Abstract- A fuzzy filter is used for the noise reduction of images which is corrupted by additive noise. Fuzzy filter consists of two stages, first stage computes a fuzzy derivative for eight different directions, and the second stage uses these fuzzy derivatives to perform fuzzy smoothing by weighting the contributions of neighboring pixel values. These stages are based on fuzzy rules which make use of membership functions. The shape of the membership functions is adapted according to the remaining noise level after each iteration, making use of the distribution of the homogeneity in the image. Fuzzy filter can be applied to effectively reduce heavy noise. A statistical model for the noise distribution can be incorporated to relate the homogeneity to the adaptation scheme of the membership functions. Experimental results are also show the feasibility of the proposed approach. These results are also compared to other filters by numerical measures and visual inspection.
Index Terms- Additive noise, fuzzy logic, impulse noise, fuzzy image filtering, noise reduction.
INTRODUCTION
The concept of Fuzzy Logic was introduced by Lotfi Zadeh in 1965, a professor at the University of California at Berkley. Fuzzy Logic is a simple, rule-based IF X AND Y THEN Z approach to a problem solving control problem rather than attempting to model a system mathematically. Human beings make decisions based on rules. Although, we may not be aware of it, we make are all the dicisions based on computer like if-then statements. If the weather is fine, then we may decide to go out. If the forecast says the weather will be bad today, but fine tomorrow, then we make a decision not to go today, and postpone it till tomorrow. Rules are associate ideas and relate one event to another. Fuzzy machines, always tend to mimic the behavior of man and it also work the same way.
The application of fuzzy techniques is a promising research field in image processing. Fuzzy techniques have been already applied in several domains of image processing like filtering, interpolation, morphology. It have some practical applications like industrial and medical image processing. In this paper, we will focus on fuzzy techniques for blurring and noise removal. Now a day several fuzzy filters for noise reduction have been developed, e.g., the well-known FIRE-filter from, the weighted fuzzy mean filter from, and the iterative fuzzy control based filter. Mostly fuzzy techniques used for image noise reduction mainly deal with fat-tailed noise like impulse noise. These fuzzy filters are able to outperform rank-order filter schemes (such as the median filter).These fuzzy techniques are not specifically designed for Gaussian noise or do not produce convincing results when applied to handle this type of noise.
The fuzzy filter is explained for each pixel that is processed, first is the filter estimates a ‘fuzzy derivative’ in order to be less sensitive to local variations due to image structures such as edges; second is the membership functions are adapted accordingly to the noise level to perform ‘fuzzy smoothing,’ a set of 16 fuzzy rules is used for correction term. These rules make use of the fuzzy derivative as input. Fuzzy sets are represent the properties small, positive, and negative. While the membership functions for positive, and negativeare fixed, the membership function for small is adapted after each iteration. The adaptation scheme is briefly explained can be combined with a statistical model for the noise.
II. SYSTEM ANALYSIS
A new fuzzy filter is used for the noise reduction of images corrupted with additive noise. The filter consists of two stages. The first stage computes a fuzzy derivative for eight different directions and the second stage uses these fuzzy derivatives to perform fuzzy smoothing by weighting the contributions of neighboring pixel values. Both stages are based on fuzzy rules which make use of membership functions. The filter can be applied iteratively to effectively reduce heavy noise. The shape of the membership functions is change according to the remaining noise level after each iteration.
The main objective of this system is to focus on fuzzy techniques for image filtering. Now a day several fuzzy filters for noise reduction have been developed. These technique deals with fat-tailed noise like impulse noise and median filter. These system presents a new technique for filtering which is filter the images very easly at very high speed. The system first estimates a ‘fuzzy derivative’ in order to image structures such as edges and second, the membership functions are used accordingly to the noise level to perform ‘fuzzy smoothing’ For each pixel that is processed, a set of 16 fuzzy rules is fired to determine a correction term. These rules make use of the fuzzy derivative as input.
III. SYSTEM DESIGN
1. Mean Filter
We know that Mean filtering is a simple, intuitive and easy to implement method of smoothing images which is reducing the amount of intensity variation between one pixel and the next. It is often used to reduce noise in images. The idea of mean filtering is simply to replace each pixel value in an image with the mean (‘average’) value of its neighbors, including itself. It has the effect of eliminating pixel values which are unrepresentative of their surroundings. Mean filtering is usually thought of as a convolution filter. Like other convolutions it is based around a kernel, which represents the shape and size of the neighborhood to be sampled when calculating the mean. We used a 3??3 square kernel, as shown in Fig, although larger kernels (e.g. 5??5 squares) can be used for more severe smoothing. Computing the straightforward convolution of an image with this kernel byuse of the mean filtering process.
5 4 6
2 1 9
8 3 7
5 + 4 + 6 + 2 + 1 +
9 + 8 + 3 + 7 = 45
45 / 9 = 5
* * *
* 5 *
* * *
Center value
(previously 1)
is replaced by
the mean of all
nine values (5).
2. Median Filter
The median filter is normally used to reduce noise in an image similar to like as the mean filter. It does a better job than the mean filter of preserving useful detail in the image. Like the mean filter, the median filter considers each pixel in the image in turn and looks at its nearby neighbors to decide whether or not it is representative of its surroundings. Instead of simply replacing the pixel value with the mean of neighboring pixel values, it replaces it with the median of those values. The median is calculated by first sorting all the pixel values from the surrounding neighborhood into numerical order and then replacing the pixel with the middle pixel value. (If the neighborhood value contains an even number of pixels, the average of the two middle pixel values is used.) Fig an example of median filter
110 112 142 126 121
115 113 116 128 211
119 156 180 143 166
191 165 132 155 182
122 144 172 182 151
Neighbourho-od values:
113, 116, 128, 132, 143, 155, 156, 165, 180
Median value: 143
Fig: As can be seen the central pixel value of 180 is rather unrepresentative of the surrounding pixels and is replaced with the median value: 143. A 3??3 square neighborhood is used here larger neighborhoods will produce more severe smoothing.
3. Fuzzy Filter Module
The fuzzy filter is explained for each pixel that is processed, first is the filter estimates a ‘fuzzy derivative’ in order to be less sensitive to local variations due to image structures such as edges; second is the membership functions are adapted accordingly to the noise level to perform ‘fuzzy smoothing,’ a set of 16 fuzzy rules is used for correction term. The idea behind the rules is the following: if no edge is assumed to be present in a certain direction, the (crisp) derivative value in that direction can and will be used to compute the correction term. The first part (edge assumption) can be using the fuzzy derivative value and the second part (filtering) we will have to distinguish between positive and negative values.
(a) Fuzzy Derivative Estimation
In this estimating derivatives and filtering can be same as a chicken-and-egg problem, because for filtering we want a good indication of the edges, while to find these edges we need filtering. In our approach, we start by looking for the edges. We try to provide a robust estimate by applying fuzzy rules. Consider the neighborhood of a pixel as displayed in Fig. A simple derivative at the central pixel position (x,y) in the direction D (D ‘ dir= {NW,W,SW,S,SE,E,NE,N}) is defined as the difference between the pixel at (x,y) and its neighbor in the direction D. This derivative value is denoted by D(x,y).
x
NW N NE
W (x,y) E
SW S SE
NW N NE
W (x,y) E
SW S SE
y
(a) (b)
Fig: (a) Neighbour of a central pixel(x,y), (b) Pixel value gray are used to compute ‘Fuzzy Derivative’ of the central pixel (x,y) of th NW direction.
Fig: These three membership function are used to calculate fuzzy derivative.
Direction Position Set w.r.t (x,y)
NW
W
SW
S
SE
E
NE
N (x-1,y-1)
(x-1,y)
(x-1,y+1)
(x,y+1)
(x+1,y+1)
(x+1,y)
(x+1,y-1)
(x,y-1) {(-1,1) (0,0) (1,-1)}
{(-1,1) (0,0) (0,-1)}
{(0,1) (0,0) (-1,-1)}
{(1,1) (0,0) (-1,0)}
{(1,-1) (0,0) (-1,1)}
{(0,-1) (0,0) (0,1)}
{(-1,-1) (0,0) (1,1)}
{(-1,0) (0,0) (1,0)}
The table used to calculating for fuzzy derivatives in each direction.
These method used to find the small value for fuzzy derivative.
if ({ is small and ??NW(x-1,y+1) is small or
({??NW(x,y) is small and ??NW(x+1,y-1) is small
({??NW(x,y) is small and ??NW(x+1,y-1) is small then ??FNW(x,y) is small.
Now the above rule use to calculate the value of fuzzy derivative ??FNW(x,y) for the pixel (x,y) in the NW direction.
(b) Fuzzy Smoothing
To compute the correction term for the processed pixel value, we use a pair of fuzzy rules for each direction. The idea behind the rules is: if no edge is assumed to be present in a certain direction, the (crisp) derivative value in that direction can and will be used to compute the correction term. The first part (edge assumption) can be realized by using the fuzzy derivative value, for the second part (filtering) we will have to distinguish between positive and negative values.
This formula for rescaling the mean truthness we use the above formula, where dir contain direction and L represents the numbers of gray levels and correction term is ‘.
IV. IMPLEMENTATION AND RESULT
Implementation is the stage where the theoretical design is turned into a working system and it giving confidence on the new system for the users, where the user will be work efficiently and effectively. It this phase involves careful planning, investigation of the current System and its constraints on implementation and design of methods to achieve the change over methods. This phase is useful for education and training of users. For complex system implementation, the more involved will be the system analysis and the design effort. Implementation is the final and important phase and also the most critical stage in achieving a successful new system and in giving the users confidence.
Example : Code For Fuzzy Derivative Function
function FD = MyFuzzyDerivative(mm)
FD = zeros(1,8);
FD(1) = mm (1,1)-mm(2,2);
FD(2) = mm (1,2)-mm(2,2);
FD(3) = mm (1,3)-mm(2,2);
FD(4) = mm (2,1)-mm(2,2);
FD(5) = mm (2,3)-mm(2,2);
FD(6) = mm (3,1)-mm(2,2);
FD(7) = mm (3,2)-mm(2,2);
FD(8) = mm (3,3)-mm(2,2);
% Here mm define a matirix of 3*3 size
mm = [1 2 3; 4 5 6; 7 8 10];
mm = rand(3, 6);
bb = blkproc(mm,[3 3], @MyFuzzyDerivative)
After run Result is :
mm =
0.7922 0.0357 0.6787 0.3922 0.7060 0.0462
0.9595 0.8491 0.7577 0.6555 0.0318 0.0971
0.6557 0.9340 0.7431 0.1712 0.2769 0.8235
bb =
Columns 1 through 9
-0.0569 -0.8134 -0.1704 0.1104 -0.0914 -0.1934 0.0849 -0.1060 0.3604
Columns 10 through 16
0.6742 0.0143 0.6236 0.0653 0.1394 0.2451 0.7916
Example : To show the membership functions for Small properties.
clearall; close all; clc;
k = -255:255;
aa = trimf(k, [-255 0 255]);
plot (aa, ‘r–‘);
x =27;
y = trimf(x, [-255 0 255])
After run Result is :
y =
0.8941
Fig: Membership function, Small
Example : Code For Fuzzy Derivative Function
function NW = MyFuzzyDerivative(mm)
NW = zeros(1,3);
NW(1) = mm (1,1)-mm(2,2);
NW(2) = mm (-1,2)-mm(2,2);
NW(3) = mm (2,-1)-mm(2,2);
Example : Here we finde small prop, we use AND (Intrsection/Min) and OR (Union/Max) operation.
mm =rand(3,6)
NW = blkproc(mm,[3 3], @MyFuzzyDerivative)
disp(‘Intersection of a and b’);
iab = min(NW(1),NW(2))
disp(‘Intersection of a and c’);
iac = min(NW(1),NW(3))
disp(‘Intersection of b and c’);
ibc = min(NW(2),NW(3))
disp(‘Union of a,b and c’);
uabc=max(max(iab,iac),ibc)
After run Result is :
mm =
0.8147 0.9134 0.2785 0.9649 0.9572 0.1419
0.9058 0.6324 0.5469 0.1576 0.4854 0.4218
0.1270 0.0975 0.9575 0.9706 0.8003 0.9157
NW =
Columns 1 through 9
0.1824 0.2810 -0.3539 0.2734 -0.0855 -0.5054 -0.5348 0.3251 0.4795
Columns 10 through 16
0.4718 -0.3435 -0.3278 -0.0636 0.4852 0.3149 0.4304
Intersection of a and b
iab =
0.1824
Intersection of a and c
iac =
-0.3539
Intersection of b and c
ibc =
-0.3539
Union of a,b and c
uabc =
0.1824
V. Conclusion
This paper proposed a new fuzzy filter for additive noise re??duction. Its main feature is that it distinguishes between local variations due to noise and due to image structures, using a fuzzy derivative estimation. Fuzzy rules are fired to consider every di??rection around the processed pixel and the shape of the membership functions is adapted according to the remaining amount of noise after each iteration. Experimental results are also show the feasibility of the new filter and a simple stop criterion. The fuzzy filter is able to compete with state-of-the-art filter techniques for noise reduction. Finally, the fuzzy filter scheme is sufficiently simple to enable fast hardware implementations.
VI. REFERENCES
[1] Filtering Dimitri Van De Ville, Member,IEEE,MikeNachtegael, Dittrich Van der Weken, Etienne E. Kerre, Wilfried Philips, member, Noise Reduction by Fuzzy Image Filtering. IEEE Transaction on fuzzy systems, vol.11, No.4, August 2003.
[2] F. Russo and G. Ramponi, ‘A fuzzy operator for the enhancement of blurred and noisy images,’ IEEE Trans. Image Processing, vol. 4, pp. 1169’1174, Aug. 1995.
[3] ‘A fuzzy filter for images corrupted by impulse noise,’ IEEE Signal Processing Lett., vol. 3, pp. 168’170, June 1996.
[4] K. Arakawa, ‘Median filter based on fuzzy rules and its application to image restoration,’ Fuzzy Sets Syst., pp. 3’13, 1996.
[5] C.-S. Lee, Y.-H.Kuo, and P.-T. Yu, ‘Weighted fuzzy mean filters for image processing,’ Fuzzy Sets Syst., no. 89, pp. 157’180, 1997.
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