2-D Fir Filter Design And Its Applications In Removing Impulse Noise In Digital Image

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1 -D Fir Filter Design And Its Alications In Removing Imulse Noise In Digital Image Nguyen Thi Huyen Linh 1, Luong Ngoc Minh, Tran Dinh Dung 3 1 Faculty of Electronic and Electrical Engineering, Hung Yen University of Technology and Education, Hung Yen, Viet Nam, huyenlinh.utehy@gmail.com School of Engineering and Technology, Vinh University, Vinh, Viet Nam, minhln@vinhuni.edu.vn 3 School of Engineering and Technology, Vinh University, Vinh, Viet Nam, dungtd@vinhuni.edu.vn Abstract: This aer conducts research into the design of -D FIR filter using the REMEZ algorithm and the alications of -D FIR filter in the digital image rocessing field. In the rocedure, the author has studied the theories to create algorithm and simulation utilizing MATLAB software to design -D FIR filter and therefore, tested its alications in restoring digital image having imulse noise. Keywords: DSP, -D FIR filter, REMEZ, image rocessing. I. INTRODUCTION In most areas, the filter is erceived as the heart of the system. In another word, there is not a system that does not use the filter. The use of filters designed on the basis of signal rocessing has been of very early interest. Previously, derived from signal analysis and signal rocessing in a time domain, the authors studied the design of the 1-D digital filter. Since then, it has been develoed into the -D digital filter for signal rocessing in the satial domain. There are two tyes of digital filters: finite imulse resonse (FIR and infinite imulse resonse (IIR. Most of the authors are interested in the design of the FIR filter as the FIR filter is a non-resonsive, finite and stable filter. Hence, the design is simler. The FIR filter design can be achieved by methods such as FFT, window method and otimal aroximation method of the REMEZ algorithm. For the 1-D FIR filter design, a number of authors are attentive in the REMEZ algorithm based on the Chebyshev s otimal aroximation theory. Nevertheless, when designing a -D FIR filter or develoing a multi-dimensional FIR filter (M-D, the use of the REMEZ algorithm meets a number of hindrances due to the set of cos functions used in D aroximation ( M-D that does not satisfy the conditions in the domain of interest. Furthermore, the technique of utilizing modified algorithms for the exansion of -D (MD of Chebyshev s aroximation calculations is quite comlex. In the image rocessing field, a large number of studies have alied filter tyes to resolve roblems including imulse noise. For the imulse noise removal, they use nonlinear filters such as the Median filter. However, this aer chooses the -D FIR filter using REMEZ algorithm, which is a linear filter, to handle the imulse noise in digital images. This is a quite bold idea to be considered as a ilot study. Thus, this toic has a real scientific significance. II. THE -D FIR FILTER A. Overview of the -D FIR filter The -D FIR filter is a linear, discrete, and invariant system in the satial domain (m, n. The D differential equation that defines the invariant linear filter has the following form: Available 1

2 N 1 N 1 m = n = 1 1 M 1M 1 m = n = 1 1 ( a ( m, n y m m, n n m, n 1 1 = b ( m, n x m m, n n where m, n x( m, n is the inut signal, y( m, n coefficients that define the filter s transfer function. In the lane fraction: H( z, z = ( z, z M1 1M 1 m= n= N1 1N 1 m= n= is the outut signal, and a( m, n b( m, n (.1, are, the filter (.1 is described by the transfer function in the form of a rational b( m, n z a( m, n z z m n z m n If a(,=1 and a(m,n= for m, n # (. is written as: M1 1M 1 m n = m= n= H( z, z b( m, n z z (.3 In that case, the difference equation has the form: M1 1M 1 m= n= y( m, n = b ( m, n x m m, n n (.4 m, n (.3 and (.4 are the transfer function and the difference equation of the -D FIR filter, resectively. B. Designing the -D FIR filter based on the REMEZ algorithm. The -D FIR filter is designed according to the REMEZ algorithm and is based on the otimal aroximation of Chebyshev. The Chebyshev otimal -D filter design means that the filter criteria allow for changes to minimize the max errors in the aroximate areas. The actual FIR filter characterized by the technical arameters in the constant frequency domain ω has four main arameters: - : assband rile s s : stoband rile : limited frequency (amlitude frequency of assband : limited frequency (amlitude frequency of stoband For one actual FIR filter, the riles do not exceed in assband and s in stoband. (. Available

3 Fig.1 Filter criteria In fact, when constructing the filter, attenuations in the assband ( into account to calculates the riles, s A according to the formula (.5: and the stoband ( A s are taken A / 1 1 = A / As / = 1 (.5 The otimal design indicates that the energy of the lateral wave is minimal, the riles in the assband, the transition band ga, and the stoband are small; in other words, it is as ideal as ossible. The otimasl FIR filter design leads to the Chebyshev aroximation roblem. Since there exists the sinusoidal oscillation around the ideal function in Chebyshev norm, it is ossible to minimize the maximum absolute values which can be called as the otimal aroximation. The otimization here means the aroximation of the filter criteria. With the otimization rogram on the comuter, the maximum aroximate error will be minimal. Parks and Mc.Clellan offered the otimal method based on REMEZ algorithm. This is known to be the most widely used and efficient FIR filter design tool. The REMEZ exchange algorithm is based on the Chebyshev aroximation roblem; Indeendent oints are sread on the frequency axis in an aroriate way so that the maximum aroximate error reaches its smallest. The smaller the rile can be, the closer it can get to the ideal state. General rinciles of the method are illustrated as follows: + Choose the ideal filter, which means to select the frequency resonse of the filter Ae (, then select the weighting function W( e of the N filter. (based on secifications of the actual filter, finally, ick out the level + Determine the aroximate roblem: find the aroximation of the weighting function W ( e frequency resonse of the strain filter Ae ( và P( e + Solve the aroximation roblem utilizing the REMEZ exchange algorithm. + Calculate the coefficients of the filter. The algorithm rocedure is secifically demonstrated as follows: - Ste 1: Assume that a low ass filter is designed with s. + Choose + Choose W( e s, the is the olynomial of the cos n function. 1 in stoband Ae = (.6 in assband s = in stoband = (.7 1 in assband where = max, (since this filter has s = = s s + Choose N (N is on odd number with FIR filter tye 1: the Kaiser aroximation formula is emloyed to find the filter level: log s 13 N 14.6( / s - Ste : Determine the aroximate roblem. ( e = W ( e. A e P e (.8 (.9 Available 3

4 Consider the FIR filter tye 1: with a known fixed function Q( e 1 ( e = ( e Q( e = ( e Ae = = A ( e Q( e W W. W A e Polynomial: N 1 ' n= =. Then: (.1 (.11 P e = n cos n (.1 where: '( n = ( n is a set of coefficients and ut N 1 R = = R P( e ( n cos n (.13 n= - Ste 3: Use the REMEZ exchange algorithm to solve the aroximate roblem. According to the alternation theorem, in the frequency domain, it is requisite to select discrete frequency oint R +. In the R + set, if the initial selection frequency is i, the error reach the extreme value at these frequencies and the sign of has to be interleaved (alternating. That means the following equation has to be solve: i i = W. i i i e e A e P e = ( 1 (.14 R i and P( e = ( n cos in (.15 n= i i P( e A e i P e i ( e e must i ( 1 = (.16 W i ( 1 i + = A( e (.17 W i ( e The equation (.17 can be written in matrix form as follows: 1 cos cos... cos R 1/ W ( e Ae ( 1 cos1 cos 1... cos R1 1/ W ( e ( 1 Ae..... =.... R R 1 cos cos... cos ( 1 / W ( R R R R R e Ae R+ 1 R+ 1 1 cosr+ 1 cos R cos R R+ 1 ( 1 / W ( e Ae From the set of extreme frequencies calculated. If i i, 1 R R 1 ( + P e i and are ascertained, then, the error E( e e for, it indicates that the otimal solution is found. If ( e (.18 is for some radom frequency, a new set of extreme frequencies R+ has to be selected, in which ( e reaches the extreme, therefore, new value of is calculated according to this new set of frequencies, Available 4

5 also has to be recalculated and ( e P e e need to be reassessed. This rocess is reeated until for all frequencies in the selected frequency set. At this stage, the result is considered the otimal solution. - Ste 4: Determine the imulse resonse The identification of + Knowing P e hd ( n considering the intermediate ste hd ( n of the actual filter. can be imlemented in two ways: (according to the otimal solution, h ( n is directly comuted without ( n N 1 A( k + ( 1 os ( A k c k n with n N N N k = 1 N hd ( n = d. For FIR filter tye 1, the following formula is alied: (.19 + Knowing P e subsequently, coefficients with the remainingn (based on the otimal solution, the samle ( n P e are determined by IDFT, from which hd is taken by M oint, ( n of the filter is established. In 197, Park and McClellan wrote a rogram to design a 1-D linear hase FIR filter based on the Chebyshev aroximation standard imlemented by the REMEZ algorithm. The design of the -D FIR digital filter by REMEZ algorithm also erforms the same stes as for the 1-D FIR filter. In this design H are converted into the -D FIR rocedure, the 1-D FIR filters that have the frequency resonse digital filters with the -D frequency resonse (, ( 1, = cos = T (, H by the following transformations: H H (. where H ( is the frequency resonse of the 1-D FIR filter that has the length N and the frequency resonse hn. T (, is the discrete-sace fourier transform of the transformed matrix (, which is calculated as: T (, (, t m n, M 1N 1 ( 1m + n 1 = t m n e (.1 m= n= Hence, the imulse resonse h( m, n of the -D filter, which needs to be constructed, is the inverse Fourier transform of the frequency resonse H (, : 1 ( 1m + n h( m, n = ( 1, 4 H e d d (. In MATLAB, the ftrans function is utilized to erform the above transformations. The 1-D FIR filter design by REMEZ algorithm uses the firl or REMEZ function and emloys the syntax h = ftrans ( h1, T to convert into the -D FIR digital filter ( is the 1-D FIR filter and h is the -D FIR filter, which are obtained by the transformation T. The design of the FIR filter according to REMEZ algorithm is executed based on the algorithmic flowchart shown in Figure [1] below: h 1 Available 5

6 Fig. The algorithmic flowchart C. Alying the -D FIR filter in rocessing digital images that have imulse noise Imulse noise is a quite distinctive kind of noise that can be roduced for various reasons. It often aears as discrete bright sots and dark sots on the image. Digital images reresented by a two-dimensional matrix of real numbers or comlex numbers include a finite number of bits. A continuous two-dimensional signal in sace is referred to as a continuous image in the real number field and is denoted as f (x, y (the value of f (x, y that continuously aears in the range of (-, +. Since the image signal has the characteristic of the two-dimensional sace, it is necessary to design a -D FIR digital filter to rocess the image. The rocedure of removing imulse noise in a digital image using the -D FIR digital filter is illustrated in Figure 3. Fig.3 Simulation of OFDM system using MATLAB SIMULINK software The rocess begins by imorting the original image. Then the imulse noise is generated for the original image. The image is assed through the designed -D FIR filter. After the imulse noise image is alied through the filter, the final image results to be sharer, minimizing the noise imact on the image. III. RESULTS AND DISCUSSION From the theoretical study on the design of FIR filters according to REMEZ algorithm, the author has introduced an algorithmic flowchart and built a simulation rogram on the MATLAB software. The author rooses a rogrammable interface for designing several tyes of filters with different design criteria. Within the aer framework, the author gives only some illustrative results. Available 6

7 Fig.4 Frequency resonse amlitude of the 1-D FIR filter Figure 4 shows results of the 1-D low-ass filter. In the assband [,8Hz], the riles are very small and almost straight, which denote that it reaches the ideal filter. Utilizing a switch from the 1-D FIR filter to the -D FIR digital filter offers a more general and comrehensive outlook at the two-dimensional low-ass FIR filter as shown in Figure. Fig.5 Frequency resonse - amlitude of the -D FIR filter In order to test the image rocessing using the designed -D FIR filter, the imulse noise function is affected on the original image and therefore, roduces white sots and black sots (Figure 6. A white sot on the image that looks like a "salt" as the ixel has the maximum value (the light intensity is 55. In contrast, a black sot on the image looks like a "eer" since the ixel has a minimum value (the light intensity is zero. Anh bi tac dong nhieu xung Fig.6 An image imacted by the imulse noise Available 7

8 Processed through the designed -D FIR digital filter, the image is smoother, white sots and black sots on the image caused by the imulse noise are minimized (Figure 7. As a result, the -D FIR filter designed by REMEZ algorithm has rocessed quite Anh goc well the imulse noise image. Fig.7 The image after filter rocessing IV. CONCLUSION The aer resents simulation results on MATLAB software designed from the 1-D FIR filter to the -D FIR filter according to REMEZ algorithm and the -D FIR digital filter alication on the imulse noise removal. The most outstanding feature of this design aroach is to ground on the Chebyshev aroximation. On that account, the maximum error of the filter is minimal. The -D FIR digital filter can be alied to various roblems in the field of image rocessing as well as in a number of other areas. Nevertheless, the simulation in this aer is limited to the rocessing of the imulse noise. In the future, the author will carry out further study on the simulation of -D FIR filter alication on noise image rocessing with various tyes of noise and the sharening in image rocessing. (1 REFERENCES [1] A. Habibi and G. S. Robinson, A survey of digital icture coding, Comuter 7, May [] D.B.Harris and R.M.Mersereau, A comarison of algorithms for minimax design of two-dimensional linear hase fir digital filters, IEEE Transactions on Acoustics, Seech, and Signal Processing, [3] J.G.Fiasconaro, Two-Dimensional nonrecursive digital filters, Picture Processing and Digital Filters, T.S. Huang, Sringer-Verlag, [4] J.S.Lim, Two-Dimesional Signal and Image Processing, Prentice Hall, NJ, 199. [5] N. S. Jayant and P. Noll, Digital Coding of Waveforms, Englewood Cliffs, NJ: Prentice-Hall, [6] P. M. Cassereau, D. H. Staelin, and G. dejager, Encoding of images based on a laedorthogonal transform, to be ublished, IEEE Trans. on Communications, [7] R.Ansari and A.E.Cetin, Two-Dimemsional FIR Filters, in The Circuits and Filters Handbook, edited by W.K.Chen, IEEE and CRC Press, [8] R. J. Clarke, Transform Coding of Images, Academic Press, London, 1985 [9] Y.Kam and J.P.Thiran, Chebyshev Aroximation for Two-Dimensional Nonrecursive Digital Filters, IEEE Trans, Circuits and Systems, Available 8

9 [1] W. K. Pratt, Image Transmission Techniques, Academic Press, New York, 1979 NGUYEN THI HUYEN LINH 4-8: Vinh University, Physics Teacher Education 8-11: Hanoi University of Science and Technology, Electronics and Telecommunications Engineering 13-15: Hung Yen University of Technology and Education, Master of Electronic engineering Research work: Research image rocessing, image coding and alications. LUONG NGOC MINH 4-8: Vinh University, Physics Teacher Education 8-11: Hanoi University of Science and Technology, Electronics and Telecommunications Engineering 13-17: University of Science and Technology Beiing, China, Master of Information and Communication. Research work: Research on telecommunications systems, signal rocessing algorithms and images. TRAN DINH DUNG Available 9

10 8-13: Posts and Telecommunications Institute of Technology, Electrical and Electronics Engineering : Hanoi University of Science and Technology, Master of Power Engineering. Research work: Smart grid. Available 1