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[4] Y. S. Kung and C. M. Liaw, “A fuzzy controller improving a linear model following controller for motor drives,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 194–202, Aug. 1994. [5] C. C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller—Part I,” IEEE Trans. Syst., Man Cybern., vol. 20, pp. 404–418, Apr. 1990. [6] C. T. Lin and C. S. Lee, Neural Fuzzy Systems. Englewood Cliffs, NJ: Prentice-Hall, 1996. [7] Y. Park, U. Moon, and K. Y. Lee, “A self-organizing fuzzy logic controller for dynamic systems using a fuzzy auto-regressive moving average model,” IEEE Trans. Fuzzy Syst., vol. 3, pp. 75–82, Feb. 1995. [8] A. Suyitno, J. Fujikawa, H. Kobayashi, and Y. Dote, “Variable-structured robust controller by fuzzy logic for servomotors,” IEEE Trans. Ind. Electron., vol. 40, pp. 80–88, Feb. 1993. [9] J. S. Taur and C. W. Tao, “Nested design of fuzzy controllers with partial fuzzy rule base,” Fuzzy Sets Syst., vol. 120, pp. 1–15, May 2001. [10] L. X. Wang, Adaptive Fuzzy Systems and Control. Englewood Cliffs, NJ: Prentice-Hall, 1994. [11] Y. Yam, P. Baranyi, and C. T. Yang, “Reduction of fuzzy rule base via singular value decomposition,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 120–132, Apr. 1999. [12] J. Yen and R. Langari, Fuzzy Logic-Intelligence, Control and Information. Englewood Cliffs, NJ: Prentice-Hall, 1999. [13] J. Yen and L. Wang, “Simplifying fuzzy rule-based models using orthogonal transformation methods,” IEEE Trans. Syst., Man, Cybern., vol. 29, pp. 13–24, Feb. 1999. [14] L. A. Zadeh, “Fuzzy sets,” Inform. Contr., vol. 8, no. 3, pp. 338–353, June 1965. [15] K. Zhou, F. C. Doyle, and K. Glover, Robust and Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1996.
An Evolutionary Tabu Search for Cell Image Segmentation Tianzi Jiang and Faguo Yang
Abstract—Many engineering problems can be formulated as an optimization problem. It has become more and more important to develop an efficient global optimization technique for solving these problems. In this correspondence, we propose an evolutionary tabu search (ETS) for cell image segmentation. The advantages of genetic algorithms (GAs) and TS algorithms are incorporated into the proposed method. More precisely, we incorporate “the survival of the fittest” from evolutionary algorithms into TS. The method has been applied to the segmentation of several kinds of cell images. The experimental results show that the new algorithm is a practical and effective one for global optimization; it can yield good, near-optimal solutions and has better convergence and robustness than other global optimization approaches. Index Terms—Cell image segmentation, evolutionary tabu search (ETS), genetic algorithms (GAs), tabu search (TS).
I. INTRODUCTION In the past years, many methods for the segmentation of cell images have been presented [1]–[6]. These methods include region-based methods, threshold-based methods, etc. They use the gray information
Manuscript received September 18, 2001; revised March 22, 2002. This work was supported in part by Hundred Talents Programs of the Chinese Academy of Sciences, the Natural Science Foundation of China, Grants 60172056 and 697908001, and State Commission of Science and Technology of China, Grant G1998030503. This paper was recommended by Associate Editor I. Bloch. The authors are with the National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100080, China (e-mail:
[email protected];
[email protected]). Publisher Item Identifier S 1083-4419(02)05728-X.
Fig. 1.
675
Framework of the evolutionary tabu search.
of an image mainly without using any priori knowledge of a specific type of image. For some simple images without or with low noise, these methods may work well. However, in the presence of noise, clutter, and occlusion, it is difficult to obtain good segmentation results with these methods. The segmentation performance can be greatly improved by incorporating some a priori knowledge about the specific type of images being processed. In fact, one of the most challenging issues in medical image segmentation is to extend traditional approaches to segmentation and object classification in order to include shape information rather than merely utilize image intensity. In this correspondence, an elliptical cell contour model is introduced to describe the boundary of the cells and the cell image segmentation problem is finally transformed into an optimization problem. Conventional local search minimization techniques are time-consuming and tend to converge to whichever local minimum they first encounter. These methods are unable to continue the search after a local minimum is reached. The key requirement of any global optimization method is that it should be able to avoid entrapment in local minima and continue the search to give a near-optimal final solution whatever the initial conditions. It is well known that simulated annealing (SA) [7], [8], genetic algorithms (GAs) [9], [10], and tabu search (TS) [11], [12], meet this requirement. In [13], GAs were applied to solve this problem. In this correspondence, we develop an efficient optimization algorithm by combining evolution into TS. This method not only has the ability to find the global optimum, but also retains advantages of both TS and GAs. Experimental results show that our method is more practical, effective, and robust, compared to GAs or TS alone. The correspondence is arranged as follows. Section II gives a detailed description of the cell image segmentation problem to explain how it can be transformed into an optimization problem. In Section III, we introduce our evolutionary tabu search (ETS) method and how it can be used to solve the cell image segmentation problem. Section IV is devoted to the experimental results. In Section V, the conclusions are presented. II. CELL IMAGE SEGMENTATION PROBLEM In this section, we will give a detailed description of the cell image segmentation problem and how it can be cast as an optimization problem.
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Fig. 2. Segmentation results of hypothyroid image and robustness comparisons. (a) and (d) Evolutionary tabu search. (b) and (e) Genetic algorithms. (c) and (f) Tabu search.
A. Localization of Cells There are many cells across an image, so the first task is to locate all the cells, which can be done by finding edge points that belong to each cell respectively. We use the Canny edge detector [14], a wellknown algorithm, for the edge detection. Because of the influence of noise, there are a lot of false edges in the thresholded gradient image. However, the variation of the gray levels in the location near the cell boundary is usually large, so the edges formed due to image noise have a small number of connected image points. We can use a threshold to decide whether an edge indicates a cell boundary. If the number of the connected edge points is greater than the threshold, we think the edge represents a cell boundary. Otherwise, we think the edge is due to image noise. The threshold can be selected by experience.
B. Contour Model of Cell Image Most of the cells in the human body usually have ellipse-like boundaries, as shown in Figs. 2–5, from which we can see that the gray levels of the cells are lower than those of the background. Moreover, there is lot of noise, clutter, and occlusion in the image. In order to improve the performance of boundary detection, a priori knowledge of the cell boundary shape needs to be incorporated into the segmentation method. If a cell boundary can be analytically found, we can easily solve the segmentation problem. Therefore, the cell image segmentation can be cast as a parametric optimization problem. In order to use the priori knowledge that the cells have an ellipse-like boundary, we use an ellipse equation to describe the boundary of a
Fig. 3. Segmentation results of small intestine image and robustness comparisons. (a) and (d) Evolutionary tabu search. (b) and (e) Genetic algorithms. (c) and (f) Tabu search.
cell. The mathematical model of the cell boundary can be described as follows: [(x
0 x0 ) cos + (y 0 y0 ) sin ]2 a2
+
[(x
0 x0 ) sin + (y 0 y0 ) cos ]2 b2
= 1:
(1)
Equation (1) denotes an ellipse in the (x; y ) domain, and x0 , y0 determines the center of the ellipse; indicates the orientation of the ellipse; a and b determines the size of the ellipse. As is known, an ellipse can be determined by five different points on it. Two parameters are needed to determine the location (x0 ; y0 ), and three are needed to determine its size and orientation (a; b; ). Therefore, the segmentation task is cast as choosing five different points from the image points to determine an ellipse fitting the cell boundary optimally. C. Objective Function In our method, we first use the Canny operator [14] to find the edge points. We assume that the number of the edge points is N and each edge point has an associated index, which is a number from 1 to N . We can select five of them to represent an ellipse. Let I = (I1 ; I2 ; . . . ; I5 )
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Fig. 4. Segmentation results of red blood image and robustness comparisons. (a) and (d) Evolutionary tabu search. (b) and (e) Genetic algorithms. (c) and (f) Tabu search.
denote the index of the five edge points selected. For the sake of convenience, we assume that Ii < Ij for i < j; i; j = 1; . . . ; 5. The objective function is defined as follows: f (I )
2
=
s(rI
)
(2)
where s is the step function; s = 0 if rj is greater than or equal to the template width; and s = 1 otherwise. This objective function counts the number of points within a fixed distance of the ellipse. Moreover, it effectively matches a small template around this ellipse to the possible edge data. Therefore, our task becomes to find I 3 , which maximizes the objective function I
3
= arg max f (I ): I
Fig. 5. Segmentation results of red blood image and robustness comparisons. (a) and (d) Evolutionary tabu search. (b) and (e) Genetic algorithms. (c) and (f) Tabu search.
(3)
III. OPTIMIZATION USING EVOLUTIONARY TABU SEARCH In this section, we will describe the ideas of the ETS and how it is used to optimize the cell image segmentation problem.
generated in the next generation is determined. This second-level selection actually provides a measure of the fitness of each family. Instead of using the objective values of a single point, which might be considerable biased, we define a fitness value based on the objective values of all the children in the same family for that measure. The number of children allocated to each family for the next generation is proportional to their fitness values, but the total number of the next generation’s children is still constant. In this way, fitter individuals have better survival chance. It has been show that eventually only one family survives, which is usually the best one. It is the second-level competition that gives the measure of the regional information. In fact, the fitness value provides the information of how good the region is. If the region is found to contain a higher fitness value, we allocate more attention to search in that region. The framework of our method is shown in Fig. 1. B. Extracting Cell Boundary Using Evolutionary Tabu Search
A. Evolutionary Tabu Search (ETS) The ETS takes advantages of GA [9], [10] and TS [11], [12], and it has two level selections, which are called the first-level selection and the second-level selection, respectively. We make use of TS in the first level, and evolution ideas in the second level. The ETS technique starts with a guess of N feasible candidates called parents, chosen randomly in the search space. Initially, each parent can generate a number of children, which forms a family. The children in the same family constitute the first level selection, and we use TS to select the child as parent for the next generation. The second-level selection is the competition between the families, in which the number of children that could be
Here we explain how the ETS is used for cell image segmentation; that is, to optimize (2). Our first level selection, which is the competition in the same family, can be described as follows. Step 1) Let Ic be the parent of one family, and fc be the corresponding objective function values computed using (2). For the first generation, let Ib = Ic and fb = fc . Step 2) Use Ic to generate N T S children It1 ; It2 ; . . . ; ItNT S (see Remark) and evaluate their corresponding objective function values ft1 ; ft2 ; . . . ; ftNT S . Step 3) Arrange ft1 ; ft2 ; . . . ; ftNT S in a descending order, and de[1] [2] [NT S ] [1] . If ft is not tabu, or note them by ft ; ft ; . . . ; ft
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[1]
[1]
[1]
b , then let Ic = It and fc = ft , and go to Step [L] [L] [L] 4); otherwise, let Ic = It and fc = ft , where ft is the [NT S ] [2] that is not tabu best objective function of ft ; . . . ; ft [1] [2] [NTS ] and go to Step 4). If all ft ; ft ; . . . ; ft are tabu, go t
f
> f
to Step 2) to regenerate neighbors. Step 4) Insert new tabu element at the bottom of the tabu list. If the length of the tabu list exceeds the predetermined size, then delete the first element in the tabu list. Remark: Given a current solution Ic , one can generate a trial solution using several strategies. We use the following strategy: given a probability threshold P , for i = 1; 2; . . . ; 5, draw a random number Ri u(0; 1), where u(0; 1) is the uniform distribution on interval [0, 1]. If Ri < P , then It (i) = Ic (i); otherwise draw random point l from the set including all the Ic (i) neighboring-points and let It (i) = l. The second-level selection, which considers the region information, is presented as follows. 1) Compute the fitness of each family, which is the average objective function value of the family. 2) Sum up the fitness of each family. 3) For each family, the number of the children generated can be calculated according to the following formula: NT S
=
N0
2
N T S0
2
F
S
(4)
where number of children that will be generated for that family in the next generation; N T S0 initial number of children being generated for each family; N0 initial number of the family; F fitness of that family; S sum of the fitness of all families. NT S
IV. EXPERIMENTAL RESULTS In this section, we present our experimental results on segmenting cell images, and compare between GAs and TS methods. In the ETS algorithm, the tabu list enables the algorithm to have short-term memory; a large tabu list size allows more diversification, while a small list size makes the algorithm more forgetful. Determining the list size is a nontrivial problem. The number of families is also important. If it is too small, it will resemble TS, otherwise the convergence speed will become worse. Thus, selecting this value is a tradeoff between the final extraction result and the convergence speed. The other important parameter is the initial number of children, which also affects the convergence speed, in our experiments, N T S0 2 [2; 8]. The segmentation results of our method ETS, GAs, and TS are given out in (a), (b), and (c) in Figs. 2–5, respectively. From the segmentation results, we can see that the ETS method and GAs work fairly well. Both of these two methods can find a fairly good solution, but the TS method does not perform as well compared to ETS and GAs. To compare the robustness of the ETS method with GAs and TS methods, we execute each algorithm ten times and display the segmentation results in the same image, respectively. The experimental results are shown (c), (d), and (e) in Figs. 2–5, respectively. From the experimental results, we can see that the robustness of ETS and GAs are better than TS methods. A nearly optimal segmentation is obtained each time using ETS method. However, TS methods cannot always obtain an optimal segmentation result, which can be seen from Figs. 2–5. V. CONCLUSIONS It is very important to segment the cells in cell image analysis because it is a prerequisite to the measure of the geometric parameters
of the cells, which is useful for the pathologists to make diagnostic decisions. In this correspondence, we have proposed a novel method for the cell image segmentation. Specifically, by incorporating the idea from evolution into TS, we propose a very powerful optimization technique called ETS, which performs much better than the traditional TS method. For the cell image segmentation, ETS can discover a near-optimum solution after examining an extremely small fraction of possible solutions. In our experiments, we find that ETS yields near-optimum solution and has a good robustness. Furthermore, ETS, which takes advantages of evolution algorithm, makes TS become one special form of it. Now, we are considering the application of ETS to other problems in computer vision. ACKNOWLEDGMENT The authors are grateful to the anonymous referees for their significant and constructive comments and suggestions, which greatly improved this correspondence. T. Jiang also thanks R. Scally, an English teacher at the 21st Century English Training Center in Beijing, for carefully checking the grammar in this correspondence. Finally, the authors thank Dr. S. Z. Li at Microsoft Research Asia for checking the manuscript in its final stage. REFERENCES [1] A. Garrido and N. Perez, “Applying deformable templates for cell image segmentation,” Pattern Recognit., vol. 33, pp. 821–832, 2000. [2] T. Mouroutis, S. J. Roberts, and A. A. Bharath, “Robust cell nuclei segmentation using statistical modeling,” Bioimaging, vol. 6, pp. 79–91, 1998. [3] I. Simon, C. R. Pound, A. W. Partin, J. Q. Clemens, and W. A. Christensbarry, “Automated image analysis system for detecting boundaries of live prostate cancer cells,” Cytometry, vol. 31, pp. 287–294, 1998. [4] H. S. Wu, J. Barba, and J. Gil, “Iterative thresholding for segmentation of cells from noisy images,” J. Microsc., vol. 197, pp. 296–304, 2000. [5] , “A parametic fitting algorithm for segmentation of cell images,” IEEE Trans. Biomed. Eng., vol. 45, pp. 400–407, Mar. 1998. [6] H. S. Wu and J. Gil, “An iterative algorithm for cell segmentation using short-time Fourier transform,” J. Microsc., vol. 184, pp. 127–132, 1996. [7] E. Aarts and J. Korst, Simulated Annealing and Boltzmann Machine. New York: Wiley, 1989. [8] S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, “Optimization by simulated annealing,” Science, vol. 220, pp. 621–680, 1983. [9] D. E. Goldber, Genetic Algorithm in Search, Optimization and Machine Learning. Reading, MA: Addison-Wesley, 1989. [10] Z. Michalewicz, Genetic Algorithm Data Structures Evolution Programs. New York: Springer-Verlag, 1992. [11] D. Cvijovic and J. Klinowski, “Taboo search: An approach to the multiple minima problem,” Science, vol. 267, pp. 664–666, 1995. [12] F. Glover, “Tabu search,” in Modern Heuristic Techniques for Combinatorial Problems, C. R. Reeves, Ed. New York: Wiley, 1993. [13] F. Yang and T. Jiang, “Cell image segmentation with the kernel-based dynamic clustering and an ellipsoidal cell shape model,” J. Biomed. Inform., vol. 34, no. 2, pp. 67–73, 2001. [14] J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 679–698, Nov. 1986.
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