Department of Computer Science, Aberystwyth University, UK. 2. Faculty of Information and Control Engineering,. Shenyang Jianzhu University, Liaoning, China.
Automatic Estimation of the Number of Segmentation Groups Based on MI Ziming Zeng1,2 , Wenhui Wang3,4 , Longzhi Yang1 , and Reyer Zwiggelaar1 1
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Department of Computer Science, Aberystwyth University, UK 2 Faculty of Information and Control Engineering, Shenyang Jianzhu University, Liaoning, China Network Information Center of the Sixth Affiliated Hospital of Sun Yat-sen University, Guangzhou, China 4 Key Laboratory of Medical Image Processing, Southern Medical University, Guangzhou, China
Abstract. Clustering is important in medical imaging segmentation. The number of segmentation groups is often needed as an initial condition, but is often unknown. We propose a method to estimate the number of segmentation groups based on mutual information, anisotropic diffusion model and class-adaptive Gauss-Markov random fields. Initially, anisotropic diffusion is used to decrease the image noise. Subsequently, the class-adaptive Gauss-Markov modeling and mutual information are used to determine the number of segmentation groups. This general formulation enables the method to easily adapt to various kinds of medical images and the associated acquisition artifacts. Experiments on simulated, and multi-model data demonstrate the advantages of the method over the current state-of-the-art approaches.
1
Introduction
Tissue segmentation in magnetic resonance (MR) images is an important application area in medical image analysis, for which clustering algorithms have been proposed [1]. A remaining problem is the robust estimation of the optimal number of segmentation groups. The Akaike information criteria (AIC) algorithm [2] is widely used, however, the theoretical possible and actual results do not always agree. Lv et al. proposed a measure based on the difference of mutual information for determining the number of clusters [3]. Wang [4] improved the iteration conditions, but this approach does not work well when noise levels are increased. To overcome the problem of the sensitivity to the presence of noise, we propose a more robust method to estimate the number of segmentation groups. Mutual information (MI) is a measure of statistical correlation of two random variables. It has been widely used in multi-model medical image registration [5]. Kim et al. and Rigau et al. used MI for image segmentation [6]. Lv et al. proposed a method based on the entropy difference of mutual information to determine the number of segmentation groups. Wang et al. improved the iteration condition and obtained improved results. The complete algorithm from Wang et al. is J. Vitri` a, J.M. Sanches, and M. Hern´ andez (Eds.): IbPRIA 2011, LNCS 6669, pp. 532–539, 2011. c Springer-Verlag Berlin Heidelberg 2011
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described in Fig. 1 where k denote the number of groups, kmeans() stand for the K−means algorithm, mutual inf ormation() denotes the mutual information algorithm, ∂M I denotes the entropy difference in mutual information, ε which is defined as 0.2 is the stopping criterion. Results based on synthetic and simulated data (see Section 4) indicate that this approach does not deal well with increased noise levels. Input: Original image. k = 2 Output: The number of segmentation groups = k − 1 (1) do (2) groups label = kmeans(Original image, k); (3) M I(k) = mutual inf ormation(Original image, groups label); (4) if k = 2 (5) ∂M I(k) = M I(k); ∂ 2 M I(k) = M I(k); (6) else (7) ∂M I(k) = M I(k) − M I(k − 1); (8) ∂ 2 M I(k) = abs(∂M I(k − 1) − ∂M I(k))/∂M I(k − 1); (9) end (10) k = k + 1; (11)while (∂ 2 M I(k) < ε) Fig. 1. Wang et al.’s algorithm [4]
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The Proposed Method
In order to enable the approach to handle certain level of noise, we propose a new approach to determine the number of segmentation groups by adopting Wang et al.’s approach. Noise removal, prior to estimating the number of segmentation groups, is expected to be beneficial. However, any such approach should preserve edge information. We propose an improved anisotropic diffusion approach to deal with such issues. Then, we use a class-adaptive Gauss Markov Random Field model, a clustering approach using both individual pixel and spacial information, to cluster. From this, a mutual information-based approach is utilized to judge what is the best number of iterations and segmentation groups. Our proposed method is summarized in Fig. 2. 2.1
Noise Removing
In this step, we use anisotropic diffusion to decrease noise. It is expected this can deal with additive noise and retain edges. The traditional nonlinear diffusion filtering method was proposed by Perona and Malik [7], which is described by ∂I ∂t = div c (| ∇I |) · ∇I (1) I (t = 0)=I0 where ∇ is the gradient operator, div() is the divergence operator,|| denotes the magnitude, c (x) denotes the diffusion coefficient, and I0 is the initial image. The diffusion coefficient which was improved in [8] is defined as
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Fig. 2. Schematic representation of the developed approach
c (x) =
1 2 ln e + (x/κ)
(2)
where κ is an edge magnitude parameter. Under the control of c (| ∇I |), the model can achieve a selective smooth diffusion from the original image based on the gradient. At the edge the gradient magnitude is large and c (| ∇I |) is small and the model will perform hardly any smoothing in order to keep the edge details. When the gradient magnitude is small at a flat area, c (| ∇I |) is large and noise is removed. In addition, we have included an automatic stopping criteria in the developed approach. This is based on estimating the difference in mutual information (M I) in successive steps and stopping the diffusion when this difference (∂M I) is smaller than an empirically determined threshold value (δ). The traditional anisotropic diffusion only considers 4 directional gradient (up-down-left-right). In our work we extend the anisotropic diffusion gradient direction to eight directions to calculate the pixel gray level. 2.2
Estimation of the Number of Segmentation Groups
In the segmentation step, a class-adaptive Gaussian Markov Random Field is used. The class-adaptive penalty factor β which is anisotropic for each group is automatically calculated from the posterior probability. By incorporating both a Markov random field model and expectation-maximization into a GMRF-EM framework, we can achieve accurate and robust segmentation results. Let L={1,2,· · ·,l} denote the set of group labels. Let D={1,2,· · ·,d} denote the grey level set. Let S={1,2,· · ·,N} denote the set of indexes which contain N pixels. Image segmentation is achieved by assigning a value from L to each pixel in the image. We define the label assignment X={x = (x1 , · · · , xi , · · · , xN ) | xi ∈ L, i ∈ S} to all sites as a realization of a family of random variables defined on S.
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We also define an observation field Y={y = (y1 , · · · , yi , · · · , yN ) | yi ∈ D, i ∈ S} According to the Hammersley-Clifford theorem [6]: the prior probability P(x) satisfies a Gibbs distribution: P (X) = Z −1 exp (−U (X)) where Z is a normalisation factor. The potential function is defined as U(x) = β c∈C (Vc (x)) where C = {(i, i ) | i ∈ N, i ∈ S},c ∈ C. MRF multi-level logistic (MLL) is introduced as the prior probability for which −1 if xi = xj , j ∈ Ni (3) Vc (xi ) = 1 else where Ni denotes the eight neighborhood pixels of the ith pixel. According to [9] the penalty factor β of the ith pixel is defined as (pim − pjm )2 i∈S j∈Nd i (4) β= N/k where t denotes the number of iterations, m denotes the group, d denotes the number of directions (e.g. horizontal, vertical and two diagonal directions). pim denotes the posterior probability of pixel i. pjm denotes the posterior probability of pixel j in the neighborhood i on the direction of d. Assuming that the distribution of intensities yi follows a Gaussian distribution with parameters θl = (μl , σl ), μl ∈ μxi , σl ∈ σxi , given the group label xi = l. 1 (yi − μl )2 p (yi | xi ) = g (yi ; θl ) = exp − (5) 2σl2 2πσl2 Based on the conditional independence assumption, the joint likelihood probability is given by
1 (yi − μxi )2 − log(σ ) (6) P (Y | X) = Πi∈S p(yi | xi ) = Πi∈S √ exp − x i 2σx2i 2π
which can be written as P (Y | X) = Z −1 exp − U (Y | X) , with the likelihood energy (yi − μx )2 i U (Y | X) = U (yi | xi ) = + log(σ ) (7) xi 2σx2i i∈S
i∈S
We seek a labeling x of an image, which is an estimate of the true labeling. According to the MAP criterion: x = argmaxx∈X {P (Y | X)P (X)}. The MAP estimation is equivalent to minimizing the posterior energy function x = argminx∈X {U (Y | X) + U (X)}. Application of the EM algorithm to solve the GMRF model results in (see [10] for details) (t) (l | yi ) yi (t+1) i∈S P μl = (8) (t) (l | y ) P i i∈S
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(t+1) σl
2 =
P (t) (l | yi ) =
P(t) (l | yi ) (yi − μl )2 (t) (l | y ) i i∈S P
i∈S
g (t) (yi ; θl ) P (t) (l | xNi ) p (yi )
(9) (10)
where t denotesthe iteration number. + U (t) (X)}, is an estimate of X. The Let U ∗(t) = i∈S {U (t) (Y | X) where X termination criterion of the EM algorithm is | (U ∗(t+1) − U ∗(t) )/(U ∗(t) ) |≤ ψ. The GMRF-EM approach can be repeated depending on the number of segmentation groups/labels. For each we determine the mutual information and derive ∂M I(k) = M I(k) − M I(k − 1) and ∂ 2 M I(k) = ∂M I(k) − ∂M I(k − 1). For these first and second order derivative we introduce thresholds ε1 and ε2 respectively, which can be used as stopping criteria and will result in an estimation for the number of segmentation groups/labels. We use the two thresholds as ∂M I(k + 1) ≤ ε1 and ∂ 2 M I(k + 2) ≤ ε2 . The final segmentation result by using the GMRF-EM approach is the segmentation after k iterations.
3
Experiment and Discussion
We set κ in Eq. 2 equal to 20, the termination criterion of the EM algorithm ψ = 0.001. We create a synthetic images with Gaussian noise from 1% to 20% to estimate model parameters: resulting in δ = 0.1, ε1 = 0.07, ε2 = 0.02. We used two types of images as shown in Fig. 3. According to the termination condition (∂M I < δ), the number of iterations are calculated as 10 and 13, respectively. From 2 to 5 groups, we can see the mutual information increasing rapidly, but after 5 groups this is reduced which denotes the number of groups are equal to five. According to our termination criterions (∂M I(k + 1) ≤ ε1 and ∂ 2 M I(k + 2) ≤ ε2 ), we can easily determine that the number of segmentation groups are five for both images which is the same as the ground truth, but the numbers of groups determined by the approach of Wang et al. are 6 and 7, respectively. We did run the experiment at variable noise levels and from 1%19% noise the developed method produced the correct number of groups/labels. With increased noise levels the method will over or under estimate the number of groups/labels. Our method was also tested on simulated T1 and T2 MR data from Brain Web [11]. In Fig. 4 non-brain tissue regions have been removed using a standard approach [12]. According to our termination criteria, the anisotropic diffusion iteration numbers are determined as 1 and 2 when using an anisotropic diffusion model and mutual information. In the second graphs the ∂M I curves on T1 and T2 are 0.0404 and -0.0308 at five groups, respectively, the ∂ 2 M I curves on T1 and T2 are 0.0089 and -0.0665 at six groups, respectively. According to our termination criterion, we can automatically determine both of the MR images can be segmented into 4 groups which are the same as the ground truth. We also compared our method with the method proposed by Wang et al., for which the
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Fig. 3. Experimental results on synthetic images. The first and third rows show from left to right the original images, the original images with 15% Gaussian noise, the images after using the anisotropic diffusion model, the segmented images using our method, and segmentation results based on the GMRF-EM approach only. In the second and fourth rows, the first graphs show the mutual information (M I) curves of the filter result by using the anisotropic diffusion model for 20 interations, and the difference of mutual information curves (∂M I). The second graphs are the mutual information curves of numbers of segmentation groups and its derivative curves based on the proposed approach, while the third graphs show the equivalent based on the approach of Wang et al. [4].
results show the number of segmentation groups are determined as 6 groups for T1 and 3 groups for T2. We also segmented the images using only the GMRF-EM approach. Note that the proposed approach has also been applied to some other images from the Brain Web and other sources, but the detailed results are not given here due to space limit. Nevertheless, the results showed that our approach can
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Fig. 4. Experimental results on simulated MR data. The first (T1) and third (T2) rows show from left to right the original images with 9% noise and 40% intensity inhomogeneity, the ground truth images, the segmented images using our method, and segmentation results based on the GMRF-EM approach only. The graphs in the second and fourth rows are arranged in the same manner as those in Fig. 3.
effectively determine the number of segmentation groups by comparing them with the ground truth, which are more promising than those of Wang’s approach. In addition, as a side effect but importantly, the segmentation results obtained by our approach are better than those obtained by only using the GMRF-EM approach.
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Conclusion
In this paper, we have proposed a method which automatically determines the number of segmentation groups/labels. This is achieved by initially using an anisotropic diffusion model for noise removal which automatically estimate the number of iteration by mutual information. Subsequently, mutual information and class-adaptive Gauss-Markov modeling are used to determine the number of groups/labels. We tested our method on simulated and multi-modal images. The evaluation showed that our method based on the difference of mutual information can effectively and automatically determine the number of segmentation groups. In addition, segmentation results are improved and the methodology leads to unsupervised automatic segmentation.
References 1. Bankman, I.: Handbook of Medical Image Processing and Analysis, pp. 71–258 (2008) ISBN 978-0-12-373904-9 2. Akaike, H.: A new look at the statistical model identification. IEEE Transactions on Automatic Control 19(6), 716–723 (1974) 3. Lv, Q.W., Chen, W.F.: Image segmentation based on mutual information. Chinese Journal of Computers 29(2), 296–301 (2006) 4. Wang, W.H., Feng, Q.J., Chen, W.F.: Segmentation of brain MR images based on the measurement of difference of mutual information and Gauss-Markov random field model. Journal of Computer Research and Development 46(3), 521–527 (2009) 5. Maes, F., Collignon, A.: Multimodality image registration by maximization of mutual information. IEEE Transactions on Medical Imaging 16(2), 187–198 (1997) 6. Kim, J., Fisher, J.W., Cetin, M., Yezzi, A., Willsky, A.S.: Incorporating complex statistical information in active contour-based image segmentation. In: International Conference on Image Processing, pp. 655–658 (2003) 7. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12(7), 529–539 (1990) 8. Tong, C., Wang, S.T., Man, L.H.: Improved image denoising method based on PDE. Journal of Computer Engineering and Applications 46(15), 176–179 (2010) 9. Wang, W.H., Feng, Q.J., Liu, L., Chen, W.F.: Segmentation of brain MR images through class-adaptive Gauss-Markov random field model and the EM algorithm. Journal of Image and Graphics 13(3), 488–493 (2008) 10. Markov, S.Z.L.: Random Field Modeling in Image Analysis, pp. 19–21. Springer, Heidelberg (2001) 11. Brain Web. Montreal Neurological Institute, McGill University (2006), http://www.bic.mni.mcgill.ca/brainweb/ 12. Zhuang, A.H., Valentino, D.J., Toga, A.W.: Skull-stripping magnetic resonance brain images using a model-based level set. NeuroImage 32(1), 79–92 (2006)
