Weighted Entropy-based Measure for Image Segmentation - Core

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Procedia Engineering 41 (2012) 1261 – 1267

International Symposium on Robotics and Intelligent Sensors 2012 (IRIS 2012)

Weighted Entropy-based Measure for Image Segmentation Weng Kin Laia*, Imran M. Khanb, Geong Sen Pohc a School of Technology, TAR College, 53300 Kuala Lumpur, Malaysia Dept. of Electrical and Computer Engineering, IIUM, 53100 Kuala Lumpur, Malaysia c MIMOS Bhd., Technology Park Malaysia, 57000 Kuala Lumpur, Malaysia

b

Abstract Image segmentation is one of the fundamental and important steps that is needed to prepare an image for further processing in many computer vision applications. Over the last few decades, many image segmentation methods have been proposed, as accurate image segmentation is vitally important for many image, video and computer vision applications. A common approach is to look at the grey level colours of the image to perform multi-level-thresholding. The ability to quantify and compare the resulting segmented images is of vital importance even though it can be a major challenge. One measure used here computes the total distances of the pixels to its centroid for each region to provide a quantifiable measure of the segmented images. We also suggest an improved Zhang’s entropy measure for image segmentation based on computing the entropy of the image and segmented regions. In this paper, we will present the results from both of these approaches.

© 2012 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Centre of Humanoid Robots and Bio-Sensor (HuRoBs), Faculty of Mechanical Engineering, Universiti Teknologi MARA.

Open access under CC BY-NC-ND license.

Keywords: Segmentation, objective evaluation, entropy, information theory, minimum description length.

1. Introduction An image contains a lot of information and it has been said that “A picture is worth a thousand words”[1] And as computer power improves and their prices drop, the use of computers to automatically process and analyse digital images becomes increasingly attractive. Nevertheless, there are several fundamental steps involved in processing these images and one very important one is image segmentation. Segmentation is an essential stage in image analysis since it conditions the quality of the interpretation. This processing either consists in partitioning an image into several regions or in detecting their frontiers. The classical hypothesis is that a good segmentation result guarantees a correct interpretation. This hypothesis makes sense clearly when the grey-level of each pixel is related to the interpretation task. For example, if we consider satellite images, the localization of the different types of vegetation in the image can be achieved with a segmentation method. In this case, the relation between the segmentation and the interpretation is very close. However, much more complicated situations can be encountered. If we have an indoor image containing some objects we want to identify, a good segmentation result will determine the frontier of each object in the image. In this case, a region containing an object is not characterized by a grey-level homogeneity and the level of precision of a segmentation result affects the understanding of the image. Many segmentation methods have been proposed in the literature in the last few years [2],[3],[4],[5],[6],[7]. A major problem in segmentation is the diversity in the types of regions composing an image. Indeed, an image can be composed of

* Corresponding author. Tel.:+6 03 4145 0123; fax: +6 03 4142 3166. E-mail address: [email protected]

1877-7058 © 2012 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.07.309

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uniform, textured or degraded regions. Few segmentation methods provide good results for each type of region. While we may be able to judge the quality of the segmentation by inspection, nonetheless this visual evaluation is still subjective. Thus, the comparison of different segmentation methods is not an easy task. To overcome this, some techniques have been proposed to facilitate the visual evaluation of a segmentation result by using a coloured representation. Furthermore, different metrics have been proposed to quantify the quality of a segmentation result. In order to make an objective comparison of different segmentation methods or results, some appropriate evaluation criteria must be defined. Ideally it should not contradict but support the results from visual inspection, - especially when it comes to very obvious improvements or deteriorations of the segmentation results. The problem of segmentation is well-studied and there are a variety of approaches that may be used. Nevertheless, different approaches are suited to different types of images and the quality of output of a particular algorithm is difficult to measure quantitatively due to the fact that there may be many “correct” segmentations for a single image. We treat image segmentation as a data clustering problem and propose an improved population based stochastic optimization, namely particle swarm optimization (PSO), to compute the optimal clusters[8]. The experimental results indicate that the proposed PSO is capable of delivering improved segments as compared to the regular PSO approach. The rest of the paper is organized as follows. Section 2 reviews prior work on segmentation measures image segmentation with PSO while section 3 analyses the entropy-based segmentation measure when applied to grey level segmentation. Section 4 introduces the experimental setup and discusses the results. Finally in section 5, we put forward some conclusions of this preliminary study and recommend some areas for further work. 2. Prior Work Accurate image segmentation is important for many image, video and computer vision applications. Many image segmentation methods have been proposed over the years. Common approaches involve using the grey level colours of the image to perform multi-level-thresholding. Some of these techniques use only the grey level histogram, some use spatial details while others use fuzzy set theoretic approaches. Still others have adopted a different approach that involves performing bi-level thresholding in which only the foreground and background images are considered. Liping et al [9] proposed an approach based on a 2D entropy to segment the image based on the thresholds identified. And in recent years, segmentation methods with PSO have also been investigated. Now, the segmentation problem is viewed as an optimization problem instead. Particle Swarm Optimization (PSO) was originally modelled by Kennedy and Eberhart in 1995 after they were inspired by the social behavior of a bird flock. PSO had been used to solve a myriad of various problems. Kiran et al [10] investigated using PSO for Human Posture recognition with good recognition rates for many of the different postures. Mahamed G. H. Omran first used Particle Swarm Optimization to perform image segmentation where the problem was modelled as a data clustering problem[11]. CC Lai [12] adopted a different approach in using the PSO approach for image segmentation. He used the PSO to compute the parameters for the curve that had been determined to give the best fit for the given image histogram. Subsequently, this was used to identify the appropriate thresholds for the various segments of the given image. Kaiping et al [13] proposed an effective threshold selection method of image segmentation which was then refined with the PSO embedded into a two-dimensional Otsu algorithm[5]. Meanwhile, Tang, Wu, Han and Wang [14] adopted a different approach in using PSO whereby they used it to search for the regions that returned the maximum entropy. In our approach, we adopted the PSO to firstly identify the best thresholds and subsequently refined with pivotal alignment to optimize the regions. Nevertheless, an equally important area of research is in finding an accurate measure of the segmentation results. Without knowing whether the segmentation gives good results objectively, it will really be very difficult to come up with better methods. Briefly stated, there are two main approaches. Firstly, there those which are based on supervised evaluation criteria. These criteria generally compute a global dissimilarity measure between the ground truth and the segmentation result. They need two components. The first one is a ground truth corresponding to the best and expected segmentation result. In the case of synthetic images, this ground truth is known. In other cases (natural images), an expert can manually define this ground truth[15]. Even if these images are more realistic, one problem concerns the objectivity and variability of experts. The second component is the definition of a dissimilarity measure between the obtained segmentation result and the ground truth. In this case, the quality of a segmentation result depends on the correct classification rate of detected objects in the image[16]. This type of approach is based on local processing and is dedicated to a given application. On the other hand, there are unsupervised evaluation criteria based on information-theoretic approaches that enable the quantification quality of a segmentation result without any a priori knowledge. Liu and Yang [17]proposed an evaluation function that is based on empirical studies. However, their approach suffers from the fact that unless the image has very well defined regions with very little variation in luminance and chrominance, the resultant segmentation score would tend to lean towards results with very few segments. Subsequently, an improved approach was proposed using a modified

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quantitative evaluation with a new measure[18] to minimise such undesirable effects. Nevertheless, both these approaches were criticized for their lack of theoretical grounding of information theory as they were merely based on empirical analysis. Zhang et al [19] then came up with a measure that combines two measures - expected region entropy Here and layout entropy Hlay. This will be discussed in more details in the next section. 3. Entropy-based Measure In our approach, an image is assumed to have Lu grey levels [1, ...., Lu] spatially distributed across the image. Nonetheless, the individual distinctive grey levels can be summed up to form a histogram h(g), as illustrated in Fig. 1. The x-axis represents the luminance Lu whereas the y-axis shows the number of pixels for each of the luminance levels.

Fig 1. A histogram with 9 luminance values

Zhang’s entropy works on a segmented image I with n regions, where Vj is the set of all possible values for the luminance in region j while Lm ( R j ) represents the number of pixels in region Rj that have luminance of m, the entropy Q, i.e.,

Q = Hlay ( I ) + Here ( I ) n

L( R j )

j

SI

H lay( I ) = −¦ n

H ere ( I ) =

¦ j

log

(1) L( R j )

(2)

SI

§ · L( R j ) ¨ L j ( m) L j ( m) ¸ log ¨− ¸ SI ¨ Sj Sj ¸ v m ∈ V j © ¹

¦

(3)

SI defines the total number of pixels for image I. One measures the lack of uniformity whereas the other calculates the layout of the segmentation itself. This is a good approach as it maximizes the uniformity of the pixels in each segmented region with Here and maximizing the differences between different regions with Hlay. When the segmented region is uniform, the entropy Here would be small. On the other hand, such a development would result in an increase in the layout entropy Hlay instead. Noting that , L( R j ) (4) log = log L( R j ) − log S I SI Hence, for n-segments, L( R1 ) H lay( I ) = (log S I − log L( R1 ) ) + L( R2 ) (log S I − log L( R2 ) ) + ............ + L( Rn ) (log S I − log L( Rn ) ) SI SI SI However, L(R1)+L(R2) + L(R3) + ….. L(Rn) = = S1 + S2 + S3 + …Sn = SI.

(5)

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Similarly, for n-segments, the expected region error is, H ere ( I ) =

L ( R1 ) SI

(

)

(

ª L11 ( m) Lm1 ( m) L2 ( m) m log S1 − log L11 (m) + 1 « log S1 − log L12 (m) + ....... 1 log S1 − log L1 1 ( m) S1 S1 «¬ S1 1

(

)

mn L( R n ) ª L n ( m) º log S n − log L1n (m) + ...... Ln (m) log S n − log Lmnn (m) » « S I ¬« S n Sn ¼

(

)

(

)]

)

(6)

where mj is the total number of all possible luminance for region Rj. 1.1 Different number of segments To illustrate the layout entropy, we will compute its value for n = 3,

S1 S S S S S log 1 − 2 log 2 − 3 log 3 SI SI SI SI SI SI

3 H lay (I ) = −

§S · §S · §S S S · S = −¨¨ 1 log S1 − 1 log S I ¸¸ − ¨¨ 2 log S 2 − 2 log S I ¸¸ − ¨¨ 3 log S3 − 3 log S I ¸¸ S S S S S S I I I © I ¹ © I ¹ © I ¹

(7)

Rearranging,

S S S S S S1 log S1 − 2 log S 2 − 3 log S3 + 1 log S I + 2 log S I + 3 log S I SI SI SI SI SI SI

3 (I ) = − H lay

(8)

Because we are using the luminance of the pixel as a feature for the segmentation, therefore for n-segments,

S1 + S 2 + S3 + ......Sn = S I

(9)

Hence,

S1 S S (10) log S1 − 2 log S 2 − 3 log S3 + log S I SI SI SI Assuming region 1, 2 and 3 have m1, m2 and m3 luminance values respectively, then the expected region entropy would be, 3 H lay (I ) = −

3 (I ) = − H ere

−

m1 m1 · S1 § L11 (m) L1 (m) L12 (m) L2 (m) ¨¨ log 1 + log 1 + ..... ........ + L1 (m) log L1 (m) ¸¸ − .......... S S S I © S1 S1 S1 S1 1 1 ¹

m m L 3 ( m) L 3 ( m) ·¸ S 3 §¨ L13 ( m) L1 (m) log 3 log 3 + ..... + 3 S I ¨ S3 S3 S3 S3 ¸ © ¹

(11)

i.e. 3 H ere (I ) = −

· S1 § L11 (m) L2 (m) Lm1 (m) L1 (m) Lm1 (m) ¨¨ log L11 (m) + 1 log L12 (m) + .....+ 1 log L1m1 (m) − 1 log S1 − 1 log S1 ¸¸ S I © S1 S1 S1 S1 S1 ¹ ..... −

S3 SI

m m · L3 3 (m) § L13 ( m) L13 ( m) L23 ( m) L 3 ( m) 3 ¨ S S log log .. log S 3 ¸ − − − log L13 (m)........ + 3 log Lm ( m ) 3 3 3 ¸ ¨ S3 S3 S3 S3 S3 © ¹

(12)

mj

For segment j,

¦ L ( m) = S k j

j

(13)

k

Hence, mj

¦ k

Lkj (m) Sj

log S j = log S j

(14)

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Therefore, · S § L1 ( m) · Lm1 ( m) Lm2 ( m) S1 §¨ L11 ( m) ¸ 2 log L11 ( m) + .... 1 log L1m1 (m) ¸ − 2 ¨ 2 log L12 (m) + .... 2 log Lm 2 ( m) ¸ ¨ ¸ ¨ SI S1 SI S2 S1 S2 © ¹ © ¹ m3 · L m ( ) S S S S 3 §¨ L13 ( m) m − log L13 ( m) + .... 3 log L3 3 (m) ¸ + 1 log S1 + 2 log S 2 + 3 log S3 ¸ S I ¨ S3 S3 SI SI ¹ SI ©

3 H ere (I ) = −

· § L1 ( m) · § L1 ( m) Lm1 ( m) Lm2 ( m) m m = −¨ 1 log L11 (m) + .... 1 log L1 1 ( m) ¸ − ¨ 2 log L12 ( m) + .... 2 log L2 2 ( m) ¸ ¸ ¨ SI ¸ ¨ SI SI SI ¹ © ¹ © m3 · § L1 ( m) L ( m) S S S m −¨ 3 log L13 (m) + .... 3 log L3 3 ( m) ¸ + 1 log S1 + 2 log S 2 + 3 log S3 ¸ ¨ S SI S S S I I I I ¹ ©

Furthermore,

(15)

(16)

Lkp (m) = Lkq (m) = Lk (m) , i.e. the luminance for the same image is independent of the segments. Hence

for an image I, the total luminance is Lu i.e. m m · § L1 (m) · § L1 ( m) · § L1 ( m) Lm3 ( m) L 1 ( m) L 2 ( m) m m m ¨ 1 log L11 (m) + .... 1 log L1 1 ( m) ¸ + ¨ 2 log L12 ( m) + .... 2 log L2 2 ( m) ¸ + ¨ 3 log L13 ( m) + .... 3 log L3 3 ( m) ¸ ¸ ¨ ¸ ¨ SI ¸ ¨ SI SI SI SI SI © ¹ © ¹ © ¹ Lu

=

¦ k

Lk (m) log Lk (m) SI

(17)

Therefore, Lu

3 H ere (I ) = −

¦ k

Lk ( m) log Lk ( m) SI

+

S S S1 log S1 + 2 log S 2 + 3 log S3 SI SI SI

(18)

2 2 Similarly, we can obtain both H lay ( I ) and H ere ( I ) for n = 2. As a result, the difference between the entropies for n = 3

and n = 2 would then be,

(

3 3 2 2 H lay ( I ) + H ere ( I ) − H lay ( I ) + H ere (I )

=−

Lu k S S S S S S1 L ( m) log Lk ( m) log S1 − 2 log S 2 − 3 log S 3 + log S I + 1 log S1 + 2 log S 2 + 3 log S3 − ¦ SI SI SI SI SI SI SI k

+ where

)

Sˆ Sˆ Sˆ Sˆ1 log Sˆ1 + 2 log Sˆ2 − log S I − 1 log Sˆ1 − 2 log Sˆ2 SI SI SI SI

Lu

+

¦ k

Lk (m) log Lk (m) SI

=0

Sˆ represents the segments when n = 2.

1.2 Same number of segments We will be using the previous example to illustrate the change in the entropy values for n=3. Assume that the  is, segmentation with the same image now results in a new set of regions where the new layout entropy H lay

    3 ( I ) = − S1 log S − S 2 log S − S 3 log S + log S H lay 1 2 3 I SI SI SI

(19)

Similarly, substituting Eq(19) with the new segmentation, Lu

 3 ( I ) = − H ere

¦ k

S Lk ( m) S S log Lk ( m) + 1 log S1 + 2 log S2 + 3 log S3 SI SI SI SI

Therefore, the difference between the entropies for n = 3 is,  3 ( I ) + H  3 ( I ) − H 3 ( I ) + H 3 ( I ) H lay ere lay ere

(

)

Lu k S S S S S S L ( m) = − 1 log S1 − 2 log S2 − 3 log S3 + log S I + 1 log S1 + 2 log S2 + 3 log S3 − log Lk ( m) SI SI SI SI SI SI SI k Lu k S S1 S S S S L (m) log S1 − 2 log S 2 − 3 log S 3 + + 1 log S1 + 2 log S 2 + 3 log S 3 − log S I − log Lk ( m ) SI SI SI SI SI SI SI k

¦

¦

=0

(20)

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Hence it may be clear that a more effective measure would be a weighted entropy of the segmented results, viz. where

Q = w1Hlay ( I ) + w2 H ere ( I ) w1 ≠ w2 .

(21)

4. Particle Swarm Optimization The PSO segmentation algorithm investigated here which mimics the social behaviour of a flock of birds can be summarised into two main stages; a global search stage with pivotal alignment as a local refining stage. Within the PSO context, a swarm refers to a set of particles or solutions which could optimize the problem of clustering data. In a typical PSO system, the set of particles will fly through the search space with the aim of finding the particle position which results in the best evaluation of the given fitness function. Each particle will have its own memory which is effectively the local best (pbest) and the best fitness value out of all the pbest which is then labelled as the global best (gbest). gbest will be the reference point where the other particles will strive to achieve while searching for better solutions. The PSO can be represented by the following two equations: where

(

(

)) (

(

old old + C2 + rand2 ∗ gbest − X id Vidnew = W ∗ Vidold + C1 + rand1 ∗ pbest − X id

))

(22)

old X idnew = X id + Vidnew (23) Vid represents the velocity which is involved in the update of the movement and magnitude of the particles, Xid which represents the new position of the particles after updating, W which denotes the inertia weight, C1 and C2 are acceleration coefficients while rand1 and rand2 are random values that varies from 0 to 1. These parameters provide the necessary diversity to the particle swarm by changing the momentum of particles to avoid the stagnation of particles at the local optima. Nevertheless, PSO can still get stuck in local minima or it may take a longer time to converge to optima. Thus, we propose an additional stage involving pivotal alignment [20] to improve the local optimum results. The algorithm for the PSO with pivotal alignment is shown in fig.2.

(a) Select initial set of thresholds for n segments randomly. (b) Repeat the following until gbestt+1 does not change i) ii)

Compute new thresholds with PSO If gbestt +1 < gbestt then gbestt+1 = gbestt

(c) Pivotal Alignment -

identify centroids of current segments, for each pixel, compute the distance to centroid of current and the neighboring regions, realign the pixel to nearest region.

Fig. 2. PSO with Pivotal Alignment for image segmentation

The thresholds for the n-segments, Bi are randomly selected but ensuring that there will not be any overlaps with its neighbours, i.e. Bi = Bi+rand(b1,b2) (24) L −L rand(b1,b2) are randomly generated numbers on the interval [b1, b2]with a uniform distribution , b1 = and b2 = 2n . 2n

5. Results Firstly, we used the PSO to segment the images represented by the Brodatz suite of homogeneous images. The Brodatz suite of images is a well-known benchmark database for evaluating texture recognition algorithms[21] and the set of homogeneous Brodatz images used for testing the improved PSO is shown in Fig. 3. A homogeneous set of images allows for easy visual verification of the segmentation. Each image consists of 370 × 370 pixels and the number of distinctive regions are either 3 or 7.

Img #3 (3 segments)

Img #7 (7 segments)

Fig. 3. Colour images from the Brodatz suite

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6. Conclusions We have demonstrated via our preliminary set of experiments on homogeneous images, the effectiveness of an improved PSO with Pivotal Alignment for image segmentation. We have also compared the segmentation results with 2 different approaches. The information theoretic based measure computes a combination of expected region entropy Here and layout entropy Hlay. Detailed analysis showed that it would be more effective if the contribution from the layout portion is lesser than that from the expected region error portion. Furthermore, all two measures were also able to capture the improvements of the PSOwPA. One immediate area for further investigation is to explore PSO with Pivotal Alignment for non-homogeneous images. Nevertheless, the ability to accurately quantify the amount of improvements can be another challenge even though some of the measures investigated here may be used.

Acknowledgments The authors would like to thank MW Lim, SL Tang and CS Chan for their helpful comments and suggestions in the work reported here.

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[11] [12] [13] [14] [15]

[16] [17] [18] [19] [20] [21]

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