A FUZZY FILTER FOR IMPULSE NOISE REMOVAL USING ADAPTIVE SIZED FILTER WINDOW, AND
ADAPTIVE TRIMMING RANGE
K.Kadambavanam
1, T.Senthilnathan
21
Associate Professor and Head (Retd.), Department of Mathematics, Sri. Vasavi College, Erode, Tamil Nadu, India.
2
Associate Professor, Department of Mathematics, Erode Arts and Science College, Erode, Tamil Nadu, India.
ABSTRACT
This paper proposes a fuzzy filter to remove impulse noise from an image with a combination of (i) rank conditioned median (RCM) filter of adaptive window size, (ii) adaptive trimming range, (iii) adaptive fuzzy thresholding technique. This algorithm uses two types of filter windows, one for detecting the corrupted pixels and the other one to denoise the pixel. In this process after ranking in the RCM filter of adoptive size, the pixels lying outside the adoptive trimming range are further tested for the presence of noise in the process of fuzzy thresholding. This algorithm uses automated threshold value instead of a crisp threshold value as the level of corruption varies from pixel to pixel. The better performance of the filter is established on the basis of Peak Signal to Norse Ratio (PSNR) values calculated by comparing the original and restored images.
Keywords: Fuzzy Logic, Image Filtering, Impulse Noise, Median Filter, Noise Detection.
I. INTRODUCTION
Filtering is one of the most important operations in image processing and computer vision. Filtering is a function of values at a given location and its neighborhood of the input image. According to Markov process, near pixels are likely to have similar values. But this concept fails at locations that contain edges and image details (e.g. lines, end of lines corners, etc.). Most of the classical linear filters, like the averaging low pass filters destroy the edges, lines and other fine details of the input image. It is resulted in blurred output. The median filter is an important non-linear order statistic filter [1]. Though median filter is simple and computationally efficient it replaces all the pixels (irrespective of them being corrupted or not) of the input image with the median value of the pixels around the local neighborhood. It results in considerable lose in image details [2].
To make a tradeoff between noise reduction against detail preservation, Ko et al.[3] proposed the Centre Weighted Median (CWM) filter. In this method a weight adjustment is applied to the center pixel within the sliding filter window. It results in the median of the extended set. While the centre weight is increased, the performance of CWM filter increases in detail preservation but decreases in noise reduction.
The main aim of the entire process of filtering is to preserve the image details, and to remove the impulse noise effectively. Hence a filtering technique was applied only on the corrupted pixels and to keep the noise free pixels unchanged.
As the local conditions in some location of the image can be evaluated only vaguely, it is very difficult, if not possible to set the conditions under which a certain filter should be selected. For this reason, the filtering system should be able to perform reasoning with uncertain vague information, in which fuzzy logic is used commonly.
To improve the performance of filtering process of median filter different approaches such as the centre weighted median (CWM) filter [3], the RCM filter [4], the switching median (SM) filter [5], the tri-state median (TSM) filter [6], the multi-state median (MSM) filter [7], adaptive two pass rank order filter [8], are proposed.
To remove impulse noise many fuzzy filters [9]-[19] including Fuzzy median filter (FMF), adaptive weighted mean filter (AWMF), fuzzy weighted mean filter, fuzzy switching filters, FIRE filters have been proposed . In general, classification of fuzzy filters falls into two categories. In the first, the fuzzy filter is entirely based on fuzzy techniques. In the in the next, the fuzzy filter is based on the fuzzification of more than one classical filtering method. This article proposes a simple fuzzy filter which belongs to the latter category. The results establish that the proposed filter performs better than most of the filters studied.
The rest of this article is organized as follows: Section II contains the description of the proposed filter.
Subsequently its simulation results and discussion are presented in section III. The last section is devoted for conclusion.
II. DESCRIPTION OF PROPOSED FILTER
The proposed algorithm uses two types of neighborhood windows. The first one is “noise detector filter” with varying window size. It is used to identify the noise pixels, whereas the second one is “denoiser filter” with a fixed size of , used to denoise the noise pixels. It composed of two steps.
Step (1). The size of the noise detector filter window is initialized as ,where ‘pad’
is initialized as 1. The window is placed in such a way that is the central pixel of the filter window. The usual assumption is that the noise pixels in a filter window will fall in either end of the ordered sequence. So to eliminate the existing corrupted pixels in the filter window, the gray values of the pixels are sorted in ascending order and trimmed on either end. As a varying sized noise detector window is used, a relatively varying trimming size is also adapted. The trimming size is assigned as the subsequent value the variable “pad”.
Step (2)
.
The central pixel is checked whether it lies in the trimmed rank order set. If so, it is left unaltered.Otherwise, it is replaced by the median of denoiser filter. This process is repeated by increasing the „pad’ size by 1 until it reaches the maximum of user defined value „Max_Pad‟.
III. MATHEMATICAL MODEL FOR THE PROPOSED ALGORITHM
Let be the set of pixels in the input image and be the denoised output image, where be the coordinates of the pixel
,
and be the height and width of the3.1. Constructing Noise ‘Detector Filter’
The size of the noise detector filter window is formulated by the equation
where
,
and „pad’ is initialized as 1.Let be the position of the pixel to be checked (whether the pixel is corrupted or not). So place the filter widow in such a way that the center of the window is fixed at
.
Let denote a sample input vector of all the pixels in the neighborhood window.
: (i.e.) (1)
where is the value at the north west corner of the filter window and is the value at the south east corner of the filter window. The pixels are arranged row by row from top to bottom and left to right. The rank order set is formed as follows:
: (2)
:
whereThen the trimming range is fixed as „pad’. The rank order set is trimmed
.
:
(i.e.) Trimmed rank order set is(3)
3.2. Testing whether the input sample lies in the Trimmed Rank Order Set and denoising.
If lies in the trimmed rank order set given in the equation (3) then it is left unaltered
.
:
(i.e.)(4)
If does not lie in the trimmed set, then perform the following part:
:
(i.e.)The denoiser window
)
is defined as a sample input vector of all the pixels in the neighborhood window of size around the pixel.
:
(i.e.)(5)
where is the value at the top left corner of the denoiser window and is the value at the bottom right corner of the denoiser window. The pixels are arranged from top to bottom and left to right. The rank order set of denoiser window is formed as follows:
: (5)
where
.
Then the denoised value is defined as: . (6)
This process is repeated after increasing the „pad’ size by 1 until it reaches the maximum of user defined value
„Max_Pad‟.
Fig.(1). Flow Chart of the Proposed Denoising Process
IV. EXPERIMENT RESULTS AND ANALYSIS
This section compares the efficiency of the proposed image restoration algorithm, with the existing median type filters based on either fuzzy or non-fuzzy techniques on gray scale images of size corrupted by impulsive noise at various levels. The results are graded by measuring the Peak Signal to Noise Ratio (PSNR).
4.1 Peak Signal to Noise Ratio (PSNR)
It is one of the approximations to human perception of reconstruction quality. Higher value of PSNR indicates that the reconstruction is of higher quality. Its unit of measurement is in decibels (dB). It is derived via the mean squared error (MSE) as follows:
If X is the noise-free monochrome image of size
(m×n),
and Y is the reconstructed image from the corrupted image then, the MSE is obtained from the equation: (7)
and the PSNR is calculated by the formula: (8)
where B is the bits per sample pixel. In real images B is set as 8.
4.2 Parameter Setting
All the denoising algorithms have some crucial parameters needed to be adjusted. The parameters are noise dependent. Their selection will trivially influence on denoising. This section discusses about the parameter setting. The denoiser filter of size is used the smallest size filter will preserve image details. While identifying the noise pixel, the noise detector window of varying size is used. Depending on the maximum size of the detector window
(„Max_Pad‟)
the efficiency of denoising varies. There are two types of experiments conducted on the experimental objects.4.2.1 Type (1) Experiment on the Objects
It is conducted to test the effect of the parameter
„Max_Pad‟
in outputs. In this experiment„Max_Pad‟
is assigned values 1 through 10 in steps of 1. Results show that the PSNR of the output is varying with the parameter „Max_Pad‟. During this experiment the induced noise level is set as 5% on the input images „lena’and ‘baboon’. The graphical representation of the effect of „Max_Pad‟ is given in fig.(2).
Fig.(2) Effect of the Parameter ‘Max_Pad’ in Outputs
4.2.2 Type (2) Experiment on the ObjectsIt is conducted to compare the results obtained by the existing denoising algorithms with the proposed algorithm using PSNR. The value of „Max_Pad‟ is fixed as 10 (the optimal value has been obtained in the first experiment). The “Salt & Pepper” noise level is set at various levels ranging from 5% through 25%. Simulation of noise was performed by 100 independent runs on each level. The simulated different structures of noise in the input image are tested by RCM, MSM, FIRE, CWM, TSM, AWFM, FMF and Proposed methods. The average PSNR% values are presented in Table (1). taken for comparison.
(a) (b)
Fig (3). Test Images Used in this Paper (a) Lena Image (b) Baboon Image.
V. CONCLUSION
From the Tables (1) and (2) it is observed that as noise level increases the denoising performance is decreasing.
Comparing all the cases the proposed algorithm is performing better than the rest of the algorithms.
Table(1). Comparative Results in PSNR of Different Filtering Methods for Various Percentages of Impulse Noise in the ‘Lena’ Image.
Noise ratio
Filters 5% 10% 15% 20% 25%
Median 33.2 32.5 32.0 31.5 30.8
RCM 38.36 37.20 35.28 34.27 32.6
MSM 39.9 37.5 35.9 32.32 31.8
FIRE 35.8 33.83 30.9 28.3 24.4
CWM 37.0 34.8 32.9 30.5 29.0
TSM 39.1 37 35.7 32.0 30.9
AWFM 33.6 32.9 31.7 31.4 30.9
FMF 37.9 35.6 33.28 31.8 29.5
Proposed 44.52 40.86 36.96 33.28 29.38
Table (2). Comparative Results in PSNR of Different Filtering Methods for Various Percentages of Impulse Noise in the Baboon Image
Noise ratio
Filters 5% 10% 15% 20% 25%
Median 22.7 22.6 22.46 22.39 21.8 RCM 26.67 26.37 25.78 25.1 24.29 MSM 30.0 27.5 26.0 25.2 24.4 FIRE 26.3 25.45 24.1 22.86 20.8
CWM 28.1 26.1 24.8 24 23.6
TSM 28.8 26.7 25.5 24.5 23.7 AWFM 24.6 24.52 24.34 24.22 24.1 FMF 29.7 27.12 26.83 24.58 22.9
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Dr.K.Kadambavanam was born in Palni, Tamilnadu, India in 1956.He did his post- graduate studies at Annamalai University, Tamil Nadu, India in 1979. He completed his M.Phil. degree in Annamalai University in 1981. He obtained his Ph.D., degree from Bharathiar University, Coimbatore, Tamil Nadu, India. His area of research is Fuzzy Queueing Models and Fuzzy Inventory Models. His research interests include Solid mechanics, Random Polynomials and Fuzzy Clustering. He has 33 years of teaching experience in both U.G and P.G level and 18 years of research experience. He was the Chairman of Board of Studies (P.G) in Mathematics and Ex-Officio Member in Board of Studies (U.G) in Mathematics in Bharathiar University, Coimbatore. He served as member in : Panel of Resource Persons, Annamalai University, Annamalai Nagar, Doctorial Committee for Ph.D., program, Gandhigram Rural University, Gandhigram.
T. Senthil nathan is an Associate Professor in Post Graduate and Research Department of Mathematics, Erode Arts and Science College, Erode, Tamil Nadu, India. He received M.Sc. from Periyar E.V.R College, Trichy, Tamil Nadu, and M.Phil. from A.V.V.M. Sri.Pushpam College, Tanjavour, Tamil Nadu, India in 1985 and 1987 respectively He received M.S. (software) from Birla Institute of Technology and Science (BITS), Pilani, India in 1994. He has 25 years of experience in teaching. He is a member in Board of Studies (U.G. and P.G.) in Mathematics in Erode Arts and Science College. Her research interests include image processing, signal processing, fuzzy logic and its application.
