Tóm tắt Luận án Research image contrast enhancement based on hedge algebra

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: 1. Library of Graduate University of Science and Technology 2. 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 S- shaped gray-scale transformation function that enhances images contrast using Hedge Algebra, Proceedings of the 10 th 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. 2 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 i j i j,E H are passed to computing with word then the formula is formatted  i j i j,ehT E H should reflect the fuzzy rule system as follows: If g ra d ie n t is hight and e n tr o p y is hight then homogeneity is hight If g ra d ie n t is low and e n tr 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. 3 Chapter 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, 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  và  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 = hnh1x, 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 1. 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 và hH fm(hu) = fm(u), uX; 4 (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, )( )( )( )( yfm hyfm xfm hxfm  , that is it does not depend on specific elements and is called fuzziness measure of h, denoted by(h). Proposition 1. For each fuzziness measure fm and  which defined in Definition 1, the following statements hold: (1) fm(c - ) + fm(c + ) = 1 and ( ) ( ) h H fm hx fm x   ; (2)     1 )( qj j h  ,    p j j h 1 )(  , where ,  > 0 và  +  = 1; (3)    k Xx xfm 1)( , where Xk is set of term which has length k; (4) fm(hx) = (h).fm(x), and xX, fm(x) = fm(x) = 0; (5) Cho fm(c - ), fm(c +) và (h) where hH, x = hn...h1c  ,   {- ,+}, it is easily to calculate the fuzziness measure of x: fm(x) = (hn)...(h1)fm(c  ). (1.1) Definition 2. 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 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 và h' is positive h; (4) Sign(h'hx) = 0, if h'hx = hx. (1.2) Proposition 2. For any h and xX, if sign(hx) =+1 then hx > x and if sign(hx) = -1 then hx < x. Definition 3. 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+); (2)           )( )( )()()()()()()()( jSignj jSigni xfmx j hx j hxfm i hx j hSignxx j h  , every j, –q  j  p and j  0, where:     ,))(()(1 2 1 )(  xhhSignxhSignxh jpjj ; (3) (c-) = 0, (c-) =  = (c+), (c+) = 1, and every j is satisfy –q  j  p, j  0, we have: (hjx) = (x) +     ),()()(1 2 1 )()()( )( )( xfmhxhSignxfmhxhSign jj jSignj jSigni ij     (1.3) 5 (hjx) = (x) +     ).()()(1 2 1 )()()( )( )( xfmhxhSignxfmhxhSign jj jSignj jSigni ij     Proposition 3. xX, 0  v(x)  1. 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 . . . . . . . . . . If X1 = An1 and ... and Xm = Anm then Y = Bn (1.4) 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 – 1 fuzzy parameters of AY: (hq), ..., (h1), (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   j 1 , 1 , 1 , n L j j m i n j S x y X Y       is the linguistic hyper-surface and   j 1 o r X j 1 , , 1 , , v (x ) , ( ) [0 ,1] j j m n m Y j m x X j m y Y S v y        (1.4) sẽ được nhúng như 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 j X ij v (A ) , Y ( ) i v B ( 1, , 1,j m i n  ) 6 Snorm   jX ij 1 , 1 , v (A ) , ( ) Y i j m i n v B    can be defined by m input aggregate oprator fSnorm, v = fSnorm(u1, ..., um), v  [0, 1] và uj  [0, 1], 1,j m , satisfy conditions Y(Bi) = fSnorm(X1(Ai1), ..., Xm(Aim)), 1,i n . (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 1,j m   0 ,1 0 , 0 , ..., [0 , 1] m Sno rm b f a a  (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.    ( , ) ( , , ) ( ( , ); , , )op t op tI I i j S I a b c S func I i j a b c   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   [ 1 , 1 ] ( ; , , )a rg m axo p t b a c b H I a b c     where H is common fuzzy entropy measure. 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 4 th moment of the intensity distribution R4,ij. d. Compute homogeneity measurement of original image’s pixel (or transformed image) by assosiation operator from 4 local values. i j i j i j , m ax HO HO   (1.6) where        ij ij ij 4 ,ij ij ij ij 4 ,ij* * * 1 * 1 * 1 * 1ijHO E V H R E V H R      (1.7) đ. Compute non-homogeneity gray value of original image’s pixel (or transformed image) i j i j ( , ) W ij ( , ) W (1 ) (1 ) p q p q p q p q p q g           (1.8) e. Compute exponential amplification    a x m in ij m in ij m in a x m in * m m              (1.9) , where 1 m in a x 1 k m g g g g     , m a x 1  , gk, g1 is tops of histogram 7 g. Evaluate the contrast associated with pixel (i; j), and amplify. i j i j i j i j i j g C g      , i j' i j i j t C C   where t{0.25, 0.5} (1.10) 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: i j i j i j i j ' i j i j i j i j i j i j' i j i j' i j ' i j i j i j i j i j i j' i j i j 1 1 , 1 1 1 1 , 1 1 t t t t C C g C C g C C g C C                           (1.11) 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: m ax 2 m in ( ) ( ) lo g ( ( )) k k L k k k g L E I p g p g     , 1 1 , ( ) ( ) K k k a vg K E I E I K    where  # ( , ) ( ) * d e f k k I i j g p g M N   and convention 0*log2(0) = 0. (1.12) 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 1 , ( ) ( ) K k k a vg K H I H I K    (1.13) , where:    ,m ax ,m in 2 2 ( ) ( ) lo g ( ( )) 1 ( ) lo g (1 ( )) * ( ) k k L k k g L H I g g g g p g            , , m in , m ax , m in ( ) de f k k k g L g g L L     and convention 0*log2(0) = 0. 8 If the value of fuzzy Entropy is lower, differentiation the brightness or darkness of an pixel is higher, that is the image having bright -dark contrast height, Gray levels of Ik channel pixels are higher than the "gray" level in the middle. , m in , m ax ( , ) ( ) 2 k k k k L L I i j H I     1.2.3. Contrast measure criterion of image channel (average contrast measure value at every pixels, criterion is proposed by) Contrast measure criterion of gray-scale Ik’ compare with a original image’s channel Ik (Ik’ and Ik have the same size MxN), follow as: ' , i j ' i ,j , i j' ( , ) ( , ) ( , ) * k k k k k k I i j I i j C M I I M N       (1.14) , where {k,ij} of gray-scale image Ik is calculated by formula (1.8) of Cheng. This criterion is used to evaluate the direct contrast enhancement algorithm method. The value of contrast measure criterion is higher, the contrast of resulting image channel is enhanced stronger compared to the original image channel. Chapter 2. Transform multi-channel image and construct the s-shape