The thesis aims to study the problems of image processing under the approach of HA theory. Building a gray level transformation function in S form to enhance the contrast for color images to apply HA. New homogeneity with hedge algebra applies to enhance contrast for color images. Building a photo fading transformation that applies to multi-channel images does not lose image detail.
MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY NGUYEN VAN QUYEN RESEARCH IMAGE CONTRAST ENHANCEMENT BASED ON HEDGE ALGEBRA MATHEMATICS DOCTORAL DISSERTATION Major: Math Fundamentals for Informatics Code: 9.46.01.10 SUMMARY OF MATHEMATICS DOCTORAL DISSERTATION Ha Noi, 2018 This work is completed at: Graduate University of Science and Technology Vietnam Academy of Science and Technology Supervisor 1: Dr Tran Thai Son Supervisor 2: Assoc Prof Dr Nguyen Tan An Reviewer 1: …………………………………………………………………… ………………………………………………………………………………… Reviewer 2: …………………………………………………………………… ………………………………………………………………………………… Reviewer 3: …………………………………………………………………… ………………………………………………………………………………… This Dissertation will be officially presented in front of the Doctoral Dissertation Grading Committee, meeting at: Graduate University of Science and Technology Vietnam Academy of Science and Technology At ………… hrs …… day …… month…… year …… This Dissertation is available at: Library of Graduate University of Science and Technology National Library of Vietnam LIST OF PUBLISHED WORKS [1] Nguyen Van Quyen, Tran Thai Son, Nguyen Tan An, Ngo Hoang Huy and Dang Duy An, “A new method to enhancement the contrast of color image based on direct method”, Joural of Research and Development on Information and Communication technology, Vol 1, No 17 (37): 59-73, 2017 [2] Nguyen Van Quyen, Ngo Hoang Huy, Nguyen Cat Ho, Tran Thai Son, “A new homogeneity measure construction for color image direct contrast enhancement based on Hedge algebra”, Joural of Research and Development on Information and Communication technology, Vol 2, No 18 (38): 19-32, 2017 [3] Nguyen Van Quyen, Nguyen Tan An, Doan Van Hoa, Hoang Xuan Trung, Ta Yen Thai, “Contruct a homogeneity measurement for the color image bassed on T-norm”, Journal of Military science and Technology, No 49: 117-131, 2017 [4] Nguyen Van Quyen, Nguyen Tan An, Doan Van Hoa, Hoang Xuan Trung, Ta Yen Thai, “A method to construct an extent histogram of multi channel images and applications”, Journal of Military science and Technology, No 50: 127-137, 2017 [5] Nguyen Van Quyen, Tran Thai Son, Nguyen Tan An, Construct an Sshaped gray-scale transformation function that enhances images contrast using Hedge Algebra, Proceedings of the 10th National Conference on Fundamental and Applied Information Technology Reseach (Fair’ 10), 884-897, Da Nang, 2017 Introduction Contrast enhancement is a very important issue in processing and analysing image, is a fundamental step in analyzation and segmentation image These are mainly two categories: (1) indirect method of contrast enhancement and (2) direct method of contrast enhancement a) About indirect method There are many indirect techniques, which were proposed in references They only modifies the histogram, without using any contrast measure In recent years, many researchers have applied fuzzy set theory to develope new techniques to enhance the contrast of the image Fuzzy approach algorithms often lead to the requirement of designing a gray-scale transformation S-shape function (The function is continuous monotonous increase, decreasing the input gray level when the input is below the threshold, and increasing the value gray level input when the input is above the threshold) However, the selection of functions in the fuzzy rule inference to produce the gray-scale transformation S-shape function is not easy With the following simple fuzzy rule R1: If luminance input is dark then luminance output is darker R2: If luminance input is bright then luminance output is brighter R3: If luminance input is gray then luminance output is gray So, the fuzzy reasoning results using fuzzy sets is not obvious and it is quite difficult to obtain the appropriate gray-scale S-shape function b) About direct method For a long time to date, almost only the studies by Cheng and coworkers have followed direct approaches, the authors have been proposed a method which modify the contrast at each pixel of gray-scale image based on the definition of image’s homogeneity measure In addition, Cheng and coworkers have proposed an algorithm that uses the S-function which have parameters to transform the multi-level gray-scale input image and then enhance the image’s contrast by direct method Cheng's algorithms are the basis of the contrast enhancement of grayscale images However, these algorithms still have some limitations when applying to color images, multichannel images : (i) The resulting image after enhancement the contrast may not change the brightness of the color compared to the original image (ii) Using images that have been modified by Cheng's image modification method as input of contrast enhancement process may lose details of original image For the homogeneity measurement of pixel, Cheng proposed a way to estimate the homogeneity value of the pixel from local values Eij, Hij, Vij, R4,ij When experimenting with color images, we noticed that with this estimate, the resulting image may not be smooth Actually the pixel's homogeneity is a fuzzy value so that we can apply the fuzzy logic to get this value If local values E , H are passed to computing with word then the formula ij is formatted If T e h E ij , H g r a d ie n t ij ij should reflect the fuzzy rule system as follows: is hight and e n tr o p y is hight then homogeneity is hight If g r a d ie n t is low and e n t r o p y is low then homogeneity is low If we add rules with terms like "very", "little", "medium" etc with linguistic variables like "homogeneity", "entropy", "gradient" etc then homogeneity values can be estimated by human inference and thus it will be finer Because fuzzy set theory has no basis form between the relationships of the linguistic variable with the fuzzy sets and the order of relations between the words, it is important to consider using a fuzzy reasoning method to ensure order Through surveys, analyses and experiments we have the conclusion: Firstly, the if-then argument based on the fuzzy set is very difficult to guarantee the S shape of the gray-scale transformation function The direct contrast enhancement method of Cheng uses a transformation gray-scale function has S-shape but not Symmetric and the gray value may fall outside the gray-area value Secondly, Cheng's homogeneity measurement has still limited, for example the resulting image may not be smooth Thirdly, using Cheng's algorithm directly on the original image channel, the brightness of the resulting image may be less volatile In order to change the brightness, it is necessary to transform the original image before applying Cheng's contrast enhancement Cheng's image transform method may cause loss of detail of the original image The research topic of doctoral dissertation is: Problem 1: Designing the Gray S-type transformation and symmetry Problem 2: Constructing a local homogeneity measurement of image Problem 3: Constructing fuzzy transformation method for image without losing details of original image Chapter Overview of contrast enhancement and solving the fuzzy rule system base on Hedge Algebars This chapter presents the concepts of hedge algebra and approximate reasoning method based on hedge algebra, overview introduction of contrast enhancement methods such as some indirect and direct methods Analysis proposed use hedge algebra to improve the contrast by direct method 1.1 Hedge algebra: some basic issues 1.1.1 Some basic definition about Hedge algebra The word-domain X = Dom(X) may be assumed to be a linearly ordered set and can be formalized as a hedge algebra, denoted by AX = (X, G, H, ), where G is a set of generators, H is a set of the hedges and “”is the semantic order relation on X Assume that, In G there are constant elements 0, 1, W which are, respectively, the least, neutral and greatest words of Dom(X) We call each word value x X is term in hedge algebra If X and H are linearly ordered then AX = (X, G, H, ) is linear hedge algebra In addition, if equipped two artificial hedges with the meaning of which is taking the infimum and supremum of H(x) - the set generated from x then we get the linear complete hedge algebra, denoted by AX = (X, G, H, , , ) Because in this dissertation, we only care about linear hedge algebra, since speaking the hedge algebra also means linear hedge algebra When operand h H in x X, will have hx Every x X, denoted H(x) is set of every terms u X from x by apply hedges in H and write u = hn…h1x, where hn, …, h1 H The H set include positive hedges H+ and negative hedges H- The positive hedges increase the semantics of a term and the negative hedges decrease the semantics of a term Without loss of generality, we always assume that H- = {h-1 < h-2 < < h-q} and H+ = {h1 < h2 < < hp} 1.1.2 Measurement function in the linear hedge algebra In this section, we use linear hedge alge AX = (X, C, H, ) with C = {c-, c+} {0, 1, W} H = H- H+, H- = {h-1, h-2, , h-q} satisfy h-1 < h-2 < < h-q and H+ = {h1, h2, , hp} satisfy h1< h2 < < hp and h0 = I with I is unit operator Let H(x) be the set of elements of X generated from x by the hedges Thus, the size of H(x) can represent the fuzziness of x The fuzziness measurement of x, we denote by fm(x) is the diameter of the set (H(x)) = {f(u) : u H(x)} Definition Let AX = (X, G, H, , , ) is linear complete hedge algebra An fm: X [0,1] is said to be a fuzziness measure of terms in X if: (1) fm(c-) + fm(c+) =1 hH fm(hu) = fm(u), uX; (2) fm(x) = 0, for all x such that H(x) = {x} Especially, fm(0) = fm(W) = fm(1) = 0; (3) x,y X, h H, fm ( hx ) fm ( x ) fm ( hy ) , that is it does not depend on fm ( y ) specific elements and is called fuzziness measure of h, denoted by(h) Proposition For each fuzziness measure fm and which defined in Definition 1, the following statements hold: (1) fm(c-) + fm(c+) = and h H (2) (3) 1 (h j ) j q x X , fm ( x ) , p j 1 (h j ) fm ( h x ) fm ( x ) ; , where , > + = 1; where Xk is set of term which has length k; k (4) fm(hx) = (h).fm(x), and xX, fm(x) = fm(x) = 0; (5) Cho fm(c-), fm(c+) (h) where hH, x = hn h1c, {- (1.1) ,+}, it is easily to calculate the fuzziness measure of x: fm(x) = (hn) (h1)fm(c) Definition The function Sign : X {-1, 0, 1} is a mapping defined recursively as follows, where h, h' H and c {c-, c+}: (1) Sign(c-) = -1, Sign(c+) = 1; (2) Sign(hc) = -Sign(c) if h is negative c; Sign(hc) = Sign(c) if h is (1.2) positive c; (3) Sign(h'hx) = -Sign(hx), if h'hx hx and h' is negative h; Sign(h'hx) = Sign(hx), if h'hx hx h' is positive h; (4) Sign(h'hx) = 0, if h'hx = hx Proposition For any h and xX, if sign(hx) =+1 then hx > x and if sign(hx) = -1 then hx < x Definition Let fm be a fuzziness measure on X A semantically quantifying mapping (SQM) v on X (associated with fm) is defined as follows: (1) (W) = = fm(c-), (c-) = – .fm(c-) = .fm(c-), (c+) = +.fm(c+); (1.3) Sign ( j ) ( h ) fm ( x ) ( h x ) ( h x ) fm ( x ) , (2) ( h j x ) ( x ) Sign ( h j x ) ij Sign ( j) i j j every j, –q j p and j 0, where: (h j x) - 1 Sign ( h j x ) Sign ( h p h j x )( ) , ; (3) (c ) = 0, (c-) = = (c+), (c+) = 1, and every j is satisfy –q j p, j 0, we have: (hjx) = (x) + Sign ( h j x ) j Sign ( j ) i Sign ( j ) ( h i ) fm ( x ) 1 Sign ( h j x ) ( h j ) fm ( x ), (hjx) = (x) + Sign (h j x) j Sign ( j ) i Sign ( j ) ( h i ) fm ( x ) 1 Sign ( h j x ) ( h j ) fm ( x ) Proposition 3. xX, v(x) 1.1.3 Interpolative Reasoning using SQM Let consider the fuzzy multiple conditional Reasoning (FMCR) has form: If X1 = A11 and and Xm = A1m then Y = B1 If X1 = A21 and and Xm = A2m then Y = B2 (1.4) If X1 = An1 and and Xm = Anm then Y = Bn where Aij, Bi, j = 1, , m and i = 1, …, n, are not fuzzy sets, but are linguistic values The Reasoning problem is with given input Xj = A0j, j = 1, …, m, linguistic model (1.4) will assist us in finding the output Y = B0 Without a general reduction we can suppose that the input is a vectors have semantic value normalized into the interval [0,1] A0 = (a0,1, …, a0,m), a0,j [0, 1] với j = 1, 2, … m, the output is numberic value is normalized into the interval [0, 1] too FMCR problem is transposed to surface interpolation and is solved base on any interpolation method In hedge algebra, this method is done as following: Step 1: Define hedge algebra for linguistic variable Xj and Y are: AXj = (Xj, Gj, Cj, Hj, j) and AY = (Y, G, C, H, ) correlative Set of all parameters included, for j = 1, …, m: *) m+1 fuzzy parameters: j = fm(cj), and = fm(c) *) pj + qj –1 fuzzy parameters of AXj: (hj, qj), , (hj, 1), (hj, 1), , (hj, pj) *) p + q – fuzzy parameters of AY: (hq), , (h1), (h1), , (hp) In practice, these parameters can be assigned by experience or determined by an optimization algorithm such as using genetic algorithms Construct Xj and Y are SQM of hedge algebras AXj and AY of orrelative linguistic variables Xj and Y, j = 1, 2, … m Let the linguistic hyper-surface and S n o rm v X j (x j ) j 1, m SL x , vY ( y ) j j 1, m , y n i 1, n [ , 1] x j X j X j Y is j 1 m 1 (1.4) , j 1, m , yY nhúng n points Ai = (Ai1, …, Aim, Bi) after that, (1.4) describe the linguistic hyper-surface SL in X1 … Xm Y Vector (X1, …, Xm, Y) of SQMs, Xj, j = 1, …, m, and Y map SL is transformed into Snorm: (X1, …, Xm, Y) : SL Snorm Step 2: Define Interpolative Reasoning on Snorm Construct SQMs v (A ) , v ( B ) ( j 1, m , i 1, n ) X j ij Y i Snorm v X (A ij ) j j 1, m , vY ( B i ) can be defined by m input aggregate oprator i 1, n fSnorm, v = fSnorm(u1, , um), v [0, 1] uj [0, 1], Y(Bi) = fSnorm(X1(Ai1), , Xm(Aim)), i 1, n j 1, m , satisfy conditions (We can be used once of many Interpolative Reasoning to execute problem) Step 3: Find output B0 referred to input A0 is normalized into the interval [0, 1]: A0 = (a0,1, …, a0,m), a0,j [0, 1] for j 1, m b0 f S n o rm a , , a ,1 ,m [ , 1] (1.5) 1.2 Cheng’s contrast enhancement 1.2.1 Auto extracting argument (from multi gray-scale) with Cheng algorithm a The gray dynamic range is [a,c] is compute based on image’s histogram b The image transformation use S-function S I ( a , b o p t , c ) S fu n c ( I ( i , j ); a , b o p t , c ) I I ( i , j ) where [a, c] is gray dynamic range which have parameters are auto estimated when survey the top of histogram and are estimated based on the maximum of fuzzy entropy b a rg m a x H ( I ; a , b , c ) where H is common fuzzy entropy measure opt b [ a , c ] c Compute local arguments of original image (or transformed image) and is normalized into the interval [0,1], gradient Eij, entropy Hij, standard deviation Vij, the 4th moment of the intensity distribution R4,ij d Compute homogeneity measurement of original image’s pixel (or transformed image) by assosiation operator from local values HO (1.6) , ij ij m a x H O ij where H O ij E ij * V ij * H ij * R ,ij E ij * V ij * H ij * R ,ij (1.7) đ Compute non-homogeneity gray value of original image’s pixel (or transformed image) ij g ( p , q ) W pq (1 pq ) ij ( p , q ) W (1 pq (1.8) ) ij e Compute exponential amplification ij m in , where m in m ax g k g1 g m ax g m in * ij m in m a x m in , m ax , gk, g1 is tops of histogram (1.9) g Evaluate the contrast associated with pixel (i; j), and amplify g ij ij C ij ,C g ij ij (1.10) where t{0.25, 0.5} ij C ij ' ij t h Compute the output gray value of original image’s pixel (or transformed image) of contrast enhancement method which use a none symmetric S-function: ' g ij C ij ij C ij ' ij C ij ' C ij ij C ij ' ij C ij t , g ij ij , g ij ij (1.11) ij C ij ' ij t ij t ij C ij t i If we use the transformed image in step c to h, we need to apply the inverse transformation of the image to get the final output pixel Cheng's contrast enhancement method satisfies the law: At each pixel on which apply step c to h, if the pixel’s homogeneity is higher, the contrast degree at that pixel is lower (RCE-rule of contrast enhancement) Since a image transformation method is monotonically increase, usually preserving the edge intensity of the image and the local entropy value, the RCE rule is generally satisfied with the original image even if the direct contrast enhancement method using a transformed images 1.2 Some Criterias for image quality assessment 1.2.1 Entropy criterion, average for many image channel {I1,I2,…,IK}: Use common entropy criterion for every gray-scale image: (1.12) E (I ) E ( I ) p ( g ) lo g ( p ( g )) , E ( I ) where K L m ax k k k 1 k k k avg g L m in k def pk (g ) # I k (i, j ) g M *N 1, K K and convention 0*log2(0) = If the value of Entropy of image channel is higher the image channel can see very detail Usually, if the contrast of image channel is high, the histogram of image channel fairly equilateral, entropy is high, this is the principle modifying the histogram of indirect contrast enhancement methods 1.2.2 Fuzzy Entropy criterion, average for many image channel {I1, I2,…, IK}: Use common fuzzy entropy for every gray-scale image: (1.13) H (I ) K k H ( I 1, K ) avg k 1 K Lk , where: H (Ik ) ,m a x ( g ) lo g g Lk def g (g) ( ( g ) ) ( g ) lo g (1 ( g ) ) * p k ( g ) ,m in g L k , m in L k , m a x L k , m in and convention 0*log2(0) = , 14 Y = Y(very), , X, Y [0,1], vX, vY is the semantic quantitative function of the HA on X and Y respectively (ii) v Y ( v l o w ) 1 v X (lo w ) (iii) Function F: [0,1] [0,1] monotonically increases, continuous (the inverse of F also monotonically increases, continuously) and increases at threshold R1: if x is then y is R2: if x is c- then y is cR3: if x is W then y is W R4: if x is c+ then y is c+ R5: if x is then y is Interpretation in the numeric value domain: (a) and (b) is equivalent to: F(vX(c-)) = vY(c-), F() = , F(vX(c+)) = vY(c+) F(vX(low)) = vY(v.low), F(vX(hight)) = vY(v.high) Suy ra: F(0) = 0, F() = , F(1) = 1, F , X F 1 X 1 Y Y 1 F (t ) t , t : t (c) Equivalent to F (t ) t , t : t Conventional: When AX, AY fixed, to be neat we are also identical FHint Comment: (i) = X = Y = 0.5 X = Y = 0.5, m= vY ( v e r y lo w ) v X (lo w ) x s , x s IN T ( x s ) x s , x s Y , X FINT, where then (AX, AY, INT) is a Hint Y (ii) When m = X , Y X then The quadratic function F has the parabola graph going through points (0; 0), (X, 2Y), (, ) then set (AX, AY, F) is not a Hint Proposition 2.3: (AX, AY, F) is a Hint, where F defined as following: m=v Y (v e ry lo w ) v X (lo w ) Y 1 X , g ( x s ), x s F ( xs ) g (1 x s ), x s for function g(xs) defined: Theorem 2.1: m = g ( xs ) v Y ( v l o w ) v X (lo w ) ax s xs ax s Y X 1, , xs where a m 1 x 1 m X 15 (2.8) g c ( , X , m )( x s ), x s H in t( x s ) g c (1 , X , m )(1 x s ), x s where gc(, X, m) (xs) defined as following: =(X,m) = 1 m lo g 1 m 1 lo g 1 g c ( xs ) or xs 1 +xs xs 1 +xs X X X X 1 m lo g 1 m 1 lo g 1 X X X X 1 m lo g 1 m 1 lo g 1 X X (2.9) X X xs xs g c ( xs ) xs g c ( xs ) xs -x s -x s , xs , 0 xs , Then (AX, AY, Hint) is a Hint, besides Hint is satisfied: H in t( x s ) m ax xs ( , X ] m in H in t( x s ) [ X , ] m xs , m ax [ , X ] H in t(1 x s ) xs m in [ X ,1 ] H in t(1 x s ) xs m Comment: Parameter X represents the semantic meanings of the low input variable, usually taken from 0.3 to 0.7, where X = m then AX is identical to AY The functions g, gc in proposition 2.3 and theorem 2.1 satisfy the following properties (i) Monotonous increase, differentiable continuous on [0,1] (ii) g(, X, m)(0) = 0, g(, X, m)() = , g(, X, m)( X ) = mX g(, X, m)(xs) xs, xs (iii) g(, X, m1)(xs) g(, X, m2)(xs), xs , < m1 m2 < m ax ( , X ] g ( , X , m )( x s ) xs m in [ X , ] g ( , X , m ) (x s ) m xs Figure 2.3 Comparing the transformation function graph of Cheng and Hint between the input-output gray-scale which have been normalized in [0,1], with in luminance surrounding =0.6, βX=0.6, =0.5,t=0.5 16 Nhận xét: The contrast enhancement method according to Formula (28) satisfies the RCE rule 2.7.2 Algorithm using Hint operator In this section, we propose algorithm using Hint operator operator to enhance the contrast of the multi channels image Details of this algorithm is available as following: Thuật toán 2.3: Enhance the contrast of the multi channels image Input: K channels of image I (in the color representation), parameter C N ,C I 1, K { I , , I K } , , M x N is size of image I, 1 K, is parameter of HA Output: Image I’ = {I’1, … IK’} Step 1: Cluster image I = {I1, … IK} into C clusters using FCM algorithm Bước 2: Define the fuzzy histogram of each channel Ik, k = … K using formular (2.3) Bước 3: Compute gray value which has enhanced and normalized in [0, 1] (2.10) H in t , , g NC g k ,s ,i j ' i , j ,c s ,i j ,c X ij ij s ,ij , c c 1 NC i , j ,c c 1 Bước 4: End, return enhanced image { g ' k ,s ,ij * (Lk,max – Lk,min) + Lk, min, 1≤ k≤ K } The complexity of algorithm is O(M*N) 2.8 Experiment and evaluate Table 2.4 The result table compares the criteria values between the Cheng algorithm and proposed algorithm 2.3 Image CMR CMG CMB Eavg Havg Cheng Propose Cheng Propose Cheng Propose Cheng Propose Cheng Propose #1 0.1292 0.3067 0.2011 0.4422 0.2550 0.5537 6.0405 6.7621 0.3523 0.4966 #2 #3 #4 #5 #6 0.0166 0.0175 0.0305 0.0179 0.0305 0.0503 0.0602 0.1002 0.0909 0.0373 0.0208 0.0209 0.0370 0.0315 0.0338 (a) 0.0982 0.0988 0.1464 0.1741 0.0441 0.0361 0.0566 0.0598 0.0368 0.0410 0.0579 0.1491 0.1876 0.1973 0.0548 7.3196 7.4822 7.4586 7.3038 3.5482 7.3506 7.6852 7.8066 7.6426 3.7398 (b) 0.8212 0.7999 0.8635 0.8519 0.2850 0.7707 0.7935 0.7929 0.8419 0.2923 17 (c) (d) Figure 2.4 (a), (c) The contrast enhancement Image on S and V channels of image # 5, # using Cheng’s method and reversed transformation to RGB (b),(d) The contrast enhancement Image on S and V channels of image # 5, # using algorithm 2.3 and reversed transformation to RGB Chapter Construct new homogeneity measure base on hedge algebra and apply contrast enhancement for multi channels image This chapter presents the new homogeneous measurement construction method using interpolative reasoning method to solve fuzzy rule system of hedge algebra, which proposes an algorithm to enhance image contrast based on new homogeneous measurements 3.1 Improve Cheng's homogeneous measurement 3.1.1 Homogeneous measurement based on t-norm operators We replaced the Cheng’s formula entered by the following formula: (3.1) H O m ax E * H ,V * R ij ij ij ij ,ij and found that the formula (3.1) is suitable for a variety of RGB images (images that have been enhanced contrast when using formula (3.1) are smooth) More generally, we proposed combination operator have general type as: (3.2) H O f E ,V , H , R T T E , H , T V , R ij ij ij ij , ij ij eh ij hr ij , ij The local properties Vij, R4,ij change slowly, so that the mainly effect is combination operator T e h E ij , H ij In the type of formulas (3.2), a homogeneous measurement of the new pixel can be constructed based on a t-norm operator Tnorm of fuzzy set theory as: (3.3) H O m ax T E , H ,V * R ij n o rm ij ij ij ,ij 3.2 Construct homogeneity measurement based on Hedge Algorithm The new homogeneity measurement is constructed based on rule system The new homogeneous measure built on the rule sytem is quite simple and clear as follows: 18 Suppose G ( g r a d ie n t ), E ( e n t r o p y ) and T is language variables with the numeric linguistic value domain normalized of [0, 1] HMR( G , E , T ) is fuzzy rule sets for R1: If G very hight AND R2: If G very low AND R3: If G hight AND R4: If G low AND R5: If G hight AND R6: If G low AND R7: If G little hight AND R8: If G little low AND R9: If G hight AND E G , and E is hight Then is low Then E is hight Then E is low Then E E E E is very higt T is very hight is very low T is hight T (3.4) is low is hight Then is low Then T T is little hight is little low is little hight Then T is little hight R10: If G low AND R11: If hight G very hight AND E T , is stated as follows: is very low T is very low Then E T is very hight Then E T is little low Then E T is little low is very hight Then T is very very R12: If G very low AND E is very low Then T T is very very low Using the interpolative reasoning method of Hedge Algebra, we build the f = fSnorm function for the system (3.4) in steps as following: Step 1: Setting the hedge algebra and correlatived fuzzy arguments Denote AG = ( G , C, w, H, ), AE = ( E , C, w, H, ) AT = ( T , C, w, H, ), C = {c–, c+}, c– = low, c+ = hight, H = H- H+, H- = {little}, H+ = {very}, L ≡ little, V ≡ very Put , , ̅ , ̅ , ̅ ̅ ̅ ̅ ̅ ̅ , , where ̅ , ̅ , ̅ , ̅ , , (0, 1) The relationship of the hegde to other is defined in Table 3.1 as follows: Bản The Relations about sign of hegdes V L V + + L From the table above we have: sign(Vc ) = sign(VIc-) = sign(VI)sign(Ic-) = 1*sign(Ic-) = -1 Table 3.2 Fuzzy and SQM value tables correspond to AG, AE, AT Parameter For G For U(c+) G G U (V) G 1 G E 1 v U (low) G G E E E For E 1E T T T E T T TT 19 vU(V.low) (very low) G G vU(L.low) (little low) ̅ vU(hight) 1 G G vU(V.hight) (very hight) 1 G G G G G vU(V.V.low) (very very low) vU(V.V.hight) (very very hight) G G E E + E E E T E G G G E E E E - E T T T 3 G E TT 3 T E - ̅ T TT T T - 1 G + ̅ ̅ 2 + vU(L.hight) (little hight) TT E E T T E , where U is G , E or T Step 2: 2.1: Compute the SQMs of the left and right components of the fuzzy rule system using Table 3.3 2.2: With operator AND: A N D : [ , 1] [ , 1] , AND( G , E ) = G * E , we have the value arrays of surface interpolation points Snorm (where m = 2) of system rule (3.4) as: Table 3.3 Table of values of interpolation points based on the AND operator of the rule system (3.4) Rule index Interpolation points (x,y)[0,1]2 x y R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 AND ( ̅ ̅ AND ( ̅ ̅ AND ( ̅ ̅ AND ( ̅ ̅ AND ( ̅ ̅ AND ( ̅ ̅ AND ( ̅ ̅ AND ( ̅ ̅ AND ( ̅ ̅ AND ( ̅ ̅ AND ( ̅ ̅ AND ( ̅ ̅ From here it is possible to use a simple Snorm interpolation as linear segmentation on interpolation points Comment: Given the input semantic value pairs ( G , E ) [0, 1]2, te define output semantic values hT [0, 1] follow as: hT = fSnorm (AND( G , E )); Function fSnorm for rule system (3.4) is defined as above which denoted as THA, AND or THA when have already previous AND operator 20 Proposition 3.1: Function T H A : [ , 1] [ , 1] preserved order, mean that a , b , a ', b ' [ ,1] , a a ', b b ' T H A ( a , b ) T H A ( a ', b ') Since fSnorm is constructed from linear interpolation operator between interpolation points, it preserve the THA order on[0, 1]2 Using THA, we define a homogeneous measure of pixels as follows: (3.5) H O T T E , H ,V * R ij ij HA ij ,ij ij The technique of constructing a homogeneous measure using hegde algebra is implemented according to algorithm 3.1 as follows: Algorithm 3.1 Construct the homogeneity measurement at each pixel HA-HRM Input: Gray-scale Image I, size M x N Parameter: g, gr, ep, ep, ho, ho (0, 1) of Hedge Algebra AGr, AEp and AHo Output: Table of the homogeneous value at each pixel Step 1: Compute gradient, entropy, standard deviation and the 4th moment of the intensity distribution is normalized into the interval [0, 1] For each pixel gij 1.1: Compute Eij, Hij, Vij, R4,ij using (A.1) to (A.4) operator in appendix (see [8]) 1.2: Compute E E , H H , V V , R R ij ij ij ij ij ij , ij , ij Step 2: With AND operator, construct function THA for HRM(AGr, AEp, AHo) rule system of language variables Gr( g r a d i e n t ), Ep( e n t r o p y ) and Ho(homogeneity) uzzy measure parameters g, gr, ep, ep, ho, ho (0,1) Step 3: Compute the homogeneous measure value at each pixel 3.1: For each pixel gij Compute E H T E , H H O m i n E H , V * R , 3.2: Normalized ij HA ij ij For each pixel gij compute ij ij = ij ij , ij H O ij m a x { H O ij } Return: {βij} Algorithm 3.1 have complexity is O(M*N) 3.3 Enhancing color image contrast with the proposed homogeneity measurement Algorithm 3.2: Enhance color image contrast using homogeneity measurement HA-HRM Input: Color image I in a color representation RGB, size M x N Parameter C N , C , threshold fcut (fcut > 0, small enough), d (d x d is size of window) 21 Output: Color image RGB Inew, CMR, CMG, CMB, Eavg , Havg Bước 1: Let (IH, IS, IV) a color representation of image I in color domain HSV Quantify IS, IV as gray-scale images Step 2: With input data is a image channel combination (IS, IV), parameter C and threshold fcut , perform FCM clustering to estimate C gray-scale dynamic ranges [Bk,1,c, Bk,2,c] with k{S, V} (see (2.4) formula) Step 3: Identify transformed images method FS, FV of IS, IV using (2.5) formula Step 4: 4.1: Calculates the homogeneous values of FS, FV using HA-HRM 4.2: Compute non-homogeneity gray values {δS, ij}, {δV, ij}, exponential amplifications {S, ij}, {V, ij} at each pixel of FS and FV channels Step 5: Compute the contrast value and define the new gray channels of the FS and FV channels F as follow: I ,F I S S ,new V V ,new For FS FS(IS) and FV FV(IV) channels: compute contrast C S ,ij F S ( g S ,ij ) ij ( F S ) F S ( g S ,ij ) ij ( F S ) , C V ,ij FV ( g V ,ij ) ij ( FV ) FV ( g V ,ij ) ij ( FV ) (3.6) compute the new gray-scale values of S and V channels I S ,n e w 1 S ,ij 1 (i, j ) 1 S ,ij 1 t S ,ij , g S ,ij S ,ij C S ,ij I V ,n e w t S ,ij C S ,ij t S ,ij V ,ij C V ,ij V ,ij , g V ,ij V ,ij t V ,ij C V ,ij (i, j ) t S ,ij C V ,ij , g V ,ij V ,ij t V ,ij V ,ij C V ,ij t t S ,ij C S ,ij , g S ,ij S ,ij C S ,ij (3.7) , Note that: the channel S is marked number k = 1, channel V is marked number k = Step 6: Convert (IH, IS,new, IV,new) in HSV color representation to RGB color representation, we have Inew Step 7: Compute objective Criterias value CM{R,G,B}, Eavg and Havg 7.1: Compute non-homogeneity gray values {δR,ij}, {δG,ij}, {δB,ij} of IR, IG and IB channels corresponding 7.2: Compute CMR, CMG, CMB ussing (1.18) formula, detail as: CM k ij I n e w , k ( i , j ) R , ij I n e w , k ( i , j ) R , ij , k {R , G , B} M *N 7.3: Compute Eavg = Eavg{Inew,R, Inew,G, Inew,B}, Havg = Havg{Inew,R, Inew,G, Inew,B} using (1.12)-(1.14) Return: Inew, CMR, CMG, CMB , Eavg , Havg Not including the FCM algorithm and interpolation reasoning method to solve Fuzzy rule system (3.4), algorithm 3.2 has the same complexity as Cheng's 22 algorithm The proposed system architecture model is constructed as shown in Figure 6: Start Input Image RGB Convert RGB to HSV Calculates homogeneous values Transform S and V channels Compute the aroundly luminance, contrast and exponential amplification values Compute the new gray-scale values of S and V channels Reverse conversion HSV to RGB End Figure 3.1 Process flow of proposed algorithms Table 3.5 The result of the criteria CMR for each image and the combination operators to compute homogeneity values Độ đo/ảnh HO1 HO HO HO HO HO HA-HRM #1 0.3692 0.3687 0.3687 0.3688 0.3687 0.3691 0.3889 #2 0.1806 0.1805 0.1805 0.1805 0.1805 0.1806 0.1813 #3 0.1688 0.1687 0.1687 0.1687 0.1686 0.1687 0.1802 #4 0.2002 0.1993 0.1993 0.1992 0.1993 0.1997 0.2065 #5 0.2132 0.2144 0.2144 0.2144 0.2144 0.2160 0.2161 #6 0.1494 0.1494 0.1450 0.1493 0.1451 0.1454 0.1505 Table 3.5 The result of the criteria CMG for each image and the combination operators to compute homogeneity values Độ đo/ảnh HO HO #1 0.3780 0.3773 #2 0.1815 0.1814 #3 0.1703 0.1701 #4 0.2016 0.2007 #5 0.2142 0.2154 #6 0.1503 0.1503 23 HO HO HO HO 0.3773 0.3774 0.3774 0.3778 0.1814 0.1814 0.1814 0.1814 0.1701 0.1701 0.1701 0.1702 0.2007 0.2006 0.2006 0.2011 0.2154 0.2154 0.2154 0.2168 0.1458 0.1502 0.1459 0.1463 HA-HRM 0.4000 0.1822 0.1814 0.2080 0.2170 0.1623 Table 3.6 The result of the criteria CMB for each image and the combination operators to compute homogeneity values Độ đo/ảnh HO #1 0.4200 #2 0.1851 #3 0.1869 #4 0.2126 #5 0.2148 #6 0.1498 HO 0.4192 0.1851 0.1868 0.2118 0.2160 0.1498 HO 0.4192 0.1851 0.1868 0.2118 0.2160 0.1451 HO HO 0.4194 0.4193 0.1850 0.1851 0.1869 0.1868 0.2117 0.2118 0.2160 0.2160 0.1497 0.1453 HO 0.4197 0.1851 0.1869 0.2121 0.2176 0.1457 HA-HRM 0.4418 0.1860 0.2048 0.2194 0.2177 0.1626 Table 3.7 The result of the criteria Eavg for each image and the combination operators to compute homogeneity values Độ đo/ảnh HO HO HO #1 6.1126 6.1129 6.1123 #2 7.2722 7.2712 7.2698 #3 7.3154 7.3127 7.3122 #4 7.5641 7.5456 7.5464 #5 7.3993 7.4098 7.4076 #6 4.2105 4.2174 4.2051 HO HO HO 6.1175 6.1101 6.1094 7.2684 7.2758 6.8158 7.3144 7.3096 6.9599 7.5462 7.5631 7.3155 7.4167 7.4168 7.2337 4.2228 4.2165 4.2239 HA-HRM 6.1509 7.3862 7.5124 7.6307 7.5833 4.3675 Table 3.8 The result of the criteria Havg for each image and the combination operators to compute homogeneity values Độ đo/ảnh HO HO HO HO HO HO #1 0.3999 0.4022 0.4021 0.4023 0.4020 0.4017 #2 0.6943 0.6944 0.6944 0.6944 0.6944 0.6944 #3 0.7198 0.7201 0.7201 0.7201 0.7201 0.7200 #4 0.7640 0.7648 0.7648 0.7648 0.7645 0.7646 #5 0.7891 0.7909 0.7909 0.7909 0.7909 0.7904 HA-HRM 0.3984 0.6943 0.7197 0.7610 0.7907 #6 0.4599 0.4606 0.4424 0.4612 0.4420 0.4414 0.4501 Table 3.4 to Table 3.8 shows the empirical results of images # - # 6, the value of objective criteria of direct contrast enhancement on each R, G, and B channel when using combination operator from the four local characteristics 24 HO7 equilateral have result highter than when using HOk, k = 1…6 The values of criteria Eavg are highter when using HA-HRM Also, the value of objective criteria Havg when applying HA-HRM, except two cases with image # and # 6, the remaining cases have value smaller value when applying HOk, k = (a) (b) (c) (d) (e) (g) (h) (k) (l) (m) (n) (p) Figure 3.2 The experiment result of images # 1, # and # The resulting image (on the left) using Cheng's algorithm, Result image (on the right) uses algorithm 3.2 with HO7 proposed Figure 3.3 The resulting image uses algorithm 3.2 with Cheng's combination operator H O E * V * H * R for image # and the resulting image is not smooth ij ij ij ij , ij Expended direction of algorithm Expand Hint operator from rules system to the rules system as following: R1: If x is then y is R2: If x is c- then y is very c- 25 R3: If x is W then y is W R4: If x is c+ then y is very c+ R5: If x is then y is R6: If x is little c- then y is cR7: If x is little c+ then y is c+ Appendix A: Comparing with indirect methods To see the effectiveness proposed by Hint operator, we have made a comparison with some recent advanced algorithms, namely the algorithm stated in: 1) ESIHE Algorithm: K Singh, R Kapoor, "Image enhancement using exposure based sub image histogram equalization", Pattern Recogn Lett 36 (2014) 10–14 2) RICE Algorithm: K Gu, G Zhai, X Yang, W Zhang, Ch W Chen, "Automatic contrast enhancement technology with saliency preservation," IEEE Trans on Circuits and Syst for Video Technology (TCSVT), vol 25, no (2015), pp 1480-1494 3) GHMF Algorithm: K Gu, G Zhai, Sh Wang, M Liu, J Zhou, W Lin, "A general histogram modification framework for efficient contrast enhancement," in Proc IEEE Int Symp Circuits and Syst (ISCAS), pp 28162819, May 2015 4) ROHIM Algorithm: K Gu, W Lin, G Zhai, X Yang, W Zhang, C W Chen, "No-reference quality metric of contrast-distorted images based on information maximization," IEEE Trans Cybernetics, 2017 (in press.) The data set used for comparison is the popular TID2013 image file, and some other Vietnamese artwork (a) Ảnh gốc I02 (b) ESIHE (c) RICE (d) GHMF (e) ROHIM (G) Hint đề xuất Figure A.1 Result of the contrast enhancement for image I02 in TID2013 when using indirect method and hint 26 (a) Ảnh gốc I10 (b) ESIHE (c) RICE (d) GHMF (e) ROHIM (g) Hint đề xuất Figure A.2 Result of the contrast enhancement for image I10 in TID2013 when using indirect method and hint (a) Ảnh gốc I24 (b) ESIHE (c) RICE (d) GHMF (e) ROHIM (f) Hint đề xuất Figure A.3 Result of the contrast enhancement for image I24 in TID2013 when using indirect method and hint 27 (a) The original image (d) GHMF (b) ESIHE (e) ROHIM (c) RICE (g) Hint đề xuất Figure A.4 Result of the contrast enhancement for art image of artist Dương Quốc Định when using indirect method and hint (a) The original image (d) GHMF (b) ESIHE (e) ROHIM (c) RICE (g) Hint đề xuất Figure A.5 Result of the contrast enhancement for art image of artist Dương Quốc Định when using indirect method and hint (a) The original image (b) ESIHE (c) RICE (d) GHMF (e) ROHIM (g) Hint đề xuất Figure A.6 Result of the contrast enhancement for art image of artist Dương Quốc Định when using indirect method and hint 28 Figure A.1 - A.3 shows that the result image of the Hint algorithm preserving better image details than the ESIHE, RICE, GHMF and ROHIM indirect algorithms (visual observation of the marking area in the image above) Figure A.4 - A.6 also shows that the proposed algorithm 2.3 has results which preserving details and color better than the above indirect algorithm Appendix B: Classic histogram Classic histogram for a gray-scale image channel: H is I ( g ) # { (i,j) I ( i , j ) g } , where I I ( i , j ) [ L m in , L m ax ] i M ,1 j N Appendix C: The Cheng's method to estimate the gray-scale dynamic range of a image channel The gray-scale dynamic range is [a,c] where a=min{(1-f2)(g1-Lmin) + Lmin, B1}, c=max{f2(Lmax-gk) + gk, B2} , where B1, B2 are estimated as follow: Lm ax Lm ax B1 H i s ( i ) f1 i L m in H is (i ) , i L m in Lm ax H is ( i ) f i B2 H is ( i ) i L m in (b) The gray-scale dynamic range of image: a=36, bopt =123, c=131, B1=70, B2=131, gl=53, gh=130 (a) Figure C.1 Estimates a gray scale dynamic range based on the top of the histogram Appendix D: Image contrast enhancement formular of Cheng and Hint Chen ’s method proposed Hint t { ,0 } , C heng l l 1 l l 1 l l 1 l l 1 l t t t t , l [0, ] H in t l , l ( , 1] l 1 l ,l l 1 l 1 1 1 [0, ] l l l l , l ( , 1] ... Overview of contrast enhancement and solving the fuzzy rule system base on Hedge Algebars This chapter presents the concepts of hedge algebra and approximate reasoning method based on hedge algebra, ... (c) The contrast enhancement Image on S and V channels of image # 5, # using Cheng’s method and reversed transformation to RGB (b),(d) The contrast enhancement Image on S and V channels of image. .. number of iterations of the standard FCM algorithm 2.4 Transforming image channel Definition 2.2: The image channel transformation of a combination of image channel in a color representation of input