Chapter 5 Extension of Fuzzy Partition Technique in Three-Level Thresholding

Transcription

1 Chapter 5 Extension of Fuzzy Partition Technique in Three-Level Thresholding 88 Two-level thresholding segments a gray scale image into two parts: object and background. This two-class segmentationisnotsufficient in some applications. More-than-two-level thresholding is needed in these cases. We have seen from the previous chapter that the derived fast searching method is very efficient, so that we may assume that the thresholding technique based on the fuzzy partition and entropy can be extended to three-level thresholding. In this chapter we are going to investigate how the fast searching method can be applied to the three-level fuzzy partition. Also the relationship among the three membership functions in the fuzzy partition is carefully examined and a novel view is proposed. 5.1 Probability Partition and Fuzzy Partition in Three Level Thresholding Probability Partition For a considered image I(x, y) :I(x, y) G, (x, y) D, it is mentioned in Section 4.1 that an l gray levels image is characterized by the PP of the image domain D, Π l = {D 0,D 1,..., D l 1 } which is directly known from the histogram of the l level considered image. Three-level thresholding thresholds the image into three gray lev-

2 89 els based on the Π l. In this gray levels image, the domain D of the original image is classified into three parts, E b, E m and E d. E b is composed of pixels with high gray levels, E m is composed of pixels with middle gray levels and E d of low level pixels. Π 3 = {E d,e m,e b } is an unknown probabilistic partition of D with a probability distribution of p d, p m and p b, i.e., p d = P (E d ), p m = P (E m ),andp b = P (E b ) Fuzzy C-Partition Different types of fuzzy membership functions have been used in fuzzy logic control. For three-level thresholding, we use the simplest function which is monotonic to approximate the memberships of bright µ b and dark µ d, and the membership of medium µ m is dependent on µ b and µ d. The membership functions have four parameters a 1, c 1, a 2 and c 2. In other words, the two thresholds t 1 and t 2,forthree-level thresholding are dependent on a 1,c 1,a 2 and c 2 (See Figure 5.1). Let t 1 = 1 2 (a 1 + c 1 ),t 2 = 1 2 (a 2 + c 2 ), where 0 a 1 c 1 255, 0 a 2 c are four unknown parameters, a 2 a 1 and c 2 c 1. For each k =0, 1,...,255, let D kd = {(x, y) :I(x, y) t 1, (x, y) D k }, D km = {(x, y) :t 1 <I(x, y) t 2, (x, y) D k },

3 90 D kb = {(x, y) :I(x, y) >t 2, (x, y) D k }. If a 1,c 1,a 2, and c 2 are selected then Π k = {D kd,d km,d kb } isappofd k with the probabilistic distribution p kd P (D kd )=p k p d k, p km P (D km )=p k p m k,and p kb P (D kb )=p k p b k,where p d k = P M x=1 P N y=1 S kd(x, y)/s k p m k = P M x=1 P N y=1 S km(x, y)/s k p b k = P M x=1 P N y=1 S kb(x, y)/s k (5.1) ½ 1 I(x, y) t1 S kd (x, y) = 0 otherwise,for(x, y) D k, S km (x, y) = ½ 1 t1 <I(x, y) t 2 0 otherwise,for(x, y) D k, S kb (x, y) = S k = X ½ 1 I(x, y) >t2 0 otherwise and,for(x, y) D k, (x,y) D k (S kd (x, y)+s km (x, y)+s kb (x, y)). It is clear that p d k is the conditional probability of a pixel that is classified into the class d (dark) under the condition that the pixel belongs to D k.theconditional probabilities of a pixel belonging to classes m (median) and b (bright) respectively are p m k and p b k. Hence, if the thresholds t 1 and t 2 canbeobtainedthenthepp,π 3 of D is given as

4 91 E d = [ 255 k=0 D kd,e m = [ 255 k=0 D km,e b = [ 255 k=0 D kb. BasedonBaye sformula, p d P (E d )= P 255 k=0 P (D kd) = P 255 k=0 P (D k) P (E d D k )= P 255 k=0 p k p d k p m P (E m )= P 255 k=0 P (D km) = P 255 k=0 P (D k) P (E m D k )= P 255 k=0 p k p m k p b P (E b )= P 255 k=0 P (D kb) = P 255 k=0 P (D k) P (E b D k )= P 255 k=0 p k p b k (5.2). Note that p d k + p m k + p b k =1for k =0, 1,..., 255. Ifa 1,c 1,a 2, and c 2 are given, then {E d,e m,e b } is a PP of D. In order to find the parameters a 1, c 1, a 2 and c 2, we consider three membership functions: µ d =[µ d (0),..., µ d (255)] T, µ m =[µ m (0),..., µ m (255)] T and µ b =[µ b (0),...,µ b (255)] T which are definedontheuniversalg = {0, 1,..., 255},where µ d (k) =p d k, and µ m (k) =p m k µ b (k) =p b k.

5 92 Obviously, µ b (k)+µ m (k)+µ w (k) =1,k=0, 1,..., 255. So {µ b,µ m,µ w } isafuzzy 3-partition. The Equation 5.2 is rewritten as X255 X255 X255 p d = p k µ d (k),p m = p k µ m (k),p b = p k µ b (k). (5.3) k=0 k=0 k=0 The fuzzy 3-partition is given in the form shown in Figure 5.1 depending on parameters a 1,c 1,a 2, and c Membership Functions in Three Level Fuzzy Partition A different view on the shape of membership functions and its application on fuzzy three partition is proposed here. In the literature [55] it is assumed that c 1 <a 2. Under this assumption, the membership functions of dark and bright µ d and µ b do not intersect. For each gray scale, there exists only two possibilities. The searching of a 2 and c 2 is limited by this assumption, which will lead to an unoptimal result. For example, if c 1 <a 2 is required, the searching of a 2 and c 2 are limited in the range [c 1, 255]. Thelargerc 1 is, the smaller the searching range for a 2 and c 2.The restriction of a 2 >c 1 is not reasonable because as long as the equation P c i=1 µ ik =1 k is satisfied, a 2 can be smaller or greater than c 1. The membership function of the median µ m can be adjusted to make sure that equation P c i=1 µ ik =1 k is held. Basedonthedarkandbrightmembershipfunctions, the relationship between a 1, c 1,

6 membership membership 93 µ 1 µd µm µb 0.5 (a) 0 a1 t1 c1 a2 t2 c2 255 x (gray scale) µ 1 µd µm µb 0.5 (b) 0 a1 t1 a2 c1 t2 c2 255 x (gray scale) Figure 5.1: Two cases of the relationship among the membership functions in Fuzzy-3 Partition. (a): traditional view; (b) new view.

7 94 a 2 and c 2 is a 2 a 1 and c 2 c 1. Experiments show that better thresholded images are obtained when c 1 <a 2 is not required. The fuzzy 3-partition shown in Figure 5.1(a) is formed by the following three membership functions: µ d (k) = 1 k a 1 k c 1 a 1 c 1 a 1 <k c 1 0 k>c 1, (5.4) µ m (k) = µ b (k) = 0 k a 1 k a 1 c 1 a 1 a 1 <k c 1 1 c 1 <k a 2 k c 2 a 2 c 2 a 2 <k c 2 0 k>c 2, (5.5) 0 k a 2 k a 2 c 2 a 2 a 2 <k c 2 1 k>c 2, (5.6) in which the parameters a 1,c 1,a 2 and c 2 satisfy the condition: a 1 c 1 <a 2 c 2. The fuzzy 3-partition shown in Figure 5.1(b) is formed by the three membership functions given below: µ d (k) = 1 k a 1 k c 1 a 1 c 1 a 1 <k c 1 0 k>c 1, (5.7) µ m (k) = µ b (k) = 0 k a 1 k a 1 c 1 a 1 a 1 <k a 2 k a 1 c 1 a 1 k a 2 c 2 a 2 a 2 <k c 1 k c 2 a 2 c 2 c 1 <k c 2 0 k>c 2, (5.8) 0 k a 2 k a 2 c 2 a 2 a 2 <k c 2 1 k>c 2, (5.9)

8 95 where the four parameters a 1,c 1,a 2 and c 2 satisfy: a 1 c 1, a 2 c 2. In this case, it is possible that c 1 a 2 or c 1 >a 2.Whena 2 <c 1, a pixel whose gray level lies in the range of [a 2,c 1 ] partly belongs to three classes. 5.3 Searching Algorithms Simulated Annealing Algorithm In the literature [56] [55], the simulated annealing algorithm [58] is applied to search parameters a 1,c 1,a 2 and c 2. The simulated annealing algorithm contains four aspects: 1. A concise description of the configuration of the system, 2. A random generator of moves or rearrangements of the elements in a configuration, 3. A quantitative objective (cost) function containing the trade-offs that have to be made, which evaluates the fitness of a configuration, 4. An annealing schedule of the temperature and length of time for which the system is to be evolved, by which the annealing process can be controlled.

9 96 The annealing schedule may be developed by trial and error for a given problem, or may consist of a warming process until the system is obviously melted, then cooling in slow stages until diffusion of the components ceases. For this particular problem, fuzzy partition which has the maximum entropy value is the required state. And the cost function is the difference between a constant and the entropy of the current state. The procedure given in the literature [56] that is to find the optimal state whose cost function is minimized is composed of the following four steps : 1. Randomly generate an initial state State init, and let the current state State cur = State init, 2. Set the initial temperature T init and let current temperature T cur = T init, 3. Repeat the following searching procedure until the termination condition is met: (a) Select a move from the move-set randomly, apply to State cur and get the new state State new, (b) Compute the change of cost according to the cost function: E = Cost(State new ) Cost(State cur ) (c) If E 0, then the new state is a better state, replace the current state State cur with a new state State new, If E >0, then the new state is a worse state, calculate moving probability p = e E/Tcur,ifp>p ran (p ran is a random number in range [0, 1] ),

10 97 replace the current state State cur with the new state State new ;otherwise, retain the current state. (d) Renew the temperature T cur = next(t cur ) according to the cooling schedule. 4. Return State cur. The configuration representation for three-level thresholding is a set of parameters: (a 1,c 1,a 2,c 2 ),where0 a 1 <c 1 <a 2 <c They have eight move-sets: M 0 :(a 1,c 1,a 2,c 2 ) (a 1 1,c 1,a 2,c 2 ), M 1 :(a 1,c 1,a 2,c 2 ) (a 1 +1,c 1,a 2,c 2 ), M 2 :(a 1,c 1,a 2,c 2 ) (a 1,c 1 1,a 2,c 2 ), M 3 :(a 1,c 1,a 2,c 2 ) (a 1,c 1 +1,a 2,c 2 ), M 4 :(a 1,c 1,a 2,c 2 ) (a 1,c 1,a 2 1,c 2 ), M 5 :(a 1,c 1,a 2,c 2 ) (a 1,c 1,a 2 +1,c 2 ), M 6 :(a 1,c 1,a 2,c 2 ) (a 1,c 1,a 2,c 2 1), M 7 :(a 1,c 1,a 2,c 2 ) (a 1,c 1,a 2,c 2 +1). The cost function is defined as: Cost(X) =3 H(U; X) =3 H(U; a 1,c 1,a 2,c 2 ), where H( ) is the entropy,

11 98 H(U; a 1,c 1,a 2,c 2 )= p d lg p d p m lg p m p b lg p b, p d, p m and p b are given by Equation 5.2.The cooling schedule is T n+1 = αt n,where α is the cooling rate. 0 < α < 1. The simulated annealing algorithm provides us with an insight into an intriguing instance of artificial intelligence in which the computer has arrived almost uninstructed at a solution that might have been thought to require the intervention of human intelligence. It is almost uninstructed, but not completely uninstructed, because the cost function provides some instruction to make the states incline to move to the states of less cost (downhill move). At the same time, the state can also move uphill in a certain range of temperatures, which ensures that the process will not get stuck at a local optimum. The higher the temperature is, the more possible it is to move uphill. But as we can see this search is nearly blind. The number of all possible states is huge and state movement is selected randomly. The initial state and the cooling schedule need to be set by trial and error for individual problems. If the initial state is close to the optimal state, and the cooling rate is correctly set, the optimal state is reached quickly. If the initial state is far away from the optimal state, it takes a long time to reach the optimal state. On the other hand, if the initial state is close to the optimal state, but the cooling rate is set at too slow, it will miss the optimal state and stop at a worse state. There is no guarantee that the process will stop at the best state. Every run will give a different result because the state movement is selected randomly. Thus, the performance of the simulated

12 99 annealing algorithm is not steady, and also the trial settings of the initial state and cooling schedule are complicated Genetic Algorithm The genetic algorithm is important in dealing with problems of optimization in studies of complex adaptive systems, especially in the economics field. John Holland is known as the father of genetic algorithms. The idea behind this algorithm is to improve the understanding of a natural adaptation process, and to design artificial systems having properties similar to natural systems [61]. The basic idea is as follows: the genetic pool of a given population potentially contains the solution, or a better solution, to a given adaptive problem. This solution is not "active" because the genetic combination on which it relies is split between several subjects. Only the association of different genomes can lead to the solution. A genetic algorithm maintains a set of possible solutions which are encoded as chromosomes [63]. Like physiological reproduction, the algorithm generates the next generations by crossovers and mutations. The crossover strategy involves the elitist model which significantly improves the performance on the uni-modal surface problems. For a particular problem the algorithm must have the following five components: A genetic representation for a potential solution to the problem, or coding method. The problem parameters are represented as a finite string over some

13 100 alphabet (usually 0 and 1) which are called chromosomes. Each chromosome represents a solution to the problem. A way to create an initial population of potential solution, An evaluation function that mimics the role of the environment, rating solutions in terms of their fitness, called object function. An object function is a fitness measure of the solution represented by each chromosome. Its value tells to what extent the chromosome satisfies the final goal. A genetic operation that alters the composition of children, which is a method that changes the chromosomes if the object function does not satisfy the final goal. Values for various parameters which the genetic algorithm uses (population size, probabilities of applying genetic operators, etc.) The genetic algorithm is similar to the simulated algorithm. Such an algorithm does not guarantee success. The stochastic system and a genetic pool may be too far from the solution, or for example, a convergence which is too fast may halt the process of evolution. These algorithms are nevertheless extremely efficient, and are used in fields as diverse as the stock exchange, production scheduling or programming of assembly robots in the automotive industry. This method cannot produce the param-

14 101 eters which best fit the objective function directly. The optimal result can be obtained by choosing from the results of many runs Fast Search Procedure The basic concept of our method is to search a set of parameters (a 1,c 1,a 2,c 2 ) which satisfy p d (a 1,c 1,a 2,c 2 )=p m (a 1,c 1,a 2,c 2 )=p b (a 1,c 1,a 2,c 2 )= 1 3. Basedontheassumptionthata 1 a 2 and c 1 c 2, and referring to the fuzzy membership function, p d is the only function of (a 1,c 1 ). Sowecanstartwithsearching (a 1,c 1 ) which satisfy X255 p d = µ d (k) h k = 1 3, k=0 where µ d (k) is defined by Equation 5.4 or 5.7. The search of a 1 and c 1 is independent of a 2 and c 2. We can see that the equation for µ b (k) is a function of (a 2,c 2 ). That means (a 2,c 2 ) can be searched independently of (a 1,c 1 ) by letting X255 p b = µ b (k) h k = 1 3. k=0 However, because of the limitation of a 1 a 2 and c 1 c 2, the search of (a 2,c 2 ) can not be independent of (a 1,c 1 ). Whenever a set of (a 1,c 1 ) is obtained, the search range of (a 2,c 2 ) is a 1 a and c 1 c 2 255, whilea 2 c 2.Ifwestart with the search of (a 2,c 2 ), the search of (a 1,c 1 ) will depend on the search result of (a 2,c 2 ).

15 102 Since the histogram function h k is discrete, it is not guaranteed that the set which gives exactly p d = p m = p b = 1 3 can be obtained. In general cases, it is expected that a set of (a 1 (1),c 1 (1)), (a 1 (2),c 1 (2)),..., (a 1 (s),c 1 (s)) will be obtained which satisfy p d 1 ε, whereε isagivensmallpositivenumber. For 3 each set of (a 1 (i),c 1 (i)), there exists a serial set (a 2 (1),c 2 (1)) i, (a 2 (2),c 2 (2)) i,..., (a 2 (n i ),c 2 (n i )) i which satisfy 1 pb 3 ε, such that we have n i possible sets of (a 1,c 1,a 2,c 2 ) for each (a 1 (i),c 1 (i)). Hence, there is total number of P s i=1 n i sets of (a 1,c 1,a 2,c 2 ), each one of which satisfies 1 pd 3 ε and 1 pb 3 ε. Next compute the corresponding entropy of each set, the optimal set ( ea 1, ec 1, ea 2, ec 2 ) can be found by looking for the maximum entropy. The searching procedure is described below: 1. Input image and compute the histogram of the image. 2. Search for a 1 and c 1 :Setp = 1 3. Initialize a 1 =0,c 1 =255, =1, i =0and ε = P (a) Compute F (a 1 )= a 1 h k. k=0

16 103 If F (a 1 )=p,set(a 1 (i),c 1 (i)) = (a 1,a 1 ),andi = i +1.Goto3andsearch for n i possible sets (a 2,c 2 ) based on (a 1 (i),c 1 (i)). Compute the entropy generated by each possible set (a 1,c 1,a 2,c 2 ). One possible set candidate ( ea 1 (i), ec 1 (i), ea 2 (i), ec 2 (i)) is found by setting ( ea 1 (i), ec 1 (i), ea 2 (i), ec 2 (i)) = ( ea 1, ec 1, ea 2, ec 2 ) i, where H( ea 1, ec 1, ea 2, ec 2 ) i = Max n i j=1 H(a 1(j),c 1 (j),a 2 (j),c 2 (j)) i.( ea 1, ec 1, ea 2, ec 2 ) i is known as a local optimal set and is a candidate at which the entropy has a local maximum value. If F (a 1 ) <p,setc 1,grt =255, c 1,les = a 1. Apply the fuzzy membership function to compute p d. i. Compute p d = 255 P µ d (k) h k basedonequation5.3. k=0 (A) If p d p <ε, (a 1 (i),c 1 (i)) = (a 1,c 1 ), i = i +1.Goto3to search a serial set of (a 2,c 2 ) based on (a 1 (i),c 1 (i)), find the local optimal set ( ea 1, ec 1, ea 2, ec 2 ) i.thengotod. (B) If p d <p,thennomatterhowc 1 changes there is no possible set (a 1,c 1 ) which can satisfy p d p <ε.gotod. (C) If p d >p,movec 1 along the x axis in the range [c 1,les,c 1,grt ]. Reset c 1,grt = c 1, c 1 = 1(c 2 1,les + c 1,grt ). Iter1: Computep d basedonequation5.3.

17 104 If p d p <ε,set(a 1 (i),c 1 (i)) = (a 1,c 1 ),goto3tosearchfor (a 2,c 2 ) based on (a 1 (i),c 1 (i)). Find the local optimal set ( ea 1, ec 1, ea 2, ec 2 ) i. i = i +1.GotoD. Else if p d <p,letc 1,les = c 1, c = 1(c 2 1,les + c 1,grt ),checkthe searching space s = c 1,grt c 1,les.Ifs<1, stop searching. Set (a 1 (i),c 1 (i)) = (a 1,c 1 ).Goto3andfind the corresponding local optimal set ( ea 1, ec 1, ea 2, ec 2 ) i.thengotod.elsegotoiter1. Else if p d >p,letc 1,grt = c 1, c 1 = 1(c 2 1,les + c 1,grt ).Checkthe searching space s = c 1,grt c 1,les.Ifs<1, stop searching. Set (a 1 (i),c 1 (i)) = (a 1,c 1 ).Goto3andfind the corresponding local optimal set ( ea 1, ec 1, ea 2, ec 2 ) i.gotod.elsegoonsearching, go to Iter1. (D) Let a 1 = a 1 +,ifa 1 < 255,thengoto(a). ii. If F (a 1 ) >p, then stop. 3. Search for (a 2,c 2 ) i based on (a 1 (i),c 1 (i)): The search procedure for a 2 and c 2 is similar to the search procedure for a 1 and c 1 except for the initial condition and probability p = 2. It is described as follows: 3 Search for a 2 and c 2, p = 2 3. Initialize a 2 = a 1 (i), c 2 =255, =1, j =0and ε = ; P (a) Compute F (a 2 )= a 2 h k. k=0 If F (a 2 )=p,set(a 2 (j),c 2 (j)) i =(a 2,a 2 ), j = j +1.

18 105 If F (a 2 ) <p,setc 2,grt =1,andc 2,les = Min(a 2,c 1 ).Applyfuzzy membership function to compute 1 p b = p d + p m. i. Compute r =(p d + p m )=1 p b based on Equation 5.3. (A) If r p <ε, (a 2 (j),c 2 (j)) i =(a 2,c 2 ),andj = j +1. (B) If r<p, then there is no possible set (a 2,c 2 ) which can satisfy r p <ε.gotod; (C) If r>p,movec 2 along the x axis in the range [c 2,les,c 2,grt ]. Reset c 2,grt = c 2, c 2 = 1(c 2 2,les + c 2,grt ). Iter :Compute r based on Equation 5.3. If r p <ε,set(a 2 (j),c 2 (j)) i =(a 2,c 2 ),andj = j +1; Else if r<p,letc 2,les = c 2, c 2 = 1(c 2 2,les + c 2,grt ).Checkthe searching space s = c 2,grt c 2,les.Ifs<1, stop searching. Set (a 2 (j),c 2 (j)) i =(a 2,c 2 ).ThengotoD.Elsegoonsearching, go to Iter. Else if r>p,letc 2,grt = c 2, c 2 = 1(c 2 2,les + c 2,grt ).Checkthe searching space s = c 2,grt c 2,les.Ifs<1, stop searching. Set (a 2 (j),c 2 (j)) i =(a 2,c 2 ).ThengotoD.Elsegoonsearching, go to Iter. (D) Let a 2 = a 2 +,ifa 2 < 255,thengoto3(a). If F (a 2 ) >p, then stop.

19 106 Throughout the above procedure, K sets of candidates ( ea 1 (i), ec 1 (i), ea 2 (i), ec 2 (i)) (i =1, 2,..., K) are obtained and the best thresholds are finally obtained: t 1 = 1 2 ( ea 1 + ec 1 ) t 2 = 1 2 ( ea 2 + ec 2 ), where ea 1, ec 1, ea 2 and ec 2 are determined by H( ea 1, ec 1, ea 2, ec 2 )= max k=1,...,k H( ea 1(k), ec 1 (k), ea 2 (k), ec 2 (k)). Figure 5.2 shows the flowchart of the search procedure for all possible candidates (a 1,c 1,a 2,c 2 ). 5.4 Results and Discussion To verify the efficiency of the proposed method and compare it to the simulated annealing algorithm, experiments are carried outonmanygrayimages. Figures show three images thresholded with the proposed method and the simulated annealing algorithm. Experiment results show that our method achieves good results, while the performance of the simulated annealing algorithm is not steady. The simulated annealing algorithm depends on many factors such as the selection of initial states and the cooling schedule. Even for the same setting, different run outputs a different result. It is clear that the best result selected from the many runs may not be globally the best.

20 107 Input image and compute its histogram Initialize a1,p and i. Compute F(a1) <1/3 F(a1) >1/3 Stop =1/3 Set current a1 and c1 as a candidate Search for corresponding a2 and c2 Initialize c1 and compute probability of black p(b) with Equation (7). p(b) close to 1/3? Yes No Fix a1 and move c1 along the axis till p(b) close enough to 1/3 Set current a1 and c1 as a candidate Search for corresponding a2 and c2 Increase a1 by 1 Figure 5.2: Flowchart of the search procedure for all the candidate (a 1,c 1,a 2,c 2 ).

21 108 The searching time by the proposed method and the simulated annealing algorithm is shown in Table 4.1. It can be seen that the proposed method can search the optimal set in less than one seventh of the time used by the simulated annealing algorithm. Figure 5.3(a) is a gray scale submarine image, and (b) is the three-level thresholded image with the proposed method. We can see that the main features of the submarine are preserved after three-level thresholding, such as the snow and ice around the submarine which remain as white, while the sky and sea are gray becauseitsgrayscaleisbetweenthesubmarineandthesnow. Figure5.3(c)isthe three-level thresholded image using the simulated annealing method, which is one of the results randomly selected from a number of runs. Although the simulated annealing algorithm can reduce the searching time for searching among a huge number of data, it does not guarantee that the searching will stop at the global optimal state. Furthermore, every run can give a different result. Figure 5.3(d) shows the fuzzy 3-partition overlapping the histogram. The fuzzy 3-partition for this image is (a 1,c 1,a 2,c 2 )=(22, 179, 71, 255), so thresholds are t 1 =100, t 2 =163,and a 2 <c 1 in this case. The membership function of the median class is no longer a trapezoid. We adjust it to make sure that µ d + µ m + µ b =1hold. Also we can adjust the shape of µ d and µ b. This is an issue of the membership function which is not included in this paper. The thresholds are chosen not at the intersection of µ m and µ d, µ m and µ b, but at the intersection of µ b and µ d with the membership value at 0.5. We can see that the fuzzy 3-partition separates the peaks of the histogram well.

22 109 Figure 5.4(a) shows a gray scale building image, and (b) is the three-level thresholded image using the proposed method. We can see that the trees, the building, the roof, the cable post, the chimney, the grass and the path are well separated from the background. Figure 5.4(c) is the thresholded image with the simulated annealing algorithm. This is a better result chosenfromseveral, inwhichpartsofthe building, the roof, the cable post, the chimney and the trees are lost. Figure 5.4(d) is the fuzzy 3-partition with the proposed method overlapping the histogram. The histogram has just one main peak but with the right partition, the image is correctly thresholded. The fuzzy 3-partition is (a 1,c 1,a 2,c 2 )=(103, 156, 119, 225), andthe thresholds are t 1 =129,andt 2 =172. Figure 5.5(a), (b), (c) and (d) show a gray scale father and son image, the threelevel thresholded image with the proposed method, the thresholded image with the simulated annealing algorithm and the fuzzy 3-partition with the proposed method overlapping the histogram, respectively. The fuzzy 3-partition is (a 1,c 1,a 2,c 2 )= (18, 175, 108, 188), and the thresholds are t 1 =96and t 2 =148. The outlines of father and son are obtained in both (b) and (c). But the light change in the middle of the image is also kept in the thresholded image with the proposed method while the background in image (c) is uniform. All shadows of the people in the image are kept by the proposed method, but some are lost by the simulated annealing algorithm.

23 110 image submarine building father&son Proposed Method (ms) Annealing Algorithm (ms) Table 5.1: Searching time using the proposed method and the simulated annealing algorithm. We did not test the genetic algorithm, but as can be seen from the literature [56] the genetic algorithm cannot give the best result directly. The relatively best result canbeobtainedbymakingachoicefromsomeruns. The reason that the simulated algorithm and the genetic algorithm cannot give thebestresultisthattheirsearching for the optimal state is random. It is known that the random searching algorithm will stop at one of the maxima which only, in rare cases, is the global one. The proposed method looks for the best fuzzy c-partition which has the maximum entropy by looking for the fuzzy partition which has the corresponding probability partition to the maximum entropy. This is much more straightforward. 5.5 Summary Exploiting the relationship between the fuzzy c-partition and the probability partition gives a more straightforward solution in the search for fuzzy parameters. Instead of blind searching the fuzzy c-partition which has the maximum fuzzy entropy, we aim at the partition which has the corresponding probability partition to the maximum

24 111 Figure 5.3: Submarine image. (a) Gray scale submarine image; (b) Three-level thresholded submarine image with the proposed method; (c) Three-level thresholded submarine image with the simmulated annealing method; (d) Histogram of the gray scale building image and Fuzzy 3-Partition with the proposed method. (a 1,c 1,a 2,c 2 )=(22, 179, 71, 255), t 1 =100,andt 2 =163.

25 112 Figure 5.4: Kiosk image. (a) Gray scale kiosk image; (b) Three-level thresholded kiosk image with the proposed method; (c) Three-level thresholded kiosk image with the simulated annealing algorithm, (a 1,c 1,a 2,c 2 )=(62, 67, 111, 171), t 1 =64and t 2 =141; (d) Histogram of the gray scale kiosk image and Fuzzy 3-Partition with the proposed method, (a 1,c 1,a 2,c 2 )=(103, 156, 119, 225), t 1 =129and t 2 =172.

26 113 Figure 5.5: Father and son image. (a) Gray scale father and son image; (b) Three-level thresholded father and son image with the proposed method; (c) Three-level thresholded father and son image with the simulated annealing algorithm, (a 1,c 1,a 2,c 2 )=(68, 106, 106, 241), t 1 =87and t 2 = 173; (d) Histogram of the gray scale father and son image and Fuzzy 3-Partition with the proposed method, (a 1,c 1,a 2,c 2 )=(18, 175, 108, 188), t 1 =96and t 2 =148.

27 114 fuzzy entropy. The probability partition is the weighted area on the histogram, where theweightisthemembership. The description of the fuzzy 3-partition is also discussed in this chapter. We abandon the traditional assumption that a 1 c 1 a 2 c 2 which means each gray level has a possibility of belonging to no more than two classes. It is found in our experiment that this assumption limits the search range of the parameters and gives a bad result. We adjust the membership function of the medium class and expand the search range of the parameters. The experiment results show that a better result is achieved without that assumption. In our assumption, each gray scale has a possibility of belonging to up to three classes. Based on the relationship between the fuzzy c-partition, the probability partition and the maximum entropy theory, a three-level thresholding method is derived. The proposed method can explicitly give the best fuzzy 3-partition with the maximum fuzzy entropy. It outperforms the simulated annealing algorithm and the genetic algorithm which are usually applied in solving this kind of optimal problem. Although the proposed method cannot be expanded simply to n-level (n >3) thresholding, it shows that when the membership function is a specified fuzzy c-partition with the maximum entropy, this can be obtained by finding the function with the corresponding probability partition. Compared to the simulated annealing algorithm, our method reaches the optimal point straightforwardly and there is no need to run it many times to choose the best from these many results. Also abandoning the tradi-

28 115 tional assumption that a 1 c 1 a 2 c 2 gives a more reasonable description of the membership function, and achieves a better performance.