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Int. J. Medical Engineering and Informatics, Vol. 3, No. 3, 2011
Tissue probability map constrained 4D clustering algorithm for increased accuracy and robustness in SerialMR brain image segmentation Zhong Xue* Biomedical Image Computing Lab, Department of Systems Medicine and Bioengineering, The Methodist Hospital Research Institute, The Methodist Hospital, Weill Cornell Medical College, Houston, TX, USA E-mail: [email protected] *Corresponding author
Dinggang Shen Department of Radiology and BRIC, University of North Carolina, Chapel Hill, NC, USA E-mail: [email protected]
Hai Li Biomedical Image Computing Lab, Department of Systems Medicine and Bioengineering, The Methodist Hospital Research Institute, The Methodist Hospital, Weill Cornell Medical College, Houston, TX, USA E-mail: [email protected]
Stephen T.C. Wong Department of Systems Medicine and Bioengineering, The Methodist Hospital Research Institute, Department of Radiology, The Methodist Hospital, Weill Cornell Medical College, Houston, TX, USA E-mail: [email protected]
Copyright © 2011 Inderscience Enterprises Ltd.
Tissue probability map constrained 4D clustering algorithm Abstract: The traditional fuzzy clustering algorithms might be biased due to the variability of tissue intensities and anatomical structures: clustering-based algorithms tend to over-segment white matter tissues of MR brain images. To solve this problem, we propose a tissue probability map constrained clustering algorithm and apply it to series of 3D MR brain images of the same subject at different time points for longitudinal study. The tissue probability maps consist of segmentation priors obtained from a population and reflect the probability of different tissue types. Using the proposed algorithm in the framework of the CLASSIC algorithm, which iteratively segments a series of images and estimates their longitudinal deformations, more accurate and robust longitudinal measures can be achieved. Experimental results using both simulated and real data confirmed the advantages of the proposed algorithm in longitudinal follow up studies of MR brain imaging with subtle morphological changes for neurological disorders. Keywords: image segmentation; longitudinal image processing; MR brain images. Reference to this paper should be made as follows: Xue, Z., Shen, D., Li, H. and Wong, S.T.C. (2011) ‘Tissue probability map constrained 4D clustering algorithm for increased accuracy and robustness in SerialMR brain image segmentation’, Int. J. Medical Engineering and Informatics, Vol. 3, No. 3, pp.286–298. Biographical notes: Zhong Xue is an Assistant Professor with Weill Cornell Medical College and the Director of Medical Image Analysis Lab. His research interests include medical image computing, software and algorithms for image guided diagnosis and therapy. Dinggang Shen is Associate Professor of Radiology, Computer Science, and Biomedical Engineering and Director of IDEA Lab and the medical image analysis core, University of North Carolina at Chapel Hill. His research interests include medical image analysis, computer vision, and pattern recognition. He serves as an editorial board member for four international journals. Hai Li is currently Research Fellow in The Center for Bioengineering and Informatics, TMHRI, and PhD candidate in Northwestern Polytechnical University. His research interests include biomedical image analysis and DTI image processing. Stephen T.C. Wong is Director of the Ting Tsung and Wei Fong Chao Center for Bioinformatics Research and Imaging, Director of The Center for Bioengineering and Informatics, John S. Dunn Distinguished Endowed Chair in Biomedical Engineering, Vice Chair and Chief of Medical Physics, Department of Radiology, The Methodist Hospital, and Professor of Radiology, Neurology, and Neurosciences, Weill Cornell Medical College. His research focuses on new approaches of combining bioinformatics, biophysics, and imaging for developing new biomarkers, drug cocktails, and on-the-spot diagnosis and therapy in neurological disorders and cancer.
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Introduction
Image segmentation plays an important role in quantitative analysis of MR brain images of many medical imaging applications, such as automatic tissue labelling and quantification, morphometry, image registration, and image-guided surgery (Bezdek et al., 1993; Pappas, 1992; Udupa and Samarasekera, 1996; Lim and Prefferbaum, 1989; Pham and Prince, 1999; Chen and Giger, 2004). In follow up studies, a series of 3D images of the same subject are usually captured at different time points, and the focus is to quantitatively measure the morphological changes of the whole brain or a specific region (Resnick et al., 2000; Tang et al., 2001; Freeborough and Fox, 1997) across with time. Therefore, not only spatial information needs to be effectively utilised to generate spatially smooth segmentation results (Wang et al., 2009; Colliot et al., 2006), but also longitudinal stability is critical, since the temporal changes in these applications are usually subtle and the true signal of morphological information might be overwhelmed by the measurement errors if we process the serial images individually. It is in this context that existing 3D segmentation and registration algorithms may not be able to provide sufficient power for longitudinal image computing. In addition to the measurement errors, it is also a challenge when the presence of vascular or other pathologies changes signal characteristics, i.e., tissue contrast change, thereby rendering tissue segmentation unreliable. Therefore, in order to more accurately and robustly measure the temporal signal changes, it is important to accurately align longitudinally corresponding anatomical structures and measure these anatomical changes across different time points by using the temporal information provided in the 3D image series. Recently, several 4D image processing algorithms as well as joint segmentation and registration algorithms have been proposed in the literature (Shen and Davatzikos, 2004; Wang and Vemuri, 2005; Chen et al., 2005; Ayvaci and Freedman, 2007; Droske and Rumpf, 2007; Pohl et al., 2006; Wang et al., 2006; Wyatt and Noble, 2003; Xue et al., 2006a). Serial image processing aims at calculating the image series in the 4D space wherein temporal deformations among images are estimated simultaneously (Shen and Davatzikos, 2004; Wang and Vemuri, 2005; Chen et al., 2005), while segmentation and registration can be combined together since image registration significantly benefits from segmentation and vice versa (Ayvaci and Freedman, 2007; Droske and Rumpf, 2007; Droske and Rumpf, 2007; Wang et al., 2006; Wyatt and Noble, 2003; Xue et al., 2006a). Based on the idea of joint segmentation and registration for serial image computing, we have proposed the CLASSIC algorithm for longitudinal MR brain image analysis (Xue et al., 2006a). The algorithm incorporates an iterative serial image segmentation and registration strategy in order to improve the longitudinal stability for 3D image series. It iteratively performs two steps: 1
jointly segment a series of 3D images using a 4D image-adaptive clustering algorithm based on the current estimate of the longitudinal deformations in the image series
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refine these longitudinal deformations using a 4D elastic warping algorithm (Shen and Davatzikos, 2004, 2002).
The 4D segmentation algorithm is based on an extension of the FANTASM algorithm (Pham and Prince) by using additional temporal constraints forcing temporally corresponding voxels being segmented with the same tissue type. After segmentation, the
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4D elastic warping algorithm is then applied to refine the longitudinal deformations among the image series to better estimate the longitudinal correspondence and segmentation iteratively. Our work is a departure from the earlier method described above (Xue et al., 2006a) in that we use tissue probability map as an additional constraint in the fuzzy clustering algorithm (Xue et al., 2009). Traditional clustering methods might be biased by the tissue population and intensity differences, as well as the spatial distribution of tissues or the anatomical structures since the size and intensities of different tissue classes vary greatly in different images and applications. For example, most clustering-based algorithms tend to over-segment white matter (WM) tissues of MR brain images. To solve this problem, we introduce a tissue probability map or segmentation prior constrained clustering algorithm and apply it to serial MR brain image segmentation. The tissue probability maps consist of the segmentation priors obtained from a population and reflect the probability of a voxel belonging to different tissue types. Thus, the clustering algorithm is not only driven by the image data but also constrained by the probability of different tissue types of each image location, and the increased segmentation accuracy and robustness is obtained. We applied the new segmentation algorithm in the CLASSIC framework. For convenience, in the rest of the paper we refer the new algorithm as CLASSIC-II, and the method in Xue et al. (2006a) as CLASSIC-I. Experiments are performed to segment both simulated and real longitudinal MR brain images by comparing the performance of the algorithms, in terms of obtaining temporally-consistent segmentation, capturing global and local intensity/contrast changes, as well as estimating longitudinal deformations. The results demonstrate that by using tissue probability maps more accurate segmentation can be obtained compared with CLASSIC-I where no tissue probability map constraint is applied.
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Method
2.1 The framework of the algorithm The algorithm incorporates an iterative serial image segmentation and registration strategy in order to improve the longitudinal stability for 3D image series. It iteratively performs two steps (Xue et al., 2006a). In Step 1, given the current segmentation results, the algorithm refines the underlying longitudinal deformations among the serial images using the 4D elastic image warping algorithm (Shen and Davatzikos, 2004). Notice that other registration algorithms that take an initial deformation field and refine the registration result based on the input segmented images in order to better match them, can also be easily embedded into this framework. In Step 2, given a current estimate of the longitudinal deformations necessary to align serial 3D images, the algorithm jointly segments the image series using the tissue probability map constrained 4D clustering algorithm (described in Section 2.2). The major idea is that temporal consistent segmentation is achieved for longitudinally corresponding tissues, and the use of tissue probability map constraint improves the accuracy and robustness of the image segmentation. In the first iteration, the initial segmentation of the serial images can be obtained by applying a 3D image segmentation algorithm, AdpkMean (Yan and Karp, 1995). Then the proposed 4D segmentation is used from the second iteration. The algorithm stops
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when the difference between two consequent iterations is smaller than a prescribed threshold.
2.2 The tissue probability map constrained 4D clustering algorithm 2.2.1 Tissue probability map constrained 4D fuzzy clustering algorithm Given a series of images It, t ∈ T, T = {t1, t2,…,tY} and the underlying longitudinal deformations Ft1 →t , t = t2,…,tY, the goal of the 4D segmentation is to classify the tissues into WM, grey matter (GM) and cerebrospinal fluid (CSF). The segmented images are denoted as I t( seg ) , t ∈ T. Since Ft1 →t is the deformation from I t1 to It, the corresponding point of a voxel i in image I t1 will be point Ft1 →t (i ) in image It. For simplicity, we denote the point Ft1 →t (i ) in image It as (t, i), thus x(t,i) indicates the intensity of point (t, i). From the segmentation results of the images of a population we can obtain the tissue probability map p(t, i), k and globally align it onto the space of the first image of the image series. The tissue probability map reflects the probability or segmentation prior of point (t, i) belonging to class k. This prior information can then be used as an additional constraint in the fuzzy clustering algorithm to reduce the effect of biased segmentation results. This new tissue probability map constrained 4D clustering algorithm can be formulated by minimising the following new objective function: 1 E ( μ , c) = 2 t∈T
2 2 ⎫ ⎧μ 2 ⎪ (t ,i ), k ( x(t ,i ) − ct , k ) + γ ( μ (t ,i ), k − p(t ,i ), k ) + ⎪ ⎨ 2 ⎬ (s) (t ) 2 ⎪⎭ k =1 ⎪ ⎩αμ (t ,i ),k μ (t ,i ),k + βμ (t ,i ),k μ (t ,i ),k K
∑∑∑ i∈Ω
(1)
where voxel x(t, i) (t ∈ T, i ∈ Ω) is classified into different tissue types by finding the clustering centres ct,k, the kth clustering centre of image It, and μ(t,i),k, the fuzzy membership function of x(t,i) belonging to class k. Ω is the image domain of I t1 . K is the number of tissue types. The first term in equation (1) is the standard energy term for fuzzy clustering algorithm. The second term reflects the constraint of the tissue probability map and γ is the weight for the constraint. The third and the forth terms are the spatial segmentation consistency and the temporal segmentation consistency constraints along the longitudinal deformations, respectively. α and β are the weights for 1 μ(2t ,i ),m , where N ((ts,)i ) is the spatial these two constraints. μ((ts,)i ),k = ( s )′ ( t , i ) N m M ∈ ∈ k ( t ,i ) N1
∑
∑
neighbourhood of point (t, i), N ((tt,)i ) = {(τ , i ) : | τ − t | ≤ TN } is the temporal neighbourhood
of point (t, i) along the temporal deformation, and N1 and N2 are the normalisation factors. TN is the size of the temporal neighbourhood, and Mk = {l: k = 1,…,K, and l ≠ k}. The fuzzy membership functions are subject to: K
∑μ k =1
(t ,i ), k
= 1, for all i ∈ Ω, t ∈ T .
(2)
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It can be seen that the algorithm becomes the segmentation step in the CLASSIC-I (with no spatial adaptation for clustering centres) when γ = 0, it is similar to the Robust FCM algorithm (Pham and Prince, 2004) when γ = 0 and β = 0, and it becomes the standard FCM algorithm when α = 0, β = 0, and γ = 0. Therefore, by setting properly the parameters α, β, and γ, we can apply constraints on spatial smoothness, temporal consistency (TC), and prior tissue knowledge on the 4D clustering algorithm. Using Lagrange multipliers to enforce the constraint equation (2) in equation (1), and calculating its partial derivatives with respect to μ and c, we can get the equations to iteratively update them:
μ(t ,i ,), k
⎛ 1− =⎜ ⎜ ⎝
∑ {γ p ∑ g
−1 (t ,i ),l g (t ,i ),l
K l =1
K −1 l =1 (t ,i ),l
}+γp
(t ,i ), k
⎞ ⎟ ⋅ g(−t1,i ), k , t ∈ T , i ∈ Ω, ⎟ ⎠
(3)
where g(t ,i ), k = ( x(t ,i ) − ct , k ) 2 + γ + αμ((ts,)i ), k + βμ((tt,)i ),k , and ct ,k =
∑ ∑
i∈Ω
μ(2t ,i ),k x(t ,i )
i∈Ω
μ(2t ,i ),k
(4)
The 4D clustering algorithm can be summarised as follows: 1
Set α, β, and γ neighbourhoods N ((ts,)i ) , N ((tt,)i ) , and TN, and compute fuzzy membership functions using equation (3)
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Compute clustering centroids using equation (4).
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If the algorithm converges (the difference of the values of the objective function between two iterations is smaller than a prescribed threshold), output the segmentation results, otherwise go back to Step 1.
Similar to the standard FCM algorithm, the new algorithm usually converges after ten to 20 iterations dependent on the difference values to terminate the iteration. This serial image segmentation algorithm is used in Step 2 of the CLASSIC framework as described in Section 2.1.
2.3 Estimation of tissue probability maps In this section, we explain how such segmentation prior is calculated in detail. A tissue probability map is calculated from the segmentation results of a population, i.e., all the S sample images captured from a population are first segmented using a 3D segmentation algorithm (Yan and Karp, 1995), and then all these segmented images are aligned onto the space of the template image Itemp using Shen and Davatzikos (2002). Denoting ntemp(i, k) as the number of aligned sample images whose voxel i is segmented as tissue k, the segmentation prior of class k at voxel i is defined as: pitemp = ,k
n temp (i, k ) , i ∈ Ω temp , k = 1,… , K , S
(5)
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where Ωtemp is the space of the template image, and
∑
temp K k =1 pi , k
= 1. When applying the
segmentation prior onto a series of input images It, t = t1,…,tY, we can transform the onto the space of the first image I t1 . Herein, the FSL FLIRT segmentation prior pitemp ,k programme is applied to align all the segmentation results onto the template image space, as well as to transfer the template image onto the first time point image using affine transformations. We denote the transformed prior in the first time point image domain as p(t,i),k, which is used in equation (1) as the tissue probability map. Figure 1 shows the tissue probability maps of different tissue types calculated from 150 normal adult MR brain images obtained from the ADNI dataset. In longitudinal image analysis of ADNI, there are no obvious pathological phenotypes from single time-point MR image; however, the subtle longitudinal morphological changes can provide quantitative neuroimaging markers for earlier detection of neuro-degeneration diseases. In this paper, we selected the MR images from normal adults download from the ADNI. As a result, the subjects used to test the algorithm are also in the same age range. For further application, since the morphology is different for young kids, adults, and elder adults, we would suggest that different subject groups should be used for different studies. Figure 1
The tissue probability maps of different tissue types calculated from 150 images of the ADNI dataset, (a) WM map (b) GM map (c) CSF map
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Results
3.1 Experiments on simulated longitudinal data The proposed algorithm is validated first on simulated images. The deformations of these simulated images are known in order to calculate the registration accuracy. The images are simulated as follows. First, a statistical model (Xue et al., 2006b) is trained using the 3D registration results obtained from 150 ADNI MR brain data, then the statistical simulation algorithm is used to simulate a number of new deformation fields (we simulated ten deformations) from a given template image. Then, the corresponding MR brain images can be generated by warping the segmented template image using these simulated fields. Serial images can then be simulated for each of these simulated subject images by using the atrophy simulation package (Karacali and Davatzikos, 2006). Thus, each simulated subject image consists of longitudinal atrophy in their serial images.
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Notice all these simulated images are the segmented images, and spatially correlated Gaussian noises are then added to them, where the mean intensity of WM, GM and CSF as 45, 85, and 110, respectively, and the standard deviation of the Gaussian noise is set to 15. In this way, the ground truth of the tissue types of these serial images and the longitudinal deformations of the serial images are known. Figure 2 shows some examples of the simulated images. The first row gives the template image and three warped template images using the simulated deformation fields. It can be seen that using the statistical model-based deformation simulation (Xue et al., 2006b), various realistic anatomical shapes of MR brain images can be simulated. The second row gives the template segmentation image and the corresponding warped segmentation images using the simulated fields. Because our goal is to evaluate the accuracy of image segmentation, we will simulate the MR images from the segmentation images by adding noises, so that the ground truth is known. The longitudinal data are generated by first manually selecting a point within the temporal lobe and then simulating a gradual atrophy of a spherical region around that point across with time. For each subject image, five time point images are simulated, and Figure 3 gives an example of the simulated serial images. It can be seen that atrophy on the left temporal lobe region is gradually introduced. Figure 2
(a) The template image and (b, c, d) three sample simulated subject images
(a) Figure 3
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A series of simulated images, (a) time 1 (b) time 2 (c) time 3 (c) time 4 (see online version for colours)
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After generating the simulated images, the proposed 4D segmentation algorithm is evaluated with (CLASSIC-I) and without (CLASSIC-II) the tissue probability map constraints by segmenting these datasets. The global and local brain volumes and the correct classification rate (CCR) can be calculated for each segmentation result with respect to the ground truth to evaluate the performance of the proposed algorithm. Herein, we calculated the WM and GM volumes, as well as the CCR for all the images within the previously selected spherical regions, and Figure 4(a) show the mean and standard deviation of the volumes, and Figure 4(b) gives the CCR for different subjects across different time points. It can be seen that in terms of local brain tissue volume (GM + WM) measures, similar average and standard deviation measures of the ten simulated serial images were obtained for both methods, however in terms of CCR, the CLASSIC with tissue probability map constraint is better: it yields higher CCR values. These results indicate that since the WM and GM are combined together in the volumetric measure, there might be some difference in the segmentation results for different tissues. In fact, if calculating the WM and GM volumes separately (see Figure 5), we can see more accurate measures of different tissue volumes by using the tissue probability map in the 4D clustering algorithm. In summary, the experiments using the simulated serial images indicate more accurate segmentation by applying the tissue probability map constraints in the 4D fuzzy clustering algorithm. Figure 4
(a) The mean and standard deviation values of the local volumes (cc = cubic centimetre) and (b) the CCRs inside a spherical neighbourhood manually drawn in the temporal lobe, for ten simulated series of images
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There are three important parameters in this proposed algorithm, namely α, β and γ. α is the weighting factor for spatial smoothness constraint, β is the parameter for temporal constraint, and γ is the weight for the prior knowledge. We adopted a method to gradually tune these parameters one by one. First, β and γ were set to zero and a proper α was selected by calculating the CCR using simulated images, and then β was determined followed by γ. Finally, the parameters we used in the experiments are α = 150, β = 170, γ = 60. These parameters are also applied to the real longitudinal images in the following experiments.
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The volumes of (a) GM and (b) WM are plotted for one series of images
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3.2 Experiments on longitudinal images of Alzheimer’s disease We also applied the proposed algorithm to ADNI (http://www.loni.ucla.edu) data. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD). In this preliminary experiment, we applied the algorithm to 30 ADNI datasets (MCI), and the goal is to evaluate the performance of the algorithm on real data. For all the original MR brain data, the FSL BET skull stripping programme has been used to remove the skulls, and some manual corrections were conducted for larger skulls that have not been removed. Then the proposed 4D segmentation algorithm was used to segment these serial images. Figure 6 shows an example of the segmentation results of a subject in the ADNI dataset using CLASSIC-II. It can be seen that the segmentation results are temporally stable while still maintaining the anatomical details and differences across different time points. In order to quantitatively analyse the segmentation results, we used a TC factor to calculate the TC of the segmentation results. Suppose x(seg t ,i ) is the segmentation result (label) of voxel x(t,i), the segmentation results of voxel x(t,i) across different time points can seg seg be denoted as x(seg t ,i ) , x(t ,i ) , … , x(t ,i ) . Denote Li as the number of label changes of the 1
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corresponding voxels across different times, the segmentation is temporally consistent if Li is small, and vice versa. Therefore, the TC of segmentation results of the image series can be measured by TC = 1 / S (Ω′) (1 − Li / (Y − 1)), where Ω′ is the voxel set of the
∑
i∈Ω′
region of interest, and S(Ω′) is the number of voxels in Ω′. Both the TC values obtained from a 3D segmentation, FAST (FSL), and CLASSIC-II are calculated and shown in Figure 7. Notice that the longitudinal deformations among the serials images are needed in order to calculate the TC values. For the 3D segmentation results, we applied two kinds of registration to get them, namely the FSL FLIRT affine registration (FSL) and the 3D hierarchical volumetric image registration (Shen and Davatzikos, 2002). For CLASSIC-II, both the longitudinal deformations and the 4D segmentation results can be obtained simultaneously. It can be seen from Figure 7
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that the TC values for 4D algorithm are much higher than those of 3D, indicating that longitudinally stable segmentation and alignment of serial images is achieved. Figure 6
An example of the 4D segmentation of the ADNI data, (a) time 1 (b) time 2 (c) time 3 (c) time 4 (see online version for colours)
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Notes: Top: the segmentation results of CLASSIC-II; bottom: overlapping of GM/WM boundary on the original images Figure 7
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The TC of the whole brain of different subjects using different segmentation algorithms
Conclusions
This paper proposed a tissue probability map constrained 4D image segmentation algorithm for longitudinal image analysis. The algorithm iteratively estimates the longitudinal deformations among the image series that reflect the underlying structural changes across time and jointly segments the serial images. By using the tissue probability map as an additional constraint in the clustering algorithm, more accurate and robust segmentation results can be obtained for serial images. Experiments with simulated MR brain images and the ADNI data have confirmed the advantages of this algorithm.
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Acknowledgements This research is partially funded by NIH G08LM008937 under STC Wong. We also thank the support from the Ting Tsung and Wei Fong Chao foundation. Dr. Shen is supported by EB006733, EB008374, EB009634, and MH088520.
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Int. J. Medical Engineering and Informatics, Vol. 3, No. 3, 2011
Tissue probability map constrained 4D clustering algorithm for increased accuracy and robustness in SerialMR brain image segmentation Zhong Xue* Biomedical Image Computing Lab, Department of Systems Medicine and Bioengineering, The Methodist Hospital Research Institute, The Methodist Hospital, Weill Cornell Medical College, Houston, TX, USA E-mail: [email protected] *Corresponding author
Dinggang Shen Department of Radiology and BRIC, University of North Carolina, Chapel Hill, NC, USA E-mail: [email protected]
Hai Li Biomedical Image Computing Lab, Department of Systems Medicine and Bioengineering, The Methodist Hospital Research Institute, The Methodist Hospital, Weill Cornell Medical College, Houston, TX, USA E-mail: [email protected]
Stephen T.C. Wong Department of Systems Medicine and Bioengineering, The Methodist Hospital Research Institute, Department of Radiology, The Methodist Hospital, Weill Cornell Medical College, Houston, TX, USA E-mail: [email protected]
Copyright © 2011 Inderscience Enterprises Ltd.
Tissue probability map constrained 4D clustering algorithm Abstract: The traditional fuzzy clustering algorithms might be biased due to the variability of tissue intensities and anatomical structures: clustering-based algorithms tend to over-segment white matter tissues of MR brain images. To solve this problem, we propose a tissue probability map constrained clustering algorithm and apply it to series of 3D MR brain images of the same subject at different time points for longitudinal study. The tissue probability maps consist of segmentation priors obtained from a population and reflect the probability of different tissue types. Using the proposed algorithm in the framework of the CLASSIC algorithm, which iteratively segments a series of images and estimates their longitudinal deformations, more accurate and robust longitudinal measures can be achieved. Experimental results using both simulated and real data confirmed the advantages of the proposed algorithm in longitudinal follow up studies of MR brain imaging with subtle morphological changes for neurological disorders. Keywords: image segmentation; longitudinal image processing; MR brain images. Reference to this paper should be made as follows: Xue, Z., Shen, D., Li, H. and Wong, S.T.C. (2011) ‘Tissue probability map constrained 4D clustering algorithm for increased accuracy and robustness in SerialMR brain image segmentation’, Int. J. Medical Engineering and Informatics, Vol. 3, No. 3, pp.286–298. Biographical notes: Zhong Xue is an Assistant Professor with Weill Cornell Medical College and the Director of Medical Image Analysis Lab. His research interests include medical image computing, software and algorithms for image guided diagnosis and therapy. Dinggang Shen is Associate Professor of Radiology, Computer Science, and Biomedical Engineering and Director of IDEA Lab and the medical image analysis core, University of North Carolina at Chapel Hill. His research interests include medical image analysis, computer vision, and pattern recognition. He serves as an editorial board member for four international journals. Hai Li is currently Research Fellow in The Center for Bioengineering and Informatics, TMHRI, and PhD candidate in Northwestern Polytechnical University. His research interests include biomedical image analysis and DTI image processing. Stephen T.C. Wong is Director of the Ting Tsung and Wei Fong Chao Center for Bioinformatics Research and Imaging, Director of The Center for Bioengineering and Informatics, John S. Dunn Distinguished Endowed Chair in Biomedical Engineering, Vice Chair and Chief of Medical Physics, Department of Radiology, The Methodist Hospital, and Professor of Radiology, Neurology, and Neurosciences, Weill Cornell Medical College. His research focuses on new approaches of combining bioinformatics, biophysics, and imaging for developing new biomarkers, drug cocktails, and on-the-spot diagnosis and therapy in neurological disorders and cancer.
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Introduction
Image segmentation plays an important role in quantitative analysis of MR brain images of many medical imaging applications, such as automatic tissue labelling and quantification, morphometry, image registration, and image-guided surgery (Bezdek et al., 1993; Pappas, 1992; Udupa and Samarasekera, 1996; Lim and Prefferbaum, 1989; Pham and Prince, 1999; Chen and Giger, 2004). In follow up studies, a series of 3D images of the same subject are usually captured at different time points, and the focus is to quantitatively measure the morphological changes of the whole brain or a specific region (Resnick et al., 2000; Tang et al., 2001; Freeborough and Fox, 1997) across with time. Therefore, not only spatial information needs to be effectively utilised to generate spatially smooth segmentation results (Wang et al., 2009; Colliot et al., 2006), but also longitudinal stability is critical, since the temporal changes in these applications are usually subtle and the true signal of morphological information might be overwhelmed by the measurement errors if we process the serial images individually. It is in this context that existing 3D segmentation and registration algorithms may not be able to provide sufficient power for longitudinal image computing. In addition to the measurement errors, it is also a challenge when the presence of vascular or other pathologies changes signal characteristics, i.e., tissue contrast change, thereby rendering tissue segmentation unreliable. Therefore, in order to more accurately and robustly measure the temporal signal changes, it is important to accurately align longitudinally corresponding anatomical structures and measure these anatomical changes across different time points by using the temporal information provided in the 3D image series. Recently, several 4D image processing algorithms as well as joint segmentation and registration algorithms have been proposed in the literature (Shen and Davatzikos, 2004; Wang and Vemuri, 2005; Chen et al., 2005; Ayvaci and Freedman, 2007; Droske and Rumpf, 2007; Pohl et al., 2006; Wang et al., 2006; Wyatt and Noble, 2003; Xue et al., 2006a). Serial image processing aims at calculating the image series in the 4D space wherein temporal deformations among images are estimated simultaneously (Shen and Davatzikos, 2004; Wang and Vemuri, 2005; Chen et al., 2005), while segmentation and registration can be combined together since image registration significantly benefits from segmentation and vice versa (Ayvaci and Freedman, 2007; Droske and Rumpf, 2007; Droske and Rumpf, 2007; Wang et al., 2006; Wyatt and Noble, 2003; Xue et al., 2006a). Based on the idea of joint segmentation and registration for serial image computing, we have proposed the CLASSIC algorithm for longitudinal MR brain image analysis (Xue et al., 2006a). The algorithm incorporates an iterative serial image segmentation and registration strategy in order to improve the longitudinal stability for 3D image series. It iteratively performs two steps: 1
jointly segment a series of 3D images using a 4D image-adaptive clustering algorithm based on the current estimate of the longitudinal deformations in the image series
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refine these longitudinal deformations using a 4D elastic warping algorithm (Shen and Davatzikos, 2004, 2002).
The 4D segmentation algorithm is based on an extension of the FANTASM algorithm (Pham and Prince) by using additional temporal constraints forcing temporally corresponding voxels being segmented with the same tissue type. After segmentation, the
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4D elastic warping algorithm is then applied to refine the longitudinal deformations among the image series to better estimate the longitudinal correspondence and segmentation iteratively. Our work is a departure from the earlier method described above (Xue et al., 2006a) in that we use tissue probability map as an additional constraint in the fuzzy clustering algorithm (Xue et al., 2009). Traditional clustering methods might be biased by the tissue population and intensity differences, as well as the spatial distribution of tissues or the anatomical structures since the size and intensities of different tissue classes vary greatly in different images and applications. For example, most clustering-based algorithms tend to over-segment white matter (WM) tissues of MR brain images. To solve this problem, we introduce a tissue probability map or segmentation prior constrained clustering algorithm and apply it to serial MR brain image segmentation. The tissue probability maps consist of the segmentation priors obtained from a population and reflect the probability of a voxel belonging to different tissue types. Thus, the clustering algorithm is not only driven by the image data but also constrained by the probability of different tissue types of each image location, and the increased segmentation accuracy and robustness is obtained. We applied the new segmentation algorithm in the CLASSIC framework. For convenience, in the rest of the paper we refer the new algorithm as CLASSIC-II, and the method in Xue et al. (2006a) as CLASSIC-I. Experiments are performed to segment both simulated and real longitudinal MR brain images by comparing the performance of the algorithms, in terms of obtaining temporally-consistent segmentation, capturing global and local intensity/contrast changes, as well as estimating longitudinal deformations. The results demonstrate that by using tissue probability maps more accurate segmentation can be obtained compared with CLASSIC-I where no tissue probability map constraint is applied.
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Method
2.1 The framework of the algorithm The algorithm incorporates an iterative serial image segmentation and registration strategy in order to improve the longitudinal stability for 3D image series. It iteratively performs two steps (Xue et al., 2006a). In Step 1, given the current segmentation results, the algorithm refines the underlying longitudinal deformations among the serial images using the 4D elastic image warping algorithm (Shen and Davatzikos, 2004). Notice that other registration algorithms that take an initial deformation field and refine the registration result based on the input segmented images in order to better match them, can also be easily embedded into this framework. In Step 2, given a current estimate of the longitudinal deformations necessary to align serial 3D images, the algorithm jointly segments the image series using the tissue probability map constrained 4D clustering algorithm (described in Section 2.2). The major idea is that temporal consistent segmentation is achieved for longitudinally corresponding tissues, and the use of tissue probability map constraint improves the accuracy and robustness of the image segmentation. In the first iteration, the initial segmentation of the serial images can be obtained by applying a 3D image segmentation algorithm, AdpkMean (Yan and Karp, 1995). Then the proposed 4D segmentation is used from the second iteration. The algorithm stops
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when the difference between two consequent iterations is smaller than a prescribed threshold.
2.2 The tissue probability map constrained 4D clustering algorithm 2.2.1 Tissue probability map constrained 4D fuzzy clustering algorithm Given a series of images It, t ∈ T, T = {t1, t2,…,tY} and the underlying longitudinal deformations Ft1 →t , t = t2,…,tY, the goal of the 4D segmentation is to classify the tissues into WM, grey matter (GM) and cerebrospinal fluid (CSF). The segmented images are denoted as I t( seg ) , t ∈ T. Since Ft1 →t is the deformation from I t1 to It, the corresponding point of a voxel i in image I t1 will be point Ft1 →t (i ) in image It. For simplicity, we denote the point Ft1 →t (i ) in image It as (t, i), thus x(t,i) indicates the intensity of point (t, i). From the segmentation results of the images of a population we can obtain the tissue probability map p(t, i), k and globally align it onto the space of the first image of the image series. The tissue probability map reflects the probability or segmentation prior of point (t, i) belonging to class k. This prior information can then be used as an additional constraint in the fuzzy clustering algorithm to reduce the effect of biased segmentation results. This new tissue probability map constrained 4D clustering algorithm can be formulated by minimising the following new objective function: 1 E ( μ , c) = 2 t∈T
2 2 ⎫ ⎧μ 2 ⎪ (t ,i ), k ( x(t ,i ) − ct , k ) + γ ( μ (t ,i ), k − p(t ,i ), k ) + ⎪ ⎨ 2 ⎬ (s) (t ) 2 ⎪⎭ k =1 ⎪ ⎩αμ (t ,i ),k μ (t ,i ),k + βμ (t ,i ),k μ (t ,i ),k K
∑∑∑ i∈Ω
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where voxel x(t, i) (t ∈ T, i ∈ Ω) is classified into different tissue types by finding the clustering centres ct,k, the kth clustering centre of image It, and μ(t,i),k, the fuzzy membership function of x(t,i) belonging to class k. Ω is the image domain of I t1 . K is the number of tissue types. The first term in equation (1) is the standard energy term for fuzzy clustering algorithm. The second term reflects the constraint of the tissue probability map and γ is the weight for the constraint. The third and the forth terms are the spatial segmentation consistency and the temporal segmentation consistency constraints along the longitudinal deformations, respectively. α and β are the weights for 1 μ(2t ,i ),m , where N ((ts,)i ) is the spatial these two constraints. μ((ts,)i ),k = ( s )′ ( t , i ) N m M ∈ ∈ k ( t ,i ) N1
∑
∑
neighbourhood of point (t, i), N ((tt,)i ) = {(τ , i ) : | τ − t | ≤ TN } is the temporal neighbourhood
of point (t, i) along the temporal deformation, and N1 and N2 are the normalisation factors. TN is the size of the temporal neighbourhood, and Mk = {l: k = 1,…,K, and l ≠ k}. The fuzzy membership functions are subject to: K
∑μ k =1
(t ,i ), k
= 1, for all i ∈ Ω, t ∈ T .
(2)
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It can be seen that the algorithm becomes the segmentation step in the CLASSIC-I (with no spatial adaptation for clustering centres) when γ = 0, it is similar to the Robust FCM algorithm (Pham and Prince, 2004) when γ = 0 and β = 0, and it becomes the standard FCM algorithm when α = 0, β = 0, and γ = 0. Therefore, by setting properly the parameters α, β, and γ, we can apply constraints on spatial smoothness, temporal consistency (TC), and prior tissue knowledge on the 4D clustering algorithm. Using Lagrange multipliers to enforce the constraint equation (2) in equation (1), and calculating its partial derivatives with respect to μ and c, we can get the equations to iteratively update them:
μ(t ,i ,), k
⎛ 1− =⎜ ⎜ ⎝
∑ {γ p ∑ g
−1 (t ,i ),l g (t ,i ),l
K l =1
K −1 l =1 (t ,i ),l
}+γp
(t ,i ), k
⎞ ⎟ ⋅ g(−t1,i ), k , t ∈ T , i ∈ Ω, ⎟ ⎠
(3)
where g(t ,i ), k = ( x(t ,i ) − ct , k ) 2 + γ + αμ((ts,)i ), k + βμ((tt,)i ),k , and ct ,k =
∑ ∑
i∈Ω
μ(2t ,i ),k x(t ,i )
i∈Ω
μ(2t ,i ),k
(4)
The 4D clustering algorithm can be summarised as follows: 1
Set α, β, and γ neighbourhoods N ((ts,)i ) , N ((tt,)i ) , and TN, and compute fuzzy membership functions using equation (3)
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Compute clustering centroids using equation (4).
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If the algorithm converges (the difference of the values of the objective function between two iterations is smaller than a prescribed threshold), output the segmentation results, otherwise go back to Step 1.
Similar to the standard FCM algorithm, the new algorithm usually converges after ten to 20 iterations dependent on the difference values to terminate the iteration. This serial image segmentation algorithm is used in Step 2 of the CLASSIC framework as described in Section 2.1.
2.3 Estimation of tissue probability maps In this section, we explain how such segmentation prior is calculated in detail. A tissue probability map is calculated from the segmentation results of a population, i.e., all the S sample images captured from a population are first segmented using a 3D segmentation algorithm (Yan and Karp, 1995), and then all these segmented images are aligned onto the space of the template image Itemp using Shen and Davatzikos (2002). Denoting ntemp(i, k) as the number of aligned sample images whose voxel i is segmented as tissue k, the segmentation prior of class k at voxel i is defined as: pitemp = ,k
n temp (i, k ) , i ∈ Ω temp , k = 1,… , K , S
(5)
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where Ωtemp is the space of the template image, and
∑
temp K k =1 pi , k
= 1. When applying the
segmentation prior onto a series of input images It, t = t1,…,tY, we can transform the onto the space of the first image I t1 . Herein, the FSL FLIRT segmentation prior pitemp ,k programme is applied to align all the segmentation results onto the template image space, as well as to transfer the template image onto the first time point image using affine transformations. We denote the transformed prior in the first time point image domain as p(t,i),k, which is used in equation (1) as the tissue probability map. Figure 1 shows the tissue probability maps of different tissue types calculated from 150 normal adult MR brain images obtained from the ADNI dataset. In longitudinal image analysis of ADNI, there are no obvious pathological phenotypes from single time-point MR image; however, the subtle longitudinal morphological changes can provide quantitative neuroimaging markers for earlier detection of neuro-degeneration diseases. In this paper, we selected the MR images from normal adults download from the ADNI. As a result, the subjects used to test the algorithm are also in the same age range. For further application, since the morphology is different for young kids, adults, and elder adults, we would suggest that different subject groups should be used for different studies. Figure 1
The tissue probability maps of different tissue types calculated from 150 images of the ADNI dataset, (a) WM map (b) GM map (c) CSF map
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Results
3.1 Experiments on simulated longitudinal data The proposed algorithm is validated first on simulated images. The deformations of these simulated images are known in order to calculate the registration accuracy. The images are simulated as follows. First, a statistical model (Xue et al., 2006b) is trained using the 3D registration results obtained from 150 ADNI MR brain data, then the statistical simulation algorithm is used to simulate a number of new deformation fields (we simulated ten deformations) from a given template image. Then, the corresponding MR brain images can be generated by warping the segmented template image using these simulated fields. Serial images can then be simulated for each of these simulated subject images by using the atrophy simulation package (Karacali and Davatzikos, 2006). Thus, each simulated subject image consists of longitudinal atrophy in their serial images.
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Notice all these simulated images are the segmented images, and spatially correlated Gaussian noises are then added to them, where the mean intensity of WM, GM and CSF as 45, 85, and 110, respectively, and the standard deviation of the Gaussian noise is set to 15. In this way, the ground truth of the tissue types of these serial images and the longitudinal deformations of the serial images are known. Figure 2 shows some examples of the simulated images. The first row gives the template image and three warped template images using the simulated deformation fields. It can be seen that using the statistical model-based deformation simulation (Xue et al., 2006b), various realistic anatomical shapes of MR brain images can be simulated. The second row gives the template segmentation image and the corresponding warped segmentation images using the simulated fields. Because our goal is to evaluate the accuracy of image segmentation, we will simulate the MR images from the segmentation images by adding noises, so that the ground truth is known. The longitudinal data are generated by first manually selecting a point within the temporal lobe and then simulating a gradual atrophy of a spherical region around that point across with time. For each subject image, five time point images are simulated, and Figure 3 gives an example of the simulated serial images. It can be seen that atrophy on the left temporal lobe region is gradually introduced. Figure 2
(a) The template image and (b, c, d) three sample simulated subject images
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A series of simulated images, (a) time 1 (b) time 2 (c) time 3 (c) time 4 (see online version for colours)
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After generating the simulated images, the proposed 4D segmentation algorithm is evaluated with (CLASSIC-I) and without (CLASSIC-II) the tissue probability map constraints by segmenting these datasets. The global and local brain volumes and the correct classification rate (CCR) can be calculated for each segmentation result with respect to the ground truth to evaluate the performance of the proposed algorithm. Herein, we calculated the WM and GM volumes, as well as the CCR for all the images within the previously selected spherical regions, and Figure 4(a) show the mean and standard deviation of the volumes, and Figure 4(b) gives the CCR for different subjects across different time points. It can be seen that in terms of local brain tissue volume (GM + WM) measures, similar average and standard deviation measures of the ten simulated serial images were obtained for both methods, however in terms of CCR, the CLASSIC with tissue probability map constraint is better: it yields higher CCR values. These results indicate that since the WM and GM are combined together in the volumetric measure, there might be some difference in the segmentation results for different tissues. In fact, if calculating the WM and GM volumes separately (see Figure 5), we can see more accurate measures of different tissue volumes by using the tissue probability map in the 4D clustering algorithm. In summary, the experiments using the simulated serial images indicate more accurate segmentation by applying the tissue probability map constraints in the 4D fuzzy clustering algorithm. Figure 4
(a) The mean and standard deviation values of the local volumes (cc = cubic centimetre) and (b) the CCRs inside a spherical neighbourhood manually drawn in the temporal lobe, for ten simulated series of images
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There are three important parameters in this proposed algorithm, namely α, β and γ. α is the weighting factor for spatial smoothness constraint, β is the parameter for temporal constraint, and γ is the weight for the prior knowledge. We adopted a method to gradually tune these parameters one by one. First, β and γ were set to zero and a proper α was selected by calculating the CCR using simulated images, and then β was determined followed by γ. Finally, the parameters we used in the experiments are α = 150, β = 170, γ = 60. These parameters are also applied to the real longitudinal images in the following experiments.
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The volumes of (a) GM and (b) WM are plotted for one series of images
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3.2 Experiments on longitudinal images of Alzheimer’s disease We also applied the proposed algorithm to ADNI (http://www.loni.ucla.edu) data. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD). In this preliminary experiment, we applied the algorithm to 30 ADNI datasets (MCI), and the goal is to evaluate the performance of the algorithm on real data. For all the original MR brain data, the FSL BET skull stripping programme has been used to remove the skulls, and some manual corrections were conducted for larger skulls that have not been removed. Then the proposed 4D segmentation algorithm was used to segment these serial images. Figure 6 shows an example of the segmentation results of a subject in the ADNI dataset using CLASSIC-II. It can be seen that the segmentation results are temporally stable while still maintaining the anatomical details and differences across different time points. In order to quantitatively analyse the segmentation results, we used a TC factor to calculate the TC of the segmentation results. Suppose x(seg t ,i ) is the segmentation result (label) of voxel x(t,i), the segmentation results of voxel x(t,i) across different time points can seg seg be denoted as x(seg t ,i ) , x(t ,i ) , … , x(t ,i ) . Denote Li as the number of label changes of the 1
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corresponding voxels across different times, the segmentation is temporally consistent if Li is small, and vice versa. Therefore, the TC of segmentation results of the image series can be measured by TC = 1 / S (Ω′) (1 − Li / (Y − 1)), where Ω′ is the voxel set of the
∑
i∈Ω′
region of interest, and S(Ω′) is the number of voxels in Ω′. Both the TC values obtained from a 3D segmentation, FAST (FSL), and CLASSIC-II are calculated and shown in Figure 7. Notice that the longitudinal deformations among the serials images are needed in order to calculate the TC values. For the 3D segmentation results, we applied two kinds of registration to get them, namely the FSL FLIRT affine registration (FSL) and the 3D hierarchical volumetric image registration (Shen and Davatzikos, 2002). For CLASSIC-II, both the longitudinal deformations and the 4D segmentation results can be obtained simultaneously. It can be seen from Figure 7
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that the TC values for 4D algorithm are much higher than those of 3D, indicating that longitudinally stable segmentation and alignment of serial images is achieved. Figure 6
An example of the 4D segmentation of the ADNI data, (a) time 1 (b) time 2 (c) time 3 (c) time 4 (see online version for colours)
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Notes: Top: the segmentation results of CLASSIC-II; bottom: overlapping of GM/WM boundary on the original images Figure 7
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The TC of the whole brain of different subjects using different segmentation algorithms
Conclusions
This paper proposed a tissue probability map constrained 4D image segmentation algorithm for longitudinal image analysis. The algorithm iteratively estimates the longitudinal deformations among the image series that reflect the underlying structural changes across time and jointly segments the serial images. By using the tissue probability map as an additional constraint in the clustering algorithm, more accurate and robust segmentation results can be obtained for serial images. Experiments with simulated MR brain images and the ADNI data have confirmed the advantages of this algorithm.
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Acknowledgements This research is partially funded by NIH G08LM008937 under STC Wong. We also thank the support from the Ting Tsung and Wei Fong Chao foundation. Dr. Shen is supported by EB006733, EB008374, EB009634, and MH088520.
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