Principle of Maximum Entropy for Histogram Transformation and Image Enhancement

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1 Principle of Maximum Entropy for Histogram Transformation and Image Enhancement Gilson A. Giraldi National Laboratory for Scientific Computing Petrópolis, RJ, Brazil Paulo S.S. Rodrigues Computer Science and Electrical Engineering Departments of the FEI Technologic University São Bernardo do Campo, SP, Brazil Abstract In this paper, we present a histogram transformation technique which can be used for image enhancement in 2D images. It is based on the application of the Principle of Maximum Entropy PME) for histogram modification. Firstly, a PME problem is proposed in the context of the nonextensive entropy and its implicit solution is presented. Then, an iterative scheme is used to get the solution with a desired precision. Finally, we perform a transformation in the intensity values of the input image which attempts to alter its spatial histogram to match the PME distribution. In the case study we tae some examples in order to demonstrate the advantages of the technique as a preprocessing step in an image segmentation pipeline. 1. Introduction Histogram modeling and transformations are important tools for image processing [5]. Point operations, such as contrast stretching, equalization and thresholding, are based upon the manipulation of the image histogram. In many applications involving image acquisition, such as medical imaging, the targets are often characterized by low contrast or non-uniform intensity patterns in the regions on interest. Therefore, enhancement algorithms are generally required as a pre-processing step for image analysis [6]. In the literature, many histogram based techniques have been proposed for image enhancement [1]. The simplest method consists in stretching the original histogram linearly to occupy the full available intensity range [5]. Histogram equalization is another nown contrast enhancing technique which tries to eep the transformed histogram as uniformly distributed as possible over the entire intensity range [9]. However, the main disadvantage of these operations is that the frontiers between the original histogram modes are not well preserved which decreases the homogeneity inside the objects of interest. Such effect is undesirable mainly for image segmentation techniques based only on the gray-level information in the image. For example, local minima of the gray-level histogram can be used to segment the image by thresholding [8]. The threshold level can be also obtained by optimizing some information measure associated, lie entropy. For instance, [4] a generalization of the classical entropy, called Tsallis entropy, is applied in a general formalism for image thresholding. In the present paper, we propose a new histogram modification technique for image enhancement. e consider the application of the Principle of Maximum Entropy PME) for histogram transformation, inspired in analogous formulation in statistical physics [10]. The solution of a PME problem is a probability distribution histogram) that maximizes the entropy subject to some constraints. Firstly, a PME problem is proposed and its implicit solution is presented. e focus on the nonextensive Tsallis entropy in this discussion due to its generality and capability to cover a larger range of applications. Then, a numerical scheme is designed based on the observation that the obtained expressions can be written as a global mapping in the histogram space. Finally, we transform the intensity values of the input so that the histogram of the output image approximately matches the PME solution. Although there are others wors using entropy in image

2 enhancement enhancement measure for parameter determination, fuzzy entropy approaches, entropy conservation techniques, among others) [1, 3, 7], the novelty of our wor is the application of a nonextensive PME for the histogram modification. In the case study, we present some examples in order to show the advantages of the proposed technique. e observe an improvement in the homogeneity inside the regions of interest in the output image. It is important to emphasize that, traditionally, enhancement is accomplished by histogram transformation that preserves the entropy, as much as possible, to avoid artifacts generation [7]. However, although we do observe artifacts in the output, the results show that such artifacts can be easily removed by simple morphological operations or they are simply cut-off by a thresholding technique to be described next. Besides, we get a considerably improvement in the final segmentation. 2. PME and Image Processing In the last decade, Tsallis [10] has proposed the following generalized nonextensive entropic form: S q = 1, 1) where is normalization factor, p i is a probability distribution and q R is called the entropic index. This expression recovers the Shannon entropy in the limit q 1. The Tsallis entropy offers a new formalism in which the real parameter q quantifies the level of nonextensivity of a physical systems [10]. In particular a general PME has been considered to find out distributions to describe such systems. In this PME, the goal is to find the maximum of S q subjected to: p i = 1, 2) i=1 i=1 e ip q i = U q, 3) where U q is a nown application dependent value and e i represent the possible states of the system in our case, the gray-level intensities). Expression 2) just imposes that p i is probability and equation 3) is a generalized expectation value of the e i if q = 1 we get the usual mean vale). The proposed PME can be solved using Lagrange multipliers α and β: F p j = q pq 1 j + α+ ) qe j p q 1 j ) i=1 e ip q i ) qp q 1 j = 0 F q = 1 q) ) ln p i 1 ) + β = ) ) i=1 e ip q i ln p i 5) ) i=1 e ip q ) i ln p i = 0, 6) subject to the constraints given by expressions 2) and 3). The equation 6) gives: 1 q) P ) ln pi) 1 P ) q 1 P ) P i=1 eipq i ln pi) P i=1 eipq i ) P P 7) If we multiplying expression 5) by p j, and sum the result over the p j, then, a simple algebra gives: p 1 q j α = q p q j. 8) j=1 If we substitute this expression in 5) and multiply by we get: q + q p q j p 1 q j +, F = 1 ) i=1 pq ) i i=1 +α p i 1 +β e ip q i U q. i=1 4) + Therefore, we have to solve the following equations: j=1 ) ) qe j ) i=1 e ip q i q = 0, 9) e can tae off the factor q in all terms, multiply by q 1) and simplify the last term using the definition of U q in expression 3) to obtain: j=1 p q j p 1 q j +q 1)β e j U q ) = 0, 10) ). ln pi)

3 From this expression, we can isolate p 1 q j which gives: )) p 1 q j = 1 j=1 pq j ) 1 ) β e j U q 11) By using this equation and a normalization, in order to guarantee that the condition 2) is satisfied, we finally get: p j = [ 1 q 1) m=1 [1 q 1) β )] 1 ej U 1 q β P q )] 1 em U 1 q P q., 12) with β defined by equation 7). Expression 12) is hard to solve because the right-hand side of it depends also on the p j. However, if the right-hand side wors as a contraction map F F x) F y) α x y, with α [0, 1)) then, we can obtain a solution through a recursive procedure [2]: where: 1 = F 1 p n 1, p n 2,..., p n ), 2 = F 2 p n 1, p n 2,..., p n ),... 13) 1 = F 1 p n 1, p n 2,..., p n ), = F p n 1, p n 2,..., p n ), F j p n 1, p n 2,..., p n ) = e stop the iteraction when: D, p n) max { i [ 1 q 1) m=1 [1 q 1) β p n i )] 1 ej U 1 q β P q )] 1 em U 1 q P q 14), i =, 2,..., 256 } < δ, 15) for some pre-defined δ R, where p n = p n 1, p n 2,..., p n ). From the contraction map theory, it is nown that the scheme 13) will converge to the solution for any starting point p 0 [2]. Therefore, we tae an input image, compute its histogram, as well as the value U q from expression 3) and solve the expression 12) through the scheme defined by equation 13). This method has the advantage of been fast and eeping the property 0 p j 1. If the map F p 1, p 2,..., p ) = F 1 p 1, p 2,..., p ),..., F p 1, p 2,..., p )) is not a contraction we can not guarantee convergence. The form of expressions 14) maes it hard to prove such property. So, in the case study of section 3, we perform a experimental analysis by just applying the iterative scheme and, in the case of convergence, perform a proper histogram. modification of the input image. So, we transform the intensity values of the input such that the histogram of the output image matches the PME solution. The usual procedure to perform this tas wors as follows [5]. Suppose a random variable u 0 with probability density p 1 u) given by the histogram of the input image. The idea is to transform the variable u in another random variable v 0 such that its probability density p 2 v) is given by the solution of the PME. To perform this tas, it is just a matter of defining the random variables: u 0 p 1 x) dx = F 1 u), v 0 p 2 y) dy = F 2 v) 16) and impose that the value v must satisfies the constraint F 2 v) = F 1 u), which gives: v u) = F 1 2 F 1 u)). 17) The so obtained random variable v is a function of the variable u and its probability density is given by p 2 v), according to the definition of F 2 v) in expressions 16). Therefore it is the desired random variable which must be re-scaled to the range [0, 255] in order properly set the intensity range of the output image. Once performed the image enhancement, we can apply a segmentation technique. The whole pipeline is summarized on Figure 1. Figure 1. Pipeline for image enhancement and segmentation. In this paper, we apply the segmentation algorithm described in [4], also called q-entropic algorithm, which wors as follows. Suppose an image with gray-levels, with probability distribution P = {p i }. So, when applying a threshold t, we can consider two probability distribution from P, one for the foreground P A ) and another one for the bacground P B ). The partitioned image has an entropy denoted by S q A+B t) which depends on the Tsallis entropy, computed by expression 1), for the foreground and bacground separately see [4] for details). As usual in entropic based thresholding algorithms [8], the best t = t opt is

a) Histogram of the original image. Histogram obtained after 9 interactions of the scheme in expression 13), with δ = 10 8 and p 0 given by the original image histogram. Figure 2.

4 the one that maximizes the information measure entropy), that means: 3. Cases Study t opt = argmax[s q A+B t)]. 18) Let us tae the image in the Figure 2. This image was used in the reference [4] to demonstrate the q-entropic algorithm. e set q = 0.5 also used in [4]) and obtain U q = through expression 3). a) Figure 3. a) Histogram of the original image. Histogram obtained after 9 interactions of the scheme in expression 13), with δ = 10 8 and p 0 given by the original image histogram. Figure 2. Gray level image for case study. Now, we set p 0 as the entries of the original histogram, pictured on Figure 3.a, and apply the iterative scheme described by expression 13). The Figure 3.b shows the solution obtained after 9 interactions, with δ = The image intensity transformation defined by expression 17) gives the output pictured on Figure 4. Figure 5 shows the segmentation obtained by the q- entropic segmentation algorithm for both the original and the transformed images. e can observe that the object is better recovered in Figure 5.b than in Figure 5.a although a pattern of artifacts is highlighted in the bacground due to the entropy maximization [7]. Such artifacts can be easily removed by an opening morphological operation [5], as we can observe in Figure 6. In this example, we observe that the method stretches the image luminance such that the intensity patterns inside the objects become more homogeneous. Such behavior is also observed in Figure 7, with tests performed on odontological X-Ray images. For all the presented tests, the number of interactions of the scheme 13) was less than 15 few seconds in a Pentium IV), with δ = 10 8, which indicates a fast convergence rate. Also, the profiles of the obtained solutions were similar to Figure 3.b. The Figure 8 shows the effect of the transformation in the segmentation. This Figure was generated by applying Figure 4. Output image obtained after histogram transformation. the q-entropic algorithm expression 18)) over the images of Figure 7. e can observe an improvement in Figures 8.b,d. The result in Figure 8.f shows undersegmented objects teeth). Certainly, we could try to improve the segmentation of the original images by adjusting the q value. However, our aim in this discussion is to show that the histogram image) transformation proposed can improve the segmentation without concerning about the q value. However, we shall discuss the effect of this parameter for the transformed image. The sensitivity respect to q value can be analyzed in Figure 9. In this case, we notice that when increasing q from q = 0.5 to q = 0.75 we also increase the image luminance

a) a) Figure 5. a)thresholding result for Figure 2. Segmentation obtained after histogram transformation. c) d) e) f) Figure 6. Opening operation applied to image pictured on Figure 5.b. everywhere.

e must deeply consider these problem in further wors. 4.

It is based on the solution of a Principle of Maximum Entropy PME) and on the transformation of the spatial histogram of the input image to match the PME distribution.

However, more experiments must be performed in order to quantify the efficiency of the technique.

Tests for X-Ray odontological images, with q = 0.5: The fist collum Figures a),c),e)) shows the input images and the second one Figures,d),f)) pictures the corresponding outputs. References [1] S.

5 a) a) Figure 5. a)thresholding result for Figure 2. Segmentation obtained after histogram transformation. c) d) e) f) Figure 6. Opening operation applied to image pictured on Figure 5.b. everywhere. Such effect may reduce the contrast between the foreground and bacground which is an undesirable results, as we can observe in Figure 9 nearby the right-hand corner, in the bottom of both images. e must deeply consider these problem in further wors. 4. Conclusions and Future ors In this paper, we propose a histogram transformation technique which can be used to improve the homogeneity of the foreground in 2D images. It is based on the solution of a Principle of Maximum Entropy PME) and on the transformation of the spatial histogram of the input image to match the PME distribution. e present some results indicating that the technique can be used as a pre-processing one for thresholding methods. However, more experiments must be performed in order to quantify the efficiency of the technique. Further directions in this wor are theoretical analysis of the global map 14) as well as strategies to avoid the observed artifact in the output image. Figure 7. Tests for X-Ray odontological images, with q = 0.5: The fist collum Figures a),c),e)) shows the input images and the second one Figures,d),f)) pictures the corresponding outputs. References [1] S. S. Agaian, B. Silver, and K. A. Panetta. Transform coefficient histogram-based image enhancement algorithms using contrast entropy. IEEE Transactions on Image Processing, 163): , [2] D. P. Bertseas and J. N. Tsitsilis. Parallel and Distributed Computation: Numerical Methods. Prentice-Hall Intern. Editions, [3] H. D. Cheng, Y.-H. Cheng, and Y. Sun. A novel fuzzy entropy approach to image enhancement and thresholding. Signal Process., 753): , [4] M. P. de Albuquerque, I. A. Esquef, and A. R. G. Mello. Image thresholding using tsallis entropy. Pattern Recogn. Lett., 259): , [5] A. K. Jain. Fundamentals of Digital Image Processing. Prentice-Hall, Inc., [6] R. Malladi and J. A. Sethian. A unified approach to noise removal, image enhancement, and shape recovery. IEEE Trans. on Image Processing, 5: , 1996.

a) c) d) e) f) a) Figure 8. Segmentation by the q-entropic algorithm, with q = 0.5: a) Input image is Figure 7.a and t o pt = 115. Segmentation of Figure 7.b with t o pt = 224.

Ogorman and L. S. Brotman. Entropy-constant image enhancement by histogram transformation. In N. Chigier and G.. Stewart, editors, App.

Image Process., 412):233 260, 1988. [9] J. A. Star. Adaptive image contrast enhancement using generalizations of histogram equalization.

6 a) c) d) e) f) a) Figure 8. Segmentation by the q-entropic algorithm, with q = 0.5: a) Input image is Figure 7.a and t o pt = 115. Segmentation of Figure 7.b with t o pt = 224. c) Input image is Figure 7.c and t o pt = 72. d)input image: Figure 7.d and t o pt = 228. e) Thresholding of Figure 7.e and t o pt = 103. f)input image: Figure 7.f and t o pt = 209. [7] L. Ogorman and L. S. Brotman. Entropy-constant image enhancement by histogram transformation. In N. Chigier and G.. Stewart, editors, App. of digital image processing VIII, volume 573, pages , January [8] P. K. Sahoo, S. Soltani, A. K. ong, and Y. C. Chen. A survey of thresholding techniques. Comput. Vision Graph. Image Process., 412): , [9] J. A. Star. Adaptive image contrast enhancement using generalizations of histogram equalization. IEEE Transactions on Image Processing, 9: , [10] C. Tsallis. Nonextensive statistics: Theoretical, experimental and computational evidences and connections. BRAZ.J.PHYS., 29, c) Figure 9. Sensitivity against q value. a) Input image. Transformed image for q = 0.5. c) Output image for q = 0.75.

Image thresholding using Tsallis entropy

Pattern Recognition Letters 25 (2004) 1059 1065 www.elsevier.com/locate/patrec Image thresholding using Tsallis entropy M. Portes de Albuquerque a, *, I.A. Esquef b, A.R. Gesualdi Mello a, M. Portes de