# Image Processing

NOISE REDUCTION BY USING FUZZY IMAGE FILTERING ABSTRACT The existing system available for fuzzy filters for noise reduction deals with fat-tailed noise like impulse noise and median filter. Only impulse noise reduction uses fuzzy filters. Gaussian noise is not specially concentrated; it does not distinguish local variation due to noise and due to image structure. The proposed system presents a new technique for filtering narrow-tailed and medium narrow-tailed noise by a fuzzy filter. The system first estimates a â€śfuzzy derivativeâ€ť in order to be less sensitive to local variations due to image structures such as edges.

Second, the membership functions are adapted accordingly to the noise level to perform â€śfuzzy smoothing. â€ť A new fuzzy filter is presented 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. 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 and effectively reduce heavy noise. In particular, 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. 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 obtained to show the feasibility of the proposed approach. These results are also compared to other filters by numerical measures and visual inspection.

Keywords: AFuzzy sets, Fuzzy filters, Fuzzy smoothing, Fuzzy derivative 1. INTRODUCTION The application of fuzzy techniques in image processing is a promising research field. Fuzzy techniques have already been applied in several domains of image processing (e. g. , filtering, interpolation, and morphology), and have numerous practical applications (e. g. , in industrial and medical image processing). In this project, we will focus on fuzzy techniques for image filtering. Already several fuzzy filters for noise reduction have been developed, e. g. , the well-known FIRE-filter from, the eighted fuzzy mean filter from, and the iterative fuzzy control based filter from. Most fuzzy techniques in image noise reduction mainly deal with fattailed noise like impulse noise. These fuzzy filters are able to outperform rank-order filter schemes (such as the median filter). Nevertheless, most fuzzy techniques are not specifically designed for Gaussian (-like) noise or do not produce convincing results when applied to handle this type of noise. Therefore, this paper presents a new technique for filtering, narrow-tailed and medium narrow-tailed noise by a fuzzy filter.

Two important features are presented: first, the filter estimates a â€śfuzzy derivativeâ€ť in order to be less sensitive to local variations due to image structures such as edges; second, the membership functions are adapted accordingly to the noise level to perform â€śfuzzy smoothing. â€ť For each pixel that is processed, the first stage computes a fuzzy derivative. Second, a set of 16 fuzzy rules is fired to determine a correction term. These rules make use of the fuzzy derivative as input. Fuzzy sets are employed to represent the properties and while the membership functions are fixed, the membership functions are adapted after each iteration.

The adaptation scheme is extensive and can be combined with a statistical model for the noise. The result of this method can be compared with those obtained by other filters. 2. FUZZY RULES 115 ISSN: 1817-3195 / E-ISSN: 1992-8615 Journal of Theoretical and Applied Information Technology , Islamabad PAKISTAN 31st May 2010. Vol. 15. No. 2. Â© 2005-2010 JATIT. All rights reserved www. jatit. org Human beings make decisions based on rules. Although, we may not be aware of it, all the decisions we make are all based on computer like if-then statements. If the weather is fine, then we may decide to go out. If the orecast 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 associate ideas and relate to one another. Fuzzy machines, which always tend to mimic the behavior of man, work the same way. However, the decision and the means of choosing that decision are replaced by fuzzy sets and the rules are replaced by fuzzy rules. Fuzzy rules also operate using a series of if-then statements. For instance, if X then A, if y then b, where A and B are all sets of X and Y. Fuzzy rules define fuzzy patches, which is the key idea in fuzzy logic.

A machine is made smarter using a concept designed by Bart Kosko called the Fuzzy Approximation Theorem (FAT). The FAT theorem generally states a finite number of patches can cover a curve as seen in the Figure 1. 1. If the patches are large, then the rules are sloppy. If the patches are small then the rules are fine. Ebrahim Mamdani in the late 70s, three steps are taken to create a fuzzy controlled machine: A) Fuzzification (Using membership functions to graphically describe a situation). B) Rule evaluation (Application of fuzzy rules). C) Defuzzification (Obtaining the crisp or actual results).

As a simple example on how fuzzy controls are constructed, consider the following classic situation: the inverted pendulum. Here, the problem is to balance a pole on a mobile platform that can move in only two directions, to the left or to the right. The angle between the platform and the pendulum and the angular velocity of this angle are chosen as the inputs of the system. The speed of the platform hence is chosen as the corresponding output. Step 1: First of all, the different levels of output (high speed, low speed etc. ) of the platform are defined by specifying the membership functions for the fuzzy sets.

The graph is shown below. Figure 2 Graph of Fuzzy Set Similarly, the different angles between the platform and the pendulum and Figure 1 Fuzzy Patches covering a curve 2. 1 Fuzzy Patches In a fuzzy system this simply means that all our rules can be seen as patches and the input and output of the machine can be associated together using these patches. Graphically, if the rule patches shrink, the fuzzy subset triangles get narrower, since novice can build control systems that beat the best math models of control theory. Naturally, it is math-free system. 2. Fuzzy Control Fuzzy control, which directly uses fuzzy rules, is the most important application in fuzzy theory. Using a procedure originated by Figure 3 Angles between platform and pendulum The angular velocities of specific angles are also defined 116 ISSN: 1817-3195 / E-ISSN: 1992-8615 Journal of Theoretical and Applied Information Technology , Islamabad PAKISTAN 31st May 2010. Vol. 15. No. 2. Â© 2005-2010 JATIT. All rights reserved www. jatit. org Figure 6 Applications of Rules â€“ Graph2 Consider the rule “if angle is zero and angular velocity is zero, the speed is zero”.

The actual value belongs to the fuzzy set zero to a degree of 0. 75 for “angle” and 0. 4 for “angular velocity”. Since this is an AND operation, the minimum criterion is used , and the fuzzy set zero of the variable “speed” is cut at 0. 4 and the patches are shaded up to that area. This is illustrated in below graph. Figure 4 Angular Velocities of Specific angles Step 2: The next step is to define the fuzzy rules. The fuzzy rules are merely a series of if-then statements as mentioned above. These statements are usually derived by an expert to achieve optimum results.

Some examples of these rules are: i) If angle is zero and angular velocity is zero then speed is also zero. ii) If angle is zero and angular velocity is low then the speed shall be low. An application of these rules is shown using specific values for angle and angular velocities. The values used for this example are 0. 75 and 0. 25 for zero and positive-low angles and 0. 4 and 0. 6 for zero and negative-low angular velocities. Figure 7 AND operation on Angle and Angular velocity Similarly, the minimum criterion is used for the other three rules.

The following figures show the result patches yielded by the rule “if angle is zero and angular velocity is negative low, the speed is negative low”, “if angle is positive low and angular velocity is zero, then speed is positive low” and “if angle is positive low and angular velocity is negative low, the speed is zero”. Figure 5 Applications of Rules â€“ Graph1 117 ISSN: 1817-3195 / E-ISSN: 1992-8615 Journal of Theoretical and Applied Information Technology , Islamabad PAKISTAN 31st May 2010. Vol. 15. No. 2. Â© 2005-2010 JATIT. All rights reserved www. jatit. org medium narrow-tailed noise by a fuzzy filter.

The system â€˘ First estimates a â€śfuzzy derivativeâ€ť in order to be less sensitive to local variations due to image structures such as edges Second, the membership functions are adapted accordingly to the noise level to perform â€śfuzzy smoothing. â€ť â€˘ Figure 8 Three Rules Result The four results overlaps and is reduced to the following resulting graph shown in Fig 1. 9. For each pixel that is processed, the first stage computes a fuzzy derivative. Second, a set of 16 fuzzy rules is fired to determine a correction term. These rules make use of the fuzzy derivative as input. Fuzzy sets are employed to represent the properties,, and While the embership functions for and are fixed, the membership function for is adapted after each iteration. 3. 1 Noise Model Figure 9 Resulting graph Step 3: The result of the fuzzy controller as of know is a fuzzy set (of speed). In order to choose an appropriate representative value as the final output (crisp values), defuzzification must be done. There are numerous defuzzification methods, but the most common one used is the center of gravity of the set as shown in Fig 1. 10. As previously mentioned, the scope of this project is limited to the study of impulse noise removal. Several distinct models exist for impulse noise. “Pure” impulsive noise, i. e. , impulsive noise in which a corrupted pixel takes on a gray scale value of either 0 or 255, is relatively simple to remove, since such maximal and minimal gray scale values occur relatively infrequently in actual images. â€˘ A more realistic and challenging, from a noise removal standpoint, is “bit error” impulsive noise. Bit error impulsive noise can be quantitatively described as follows: Given an uncorrupted two-dimensional signal, s, such that each signal value is quantized to B bits, each s( i , j ) can be written as: s( i , j ) = k1( i , j ) 2B-1 + k2( i , j ) 2B-2+… kB-1( i , j ) 2 + kB( i , j ) where km ( i , j ) all i , j. Figure 10 Defuzzification Method [0,1] for all 1 < m < B and 3. PROPOSED SYSYTEM The proposed system presents a new technique for filtering narrow-tailed and If we now assume that each coefficient of each s( i , j ) is corrupted with probability p independently of all other coefficients in s( i , j ) and of all other signal values, then the corrupted signal values, x( i , j ), are of the form: 118 ISSN: 1817-3195 / E-ISSN: 1992-8615 Journal of Theoretical and Applied Information Technology , Islamabad PAKISTAN 31st May 2010. Vol. 5. No. 2. Â© 2005-2010 JATIT. All rights reserved www. jatit. org limitations. In these variants of the median filter, the pixel value is modified only when it is found corrupted with noise. The median filter is used to remove noise from an image by replacing pixels with the middle pixel value selected from a certain window size. The median filter is very effective at removing noise while not destroying sharp edges in an image. The general idea behind the filter is to average a pixel using other pixel values from its neighborhood, but simultaneously to take care of important image structures.

The main concern of the proposed filter is to distinguish between local variations due to noise and due to image structure. In order to accomplish this, for each pixel we derive a value that expresses the degree in which the derivative in a certain direction is small. Such a value is derived for each direction corresponding to the neighboring pixels of the processed pixel by a fuzzy rule. 4. 2. 1 Fuzzy Derivative Estimation x( i , j ) = k1*( i , j ) 2B-1 + k2*( i , j ) 2B-2+… + kB-1*( i , j ) 2 + kB*( i , j ) where km*( i , j ) = { km( i , j ), with probability1-p; 1-km( i , j ), with probability p } . IMPLEMENTATION Portable Gray Map (PGM) is a standard bitmap based format consisting of a 4 lines header, and data stored in the unsigned char type, providing a maximum of 256 gray scale levels or 8-bit data per pixel. The general structure of a PGM image file The header of a PGM image file consists of: â€˘ First line containing the signature of the image file and identifies the file as PGM â€˘ Second line is the comment line â€˘ Third line provides information about number and rows and columns of data stored in the file, and â€˘ Fourth line specifies maximum gray level contained in the image 4. Mean Filter Module Estimating derivatives and filtering can be seen as a chicken-and-egg problem; for filtering we want a good indication of the edges, while to find these edges we need filtering. // calc fuzzy derivative by applying fuzzy rule for ‘small’ fuzzy set for(int t=0;t