Chapter 5 Chapter 5 MODIFIED FUZZY BASIS FUNCTION AND FUNCTION APPROXIMATION 5.1 Introduction In this chapter a Modified Fuzzy Basis Function (MFBF) is developed. This MFBF is used to reduce the Gaussian noise from an image signal. The fuzzy rule converts the membership function in to a closed and bounded space. In that space the topological properties necessary for approximation mentioned earlier, are satisfied. The studies in [18] ensure the effectiveness of fuzzy techniques in image noise reduction. An experimental study in connection with our work shows that the designed MFBF can be used more effectively to reduce noise from an image signal, so that the noisy image approximate to the original image. RGB colour space is used as the basic colour space. 89 Chapter 5 Colours in RGB space are represented by a 3-D vector with first element being red, second being green and third being blue respectively. A colour image C represented by a 2-D array of vectors where (i, j ) = xi , j defines a position in C called pixel and Ci , j ,1 = xi , j ,1 , Ci , j , 2 = xi , j , 2 and Ci , j ,3 = xi , j ,3 denotes the red, green and blue components respectively of C. The FBF Φ i, j is defined and using it this we have proved that g ( x∗ ) = yi , j , ∀i, j for n = 1 and n ≥ 2 . For MFBF the distance measure in colour space and weight in noise reduction is defined. The newly designed fuzzy filter is given by N gi , j = N ∑ ∑Φ k ,l k =− N l =− N N N ( xi , j ) xi − k , j − l ∑ ∑Φ k =− N l =− N , k ,l ( xi , j ) Here Φ k ,l is the product of membership functions. 5.2 Some Definitions 5.2.1 Definition (Distance measure in colour space) Let d be the Euclidian distance, RG-red green, NRG- Neighbour Red green. 90 Chapter 5 Let RG = ( xi , j ,1 , xi , j , 2 ) and NRG = ( xi+k , j +l ,1 , xi+k , j +l , 2 ) . Then d ( RG, NRG ) = (x i + k , j + l ,1 − xi , j ,1 ) + ( xi + k , j + l ,2 − xi , j ,2 ) 2 2 5.2.2 Definition (Weight in fuzzy noise reduction) [7] The weight function is defined the product of the membership functions. That is, Weight = Ai , j ( Dist ( RG, NRG )) × Ai , j ( Dist (RB, NRB )) = min[ Dist ( RG, NRG )), Dist (RB, NRB )] 5.2.3 Remark In the fuzzy filter developed in this chapter, the distance between two pixel positions are taken as the input vector. Small, large, very large, etc are taken as the linguistic variables. Here Φ k ,l is considerd as the weight to transform the input distance function to the centroid output space yi , j . The concept of fuzzy basis functions is related with the membership functions [5, 32]. The designed Modified Fuzzy Basis Function (MFBF) facilitates faster approximation when compared to other available filters. 91 Chapter 5 5.2.4 Example (Gaussian Fuzzy Basis Function for MISO FLS) [12] The Gaussian Fuzzy Basis Function (GFBF) can be expressed as ⎛ x−x 2 ⎞ i, j ⎟ z j exp⎜⎜ − ∑ 2 2σ i , j ⎟⎟ ⎜ i =1 ∗ ⎠ ⎝ gi, j ( x ) = 2 ⎛ x−x ⎞ M i, j ⎜− ⎟ exp ∑ 2 ⎜ 2σ i , j ⎟⎟ ⎜ i =1 ⎝ ⎠ M where xi , j , σ i , j , z j , j = 1,2,......., n, i1 , i2 ,......., in ∈ I are parameters [1]. This can also be expressed as m ( g i , j ( x∗ ) = ∑ z j Φ x − xi , j i =1 ) where ( Φ x − xi , j ) ⎛ x−x 2⎞ i, j ⎜ ⎟ exp⎜ − 2 ⎜ 2σ i , j ⎟⎟ ⎝ ⎠ = ⎛ x−x 2⎞ M i, j ⎟ exp⎜⎜ − 2 ∑ ⎜ 2σ i , j ⎟⎟ i =1 ⎝ ⎠ 5.2.5 Theorem [Necessary condition][47] Let Ω = [B, D ] be the universe of discourse and let Φ i ( x, , ai , bi , p i , qi ) be a family of triangular functions such that B = C , C i ≤ ai +1 ≤ C i +1 , ∀i = 1,2,.......N − 1 and D = CN where 92 Chapter 5 p i bi + qi ai . For two dimensional combinations of three colours pi + qi case Φ i is taken as the approximation of triangular or Gaussian Ci = membership function and satisfies the following conditions: (i )(. Φ i )1≤i≤n are combination of triangular or Gaussian shaped. (ii )(. Φ i )1≤i ≤ n are normal and complete (iii )(. Φ i )1≤i≤n are consistent in the universe of discourse (iv ).Φ1 < Φ 2 < ........ < Φ n By the interpolation problem defined in [5], we apply this in the image denoising process. 5.3 Modified membership function and approximation property 5.3.1 Definition (Centroid defuzzifier) The centroid defuzzifier is defined as N g ( x∗ ) = ∑xΦ i =1 N i ∑Φ j =1 j ( x∗ ) , ∗ j (x ) where, x∗ is the input vector of the FLF. 93 Chapter 5 5.3.2 Remark In this study the Gaussian noise is reduced from an image signal. The fuzzy rule is created with distance between colour components and its neighbour in an RGB colour space. These three constitute a triangular space, so that the topology and the related results can be applied here. The modified fuzzy filter is defined as n g (xi , j ) = m ∑∑ x i − k , j −l i =1 j =1 n m ∑∑ Φ i =1 j =1 Φ i , j ( xi , j ) i − k , j −l ( xi , j ) where Φ i, j is the product of the membership function (weight function) and the distance function. The maximum weight is equal to unity if the distance between colour pixel and its neighbouring pixel is small. This can be brought to triangular form. The Modified Fuzzy Basis Function is Φ ( xi , j ) = Φ i − k , j − l ( xi , j ) n m ∑∑ Φ i =1 j =1 i − k , j −l . ( xi , j ) 94 Chapter 5 Therefore g (x ) = ∑∑ x m i, j n i =1 j = 1 i − k , j −l Φ ( xi , j ) This is a convex linear combination of fuzzy basis functions. In this work we take x∗ = xi , j as colour pair and its neighbour. So Φ j ( x∗ ) is changed in to Φ i , j ( xi , j ) .In the approximation studies, it is observed that when j = k , the rule in which the distance is small. So the product of the membership function or weight is unity. Here Φ i , j = Φ i ,k and Φ i , j ( xi , j ) = Φ i ,k ( xi ,k ) = 1 , and if j ≠ k then Φ i , j ( xi ,k ) = 0 . This shows that the membership space is normalized. According to lemma 4.3.3 the FBF is closed and bounded. Here we can define the fuzzy topology and by the results in chapter IV, the modifications can be made in membership functions. So in this space the approximation of functions can be done and the noise in an image can be reduced by introducing the Modified Fuzzy Membership Function (MFBF). 95 Chapter 5 The following interpolation property shows the approximation property of MFBF in the image denoising process. 5.3.3 Interpolation property If f be a continuous function defined on a compact subset Ω such that f ( xi , j ) = y i , j , i = 1,2,..., n, j = 1,2,......, m , then there exists modified fuzzy basis function modelled by the fuzzy logic filter gi , j such that g ( xi , j ) = xi − j , k − l , ∀i, k Proof Case (i) When n = 1 . The RGB space is used as the basic colour space. Colours in RGB space are represented by a 3-D vector with first element being red, second being green and third being blue respectively. A colour image C can be represented by a 2-D array of vectors where (i, k ) = xi , k defines a position in C is called pixel and Ci , k ,1 = xi , k ,1 , Ci , k , 2 = xi , k , 2 and Ci , k ,3 = xi , k ,3 denotes the red, green and blue respectively components of C. That is, xi , k ∈ Ci , k and assume that for j = k the distance is small. At this point the weight function is unity. 96 Chapter 5 Let xi − j , k − l , xi + j , k + l are elements of the colour components of Ci ,k , 2 and Ci ,k ,3 respectively. Then define the distance function as xi− j ,k −l = xi ,k − d i , xi+ j ,k +l = d i − xi ,k Φ i , j ( xi , j ) = (x i + j ,k +l 2 ( xi ,k − xi − j ,k −l )( xi ,k + xi + j ,k +l ) 2 − xi − j , k − l ) =1, if the distance function and the two colour components involves in the same pixel. Then g (xi , j ) = ∑∑ xi − j , k − l Φ ( xi , j ) n m i =1 j =1 satisfies g ( xi , j ) = xi − j ,k −l , ∀i = 1, 2,...., n, j = 1, 2...., m Case (ii) When n ≥ 2 Let x∗j = (x1k , x2 k ,......, xnk ),1 ≤ k ≤ m be M distinct vectors of R n . For each xi , j define the one dimensional function Φ i, j satisfying i) If xi , j = xi , k (i.e. j = k ) then Φ i , j = Φ i , k and Φ i , j ( xi , j ) = Φ i ,k ( xi ,k ) = 1 g ( xi , j ) = xi − j , k − l ∀i = 1,2,...., n, k = 1,2...., m 97 Chapter 5 ii) If xi , j ≠ xi , k (i.e. j ≠ k ) then Φ i , j ( xi , k ) = 0 n Let Φ j (x ) = ∏Φi, j (xi,k ),∀x∗ = (x1, x2 ,....,xn ) ∈ Rn ∗ i=1 Then Φ j ( x ∗j ) = 1 and Φ j ( x k∗ ) = 0, ∀k ≠ j . The fuzzy logic system modelled by ( ) n m g x∗ = ∑∑ xi − j , k − l Φ ( x∗ ) i =1 j =1 satisfies g ( x∗ ) = xi − j , k − l , ∀i, j As we have defined in chapter I, the FLS is a decision making logic which employs fuzzy rules from the fuzzy rule base to determine a mapping from the fuzzy sets in the input space to the fuzzy sets in the output space. The defuzzifier performs a mapping from the fuzzy sets in the output space to the crisp points in the output space. In general for the better approximation of the images the centroid defuzzifier (also called center-average defuzzifier) is used and which maps the fuzzy set Φ j ( x∗ ) in the output space to the crisp point. 98 Chapter 5 5.4 Modified Fuzzy Basis Function for Different Rules The function approximation is done using fuzzy rules extracted from partitioned numerical data [57, 58]. In our work the RGB colour space is partitioned in to small distances such that for the function approximation we can apply MFBF in RGB space also. 5.4.1 Example For the red component of RGB colour space, the modified output is K gi , j ,1 = K ∑ ∑Φ k =−K i=−K K i + k , j + l ,1 ( xi , j ,1 ) xi + k , j + l ,1 K ∑ ∑Φ k =−K k =−K i + k , j + l ,1 ( xi , j ,1 ) From gi , j ,1 ,the MFBF for red colour component can be found out. 5.4.2 Modified Fuzzy Basis Function ( Φ k ,l ) for Simple rule. Consider the following rule as “If d ( RB, NRB ) is small then Φ k ,l is large (i.e. 1)”, else Φ k ,l is close to zero, where distance between RB and NRB is denote as xi. j − xi − k , j − l . 99 Chapter 5 According to fuzzy set theory, the value Φ k ,l can be estimated from the membership function in the figure shown below Fig. 5.1: Weight for a simple rule According to [58], the non linear function Φ k ,l can be approximated as a step like function in the following figure. Fig. 5.2: Approximation of weight function. 100 Chapter 5 In this case the output of the fuzzy filter is represented as n gi , j = n ∑ ∑Φ k ,l k =−n l =−n n n ( xi , j ) xi − k , j − l ∑ ∑Φ k =−n l =−n , k ,l ( xi , j ) When the image signal is flat or contains just smooth vertices the FLS is effective to remove random noise in that case Φ k ,l = 1, ∀k , l . As an image signal usually contains large amplitude or steep changes such as edges the fuzzy filter Φ k ,l = 1 makes the image blurred when it is applied to noise removal. For smoothness of edge the weight must get the values Φ k ,l = 1, k = l = 0 and =0, otherwise. In these circumferences the noise is unable to smooth well so that the value of the weight Φ k ,l should be controlled according to the local characteristics of the signal. If the pixel xi − k , j −l belongs to the same flat areas of the pixel xi , j ,the weight wk ,l must get a value of 1 and if there is an edge between these two pixels the weight must take the value 0. The amplitude of the signal difference xi. j − xi−k , j −l and xi+k . j +l − xi , j can 101 Chapter 5 be applied to separate these two states. If xi. j − xi − k , j − l and xi+k . j +l − xi , j is small then xi − k , j − l and xi + k , j + l are assumed in the same flat area (here Φ k ,l = 1 ) .If not they are assumed in different flat area ( wk ,l = 0) . 5.4.3 Remark The gradient method gives the optimal value for a parameter which minimizes certain cost function or error function by iteration [62]. In the gradient method, a parameter is updated by subtracting a value which is proportional to the gradient of the cost function. So by Gradient method, S k (n + 1) = S k (n) − α where S k (n) is the value ∂E ( n) ∂Φ k S k ( S k is the height of the k th step of the step like function) at time point n, α is a small positive coefficient and E is the mean square error. 102 Chapter 5 5.4.4 Φ k ,l for multiple rule antecedents Let the input image signal at the pixel (i, j ) be indicated as xi , j and the corresponding output be yi , j .Also denote the set of input signals around xi , j as xi − k , j − l ,−n ≤ k ≤ n,−n ≤ l ≤ n . Consider the following rule If xi. j − xi−k , j −l and xi+k . j +l − xi , j are small then Φ k ,l is large otherwise Φ k ,l close to 0. Then the membership function small has been modified which incorporates a two sided composite of two different Gaussian curve. Fig. 5.4: A form of a weight function for multiple rule antecedents. 103 Chapter 5 5.4.5 Gradient Method for multiple rule antecedents [13, 58] Assume that the height of the qth step of the step like function is indicated as S q , the output of the signal has the expression Q gi , j = N N ∑ ∑ ∑Φ q =1 k = − N l = − N Q N N klq ( xi , j ) S q xi − k , j − l ∑ ∑ ∑Φ i =1 k = − N l = − N (1) klq ( xi , j ) S q where Φ klq 1, if | x(i, j ) − x(i − k , j − l | is included in the q th region = 0, other wise If d (i, j ) = fi , j expresses the desired output signals for the pixel (i, j ) then the mean square error is given by the mean of [ y(i, j ) − f (i, j )]2 or gi , j − fi , j 2 When α small enough, the mean square error is E = gi , j − fi , j 2 When the weight is denoted as non linear function of x(i, j ) − x(i − k , j − l ) , 104 Chapter 5 the mean square error is defined as ⎤ ⎡ Q K K ⎥ ⎢ ∑ ∑ ∑ Φ klq ( xi , j ) S q xi − k , j − l q =1 k = − K l = − K ⎢ E= − fi , j ⎥ 3 K K ⎥ ⎢ ⎥ ⎢ ∑ ∑ ∑ Φ klq ( xi , j ) S q ⎦ ⎣ c =1 k = − K l = − K 2 (2) Or ⎡ 3 K K ⎤ ⎢ ∑ ∑ ∑ Φ klq ( xi , j ) S q xi − k , j − l ⎥ E = ⎢ c =1 k = − K l = − K − fi , j ⎥ D ⎢ ⎥ ⎢⎣ ⎥⎦ 2 where Q D=∑ K L ∑ ∑Φ q =1 k = − K l = − L klq ( xi , j ) S q Using (1) and (2) we find Q N N ⎛ ⎞g −f ∂E = 2⎜⎜ ∑ xi − k , j − l D − N (q ).∑ ∑ ∑ Φ klq ( xi , j ) S q xi − k , j − l ⎟⎟. i , j 2 i , j ∂S q D q =1 k = − N l = − N ⎝ ( k ,l )∈Q ( q ) ⎠ ⎛ ⎞g −f = 2⎜⎜ ∑ xi − k , j − l D − N (q ).D.g i , j ⎟⎟. i , j 2 i , j D ⎝ ( k ,l )∈Q ( q ) ⎠ ⎛ ⎞ g − fi , j = 2⎜⎜ ∑ xi − k , j − l − N (q ).g i , j ⎟⎟. i , j D ⎝ ( k ,l )∈Q ( q ) ⎠ 105 Chapter 5 Here Q(q) is the set of (k , l ) for which x(i, j ) − x(i − k , j − l ) is included in the members of q th region of the step like functions. N (q) is the number of Q(q) and the parameter x(i, j ) − x(i − k , j − l ) is included in the wklq takes a value of 1 if q th region and the parameter has a value of 0 if it is not. Therefore wq can be given by ⎛ ⎞ y (i, j ) − d (i, j ) S q (n + 1) = S q (n) + β ⎜⎜ ∑ xi − k , j − l − N (q) g i , j ⎟⎟ D ⎝ ( k ,l )∈Q (q ) ⎠ where β = 2α . The training is performed till the value S q converges for all q [2]. That is as β → 0 , S q (n + 1) and S q (n) belongs to the q th region where the basis function will have the value 1. 5.5 Observation In a fuzzy system with membership functions of product t -norm, for any given continuous function f in the compact domain Ω , there exists a fuzzy modified fuzzy basis function gi , j of g such that gi , j − f < ε . This MFBF extended to RGB colour space such that g ( xi + k , j +l ,C ) − f ( xi + k , j +l ,C ) < ε . Then 106 Chapter 5 the Fuzzy gradient method is applied for simple and multiple antecedents such that the error can be minimized. Hence the approximation of weight function as a step like function is obtained. 5.5.1 Remark For unsharp edges in multiple rule antecedents the weight function turns into 3D function Φ(dij , tk ,l ,σ i , j ) [61], where dij = 3D (x i, j − xi − k , j − l ) + (xi + k , j + l − xi , j ) , tk ,l = k 2 + l 2 (x-axis of the space), σ i , j is 2 the 2 local variance around the pixel (i, j ) = xi , j approaching the weight as a step function Φ (dij , tk ,l ,σ i , j ) (see the figure shown below).According to [19,61] this can be denoted as Φ pqr when d ij in the p th region, tk ,l in the q th region and σ i, j in the r th region. The approximation of FLF can be done only if the weight is one when d ij is in the p th region, the other two colour components being in the other two regions respectively. Otherwise the weight is equal to zero. 107 Chapter 5 5.6 Peak Signal Noise Ratio (PSNR) [50, 51] PSNR is a measure and it is useful to show that the fuzzy basis function and modified fuzzy basis function is better than any other basis functions. By using PSNR the mean square error (MSE) between the original image and the filtered image can be found out. 5.6.1 Definition [PSNR and Mean Square] The PSNR in an RGB colour space is defined as PSNR[ g (i, j , c) − f (i, j , c)] = 10 log10 [ S 2 / MSE ( g (i, j , c) − f (i, j , c)] where MSE is the mean square error and ∑∑∑ [g 3 MSE[ g i , j ,C − f i , j ,C ] = N M c =1 i =1 j =1 i , j ,C − f i , j ,C ] 2 3.N .M Here f (i, j, c) is the desired colour image and g (i, j, c) is the filtered colour image of size N.M (c = 1,2,3 corresponding to the colours red, green and blue respectively). 5.6.2 Remark From the figurers 5.5 to 5.7 it is clear that the image using MFBF is more close to the original image. Also table 5.1 and figure 5.8 show that noise removal is high when the MFBF is used. 108 Chapter 5 (a) (b) PSNR = 22.10 (c) PSNR = 26.57 (d) PSNR = 27.66 (e) PSNR =28.38 (f) PSNR = 29.33 (a) (b) PSNR = 18.59 (c) PSNR = 24.63 (d) PSNR = 24.73 (e) PSNR =25.16 (f) PSNR = 25.79 Fig. 5.6: (a) Original Peppers image (256×256) (b) Noisy image (Gaussian noise, σ = 30) (c) After applying Mean filter (3×3 window) (d) After applying Median filter (5×5 window) (e) After applying FBF of [24] with K=3 (7×7 window) and L=2 (5×5 window) (f) MFBF with K=3 (7×7 window) and L=2 (5×5 window) 109 Chapter 5 (a) (b) PSNR = 16.10 (c) PSNR = 22.03 (d) PSNR = 21.52 (e) PSNR =22.75 (f) PSNR = 23.61 Fig. 5.7: (a) Original Gantry crane image (400×264) (b) Noisy image (Gaussian noise, σ = 40) (c) After applying Mean filter (3×3 window) (d) After applying Median filter (3×3 window) (e) After applying FBF [24] with K=3 (7×7 window) and L=2 (5×5 window) (f) After applying Proposed Fuzzy filter with K=3 (7×7 window) and L=2 (5×5 window) Table 5.1 Results in PSNR of different noise removal methods. PSNR (dB) σ=5 σ = 10 σ = 20 σ = 30 σ = 40 Noisy 34.13 28.12 22.10 18.57 16.08 Mean 28.05 27.70 26.57 25.13 23.72 Median 32.31 30.81 27.66 25.02 22.95 FLF Method 34.12 31.79 28.38 25.85 23.76 MFBF Method 34.22 32.77 29.33 26.51 24.18 110 Chapter 5 Fig. 5.8: For restored image PSNR value decreasing slowly and approximating to step like function for small distance 111