a cluster pattern which willsuppress this non uniform patch. Here comes the conceptof a cluster oriented filter rather than a uniform band filter.After a certain threshold intensity, clustering is performedon frequency map except the central original component. Aqualitative analysis is performed with respect to band filteringand the results are shown in Table I. PSNR value is taken asthe quality factor [9]. More the PSNR value, more the qualityof denoising and hence more efficient the filter is.

TABLE I.

Name of Image PSNR of BP + notch filter output PSNR of CMF output

Pepper 34.44 db 36.36

Gatlin 22.11 db 22.36

Mandrill 20.53 db 22.11

Street 17.71 db 18.53

VI. CONCLUSION

In this paper, we have tried to establish a new procedureof frequency domain filtering. Conventionally a bandpass anda notch filter combination is used, where the peak noise pointand certain region around it is covered. We know that it isnot a uniform band but a cluster of points in the frequencymap. Hence simple bandpass and notch filtering wont providecomplete filtering. A certain cluster of noise component alwaysextends out of the uniform bandpass filter band. It can beobserved that most filtered output is obtained from clusteredmask filter (CMF), which is a hybrid of notch filters and thecluster mask because it not only suppresses the peak point ofspikes, it also suppresses its extended non uniform clusterednoise portions, so that the influence of that particular noisefrequency is as low as possible. Hence a higher PSNR valuecan be expected from CMF output. PSNR values (Table I)confirm the maximum efficiency of CMF.

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