By choosing different clique potential function a wide variety of distributions can be formulated as Gibbs distributions. In MLL models [7][9], the potentials for cliques containing more than one site are defined as:
is a constant depending on the type of clique. This encourages neighboring sites to have the same class label. Homogeneity is imposed on the model by assigning the same potential function to all cliques of a certain type, independent of their posi- tions in the image.
In this paper, we use a special MLL model [8] that considers only two-site cliques so that the energy function can be written as:
is the Kronecker delta fuction and is the penalty against non-equal labels on two-site cliques and inversely proportional to the signal noise ratio (SNR) of MRI data.
3 GFCM Algorithm
Given the basic theory on GRF model in last section, the definition of refusable level is proposed below.
Propositions 1: Given one pixel j and the set of its neighboring sites Nj, the prior probability of labeling the pixel to can be calculated in term of the Gibbs model presented in Section 2, can be considered as the resistance of neighbors Nj to assigning pixel j the label k. The resistance is defined as refusable level in this paper.
Because refusable level take a value between the 0 and 1 and represent the spatial constraints, it can be introduced into the objective function (1) of FCM as following,
192 Y. Feng and W. Chen
where is calculated from the hard maximum membership segmentation dur- ing the iteration.
In order to minimize the objective function (9), membership values are assigned to the pixel not only according to the distance from the centroid, but also taking into account the resistance of the neighboring pixels to the label. When minimizing (9), the mutual influences and collective role between refusable level and intensity dis- tance are described in detail as follows:
At pixel j, when the refusable level of the neighboring pixels Nj to label k is low, the high resistance caused by large distance of the intensity to centroid can be tolerated by the low refusable level. As a result, a high membership may be assigned to pixel j. If refusable level it will tolerate all resistance caused by inten- sity no matter how distant the intensity is from the centroid and the pixel will defi- nitely be assigned label k. While the reusable level of the neighboring pixels to the label k is high, it will give the intensity distance a large weight in the objective function and will encourage assigning a low value to label k.
Using Lagrange multipliers to impose the constraint in (2) and evaluating the cen- troids and membership functions that satisfy a zero gradient condition yield the two necessary conditions for to be at a minimum.
The discrete steps of Gibbs Fuzzy C-means (GFCM) Algorithm are as follows:
1.
2.
3.
4.
5.
Initial estimates of the centroids and initial segmentation;
Calculate the prior probability Pj(k) in term of (5);
Update the membership functions according to (10);
Calculate centroids using (11);
Go to step (2) and repeat until convergence.
4 Experiments and Discussions
In this section, the segmentation results of GFCM algorithm are shown. In these ex- periments, we set C=4 and q=2. The algorithms were run on a 1.2G PC system. For an image with size of 256*256, execution time of GFCM algorithm is about 5s and standard FCM algorithm requires about 3s.
The results of applying GFCM algorithm to a synthetic test image is shown in Fig- ure 1. The test image contains intensity values 0, 85, 170 and 255, and the image size is 256*256 pixels. White Gaussian noise with a standard deviation of 30 was added to the image. Figure 1a shows the test image. Figure 1b and 1c show the results of
Brain MR Image Segmentation Using Fuzzy Clustering with Spatial Constraints 193 maximum membership classification produced by standard FCM and GFCM, respec- tively. It can be seen that the result of GFCM classification is less speckled and smoother except that there is some faint distortion at edges. Therefore, the GFCM classification, under the spatial smoothing constraints of the neighborhood system, is much more robust than the traditional FCM classification.
Fig. 1. Comparison of segmentation results on a two-dimensional test image: (a) the test image, (b) FCM classification, (d) GFCM classification.
Figure 2 shows the application of GFCM to a simulated MR image from the Brainweb database. Figure 2a shows the original simulated image, 9% noise and no inhomogeneity. Figure 2b and 2c shows the segmentation result of standard FCM and GFCM, respectively and the ground truth of segmentation is shown in figure 2d.
Obviously Figure 2c is much closer to the ground truth than figure 2b and the smoother appearance of the GFCM result is evident.
Fig. 2. Comparison of segmentation results on a simulated MR image: (a) original image, (b) using FCM, (c) using GFCM, (d) the true segmentation used to simulate the MR image
The correct classification rates (CCR) of applying several different algorithms to the simulated MR images for different levels of noise were shown in Table 1. MFCM
194 Y. Feng and W. Chen
is the modified FCM algorithm [6] and PFCM is the penalized FCM algorithm [5].
With the increase of noise level, the segmentation result of standard FCM degrades rapidly. While the fuzzy clustering algorithms with spatial constraint such as GFCM, PFCM and MFCM, can overcome the problem caused by noise. Overall, the three kinds of improved FCM algorithm produce comparable results, i.e. our GFCM algo- rithm provide another novel approach to improve the performance of conventional FCM algorithm.
Figure 3 shows the application of RFCM to real MR images taken from IBSR. The algorithm has incorporated the bias field correction as described in [6]. Novel meth- ods is also addressed in [10][11][12] to correction for the intensity inhomogeneity in MR images. Figure 3a shows the original T1 weighted brain MR images, Figure 3b shows the estimated bias field and Figure 3c shows the Segmentation result of GFCM with bias field correction. The manual segmentation by medical expert is shown in figure 3d. Obviously, GFCM algorithm with the bias field correction produces classi- fication comparable to manual results of expert.
Fig. 3. GFCM applied to real T1-weighted MR images from IBSR: (a) original images, (b) estimated bias fields, (c) segmentation results of GFCM, (d) manual segmentation results
5 Conclusion
In this paper, we have described a novel extension of FCM, based on the Markov Random Fields theory, to incorporate spatial constraints. The spatial information in GFCM is extracted from the label set during segmentation. Comparison is also pre-
Brain MR Image Segmentation Using Fuzzy Clustering with Spatial Constraints 195 sented between our GFCM algorithm and traditional FCM, penalized FCM and Modi- fied FCM on synthetic image and simulated MR images. The GFCM produces com- parable results as PFCM and MFCM, while GFCM is of complete mathematical background with more promising future improvement
References
1.
2.
3.
4.
5.
6.
7.
9.
10.
11.
12.
Y. A. Tolias and S. M. Panas: On applying spatial constraints in fuzzy image clustering using a fuzzy rule-basedsystem. IEEE Signal Process. Lett. 5(10), 1998, 245–247
Y. A. Tolias and S. M. Panas: Image segmentation by a fuzzy clustering algorithm using adaptive spatially constrained membership functions. IEEE Trans. Systems, Man, Cybernet.
A 28(3), 1998, 359–369
S. T. Acton and D. P. Mukherjee: Scale space classification using area morphology. IEEE Trans. Image Process, 9(4), 2000, 623–635
A. W. C. Liew, S. H. Leung, and W. H. Lau: Fuzzy image clustering incorporating spatial continuity. IEE Proc. Visual Image Signal Process, 147(2), 2000, 185–192
D. L. Pham: Spatial Models for Fuzzy Clustering. Computer Vision and Image Understand- ing 84, pp. 285-297, 2001
M. N. Ahmed, Sameh M. Yamany, Nevin Mohamed, Aly A. Farag and Thomas Moriarity: A Modified Fuzzy C-Means Algorithm for Bias Field Estimation and Segementation of MRI Data. IEEE trans. On Medical Imaging, 21(3): 193-199, 2002
Stan Z. Li: Markov Random Field Modeling in Image Analysis. Springer, 2001, ISBN 4- 431-70309-8
Y. Zhang, M. Brady, and S. Smith: Segmentation of Brain MR Images through a Hidden Markov Random Field Model and the Expectation-Maximization Algorithm. IEEE Trans.
Medical Imaging, 20(1):45-57, 2001
S. German and D. German: Stochastic relaxation, Gibbs distribution, and the Bayesian Restoration of Images. IEEE Trans. Patter Anal. Machine Intell., Vol. PAMI-6, No. 6, pp.
721-741, 1984
Dzung L. Pham, Jerry L. Prince: Adaptive fuzzy segmentation of magnetic resonance images. IEEE Trans. On Medical Imaging, 18(9): 737-752[2], 1999
C. Z. Zhu, T. Z. Jiang: MultiContext fuzzy clustering for separation of brain tissues in magnetic resonance images. NeuroImage, Vol.18, No.3, pp. 685-696, 2003
Anatomy Dependent Multi-context Fuzzy Clustering for Separation of Brain Tissues in MR Images
C.Z. Zhu, F.C. Lin, L.T. Zhu and T.Z. Jiang
National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100080, P.R. China
{czzhu, fclin, ltzhu, jiangtz}@nlpr.ia.ac.cn
Abstract. In a previous work, we proposed multi-context fuzzy clustering (MCFC) method on the basis of a local tissue distribution model to classify 3D T1-weighted MR images into tissues of white matter, gray matter, and cerebral spinal fluid in the condition of intensity inhomogeneity. This paper is a com- plementary and improved version of MCFC. Firstly, quantitative analyses are presented to validate the soundness of basic assumptions of MCFC. Carefully studies on the segmentation results of MCFC disclose a fact that misclassifica- tion rate in a context of MCFC is spatially dependent on the anatomical posi- tion of the context in the brain; moreover, misclassifications concentrate in re- gions of brain stem and cerebellum. Such unique distribution pattern of mis- classification inspires us to choose different size for the contexts at such re- gions. This anatomy-dependent MCFC (adMCFC) is tested on 3 simulated and 10 clinical T1-weighted images sets. Our results suggest that adMCFC outper- forms MCFC as well as other related methods.
1 Introduction
In general, white matter (WM), gray matter (GM) and cerebral spinal fluid (CSF), are three basic tissues in the brain. Brain tissue segmentation of MR images means to identify the tissue type for each point in data set on the basis of information available from both MR images and neuroanatomical knowledge. It’s an important processing step in many medical research and clinical applications, such as quantification of GM reduction in neurological and mental diseases, cortex segmentation and analysis, surgical planning, multi-modality images fusion, functional brain mapping [2,9,10].
Unfortunately, intensity inhomogeneity, caused by both MR imaging device imper- fections and biophysical properties variations in each tissue class, result in various MR signal intensities for the same tissue class at different locations in the brain.
Hence intensity inhomogeneity is a major obstacle to any intensity based automatic segmentation methods and has been investigated extensively [1,4,5,6,7,8,11]. To address this issue, multi-context fuzzy clustering (MCFC) method had been proposed [11] on the basis of a local tissue distribution model in our previous work. In this paper, anatomy-dependent MCFC (adMCFC) is proposed to refine and improve the original MCFC.
G.-Z. Yang and T. Jiang (Eds.): MIAR 2004, LNCS 3150, pp. 196-203, 2004.
© Springer-Verlag Berlin Heidelberg 2004
Anatomy Dependent Multi-context Fuzzy Clustering 197 The local tissue distribution model and MCFC method are briefly summarized in Section 2. Section 3 presents adMCFC followed by experimental results in Section 4.
The final section is devoted to discussion and conclusions.
2 Original MCFC Method
Clustering context is a key concept for the local tissue distribution model and MCFC.
A context is a subset of 3-D MRI brain volume and the size of a context is defined as the number of pixels in the context. Highly convoluted spatial distributions of the three different tissues in the brain inspired us to propose the local tissue distribution (LTD) model. Given a proper context size, LTD model in any context consists of the following three basic assumptions:
(1) Each of the three classes of tissues exists with considerable proportion.
(2) All pixels belonging to same tissue class will take on similar ideal signal intensi- ties.
(3) Bias field is approximately a constant filed.
As a whole, it is conflictive to choose context size for the three assumptions simulta- neously. The following quantitative analyses are presented as a complementary ex- planation to validate the soundness of the assumptions. The simulated T1-weighted data as well as the corresponding labeled brain from the McConnell Brain Imaging Center at the Montreal Neurological Institute, McGill University [3] were used through the paper.
Firstly, an index of fractional anisotropy (FA) was presented to describe difference in proportions of the three tissue classes in a given context.
Where and are the number of pixels in a context belonging to WM, GM, and CSF respectively and is the average. Given normalized context size (NCS), N contexts or brain regions were uniformly sampled in the labeled brain image and the averaged FA among N contexts was defined as a measure of proportion difference for the given NCS [11]. When pixel amount were same for all the three tissues in a con- text, FA reached zero as the minimum; when the amount differences among the three tissues became more significant FA would increased. As a function of NCS, FA was plotted in Fig.1 where FA is decreasing when the context became larger. Accord- ingly, larger context size could guarantee assumption (1) better.
As for assumption (2) the centers of ideal signal distributions (CISD) were calcu- lated for both WM and GM in each sampled context with given NCS in the 3-D simu- lated T1-weighted data without any noise and bias field imposed. Profiles of CISD of GM in contexts at different positions in the brain were plotted in Fig 3 (a) ~ (d) corre- sponding to NCS of 0.02,0.06, 0.10 and 0.18 respectively.
In any one of the 4 figures, CISD are various in different brain regions, which im- plies the intrinsic variations of biophysical properties in GM. Moreover, such varia- tion of CISD gradually decreases when the NCS becomes larger and larger, which
198 C.Z. Zhu et al.
suggests local differences in biophysical properties of different GM structures are gradually vanished. Accordingly assumption (2) required the context as small as possible to keep the truth. As for assumption (3), however, the smaller, the better.
Fig. 1 FA distribution with NCS Fig. 2 MCR distribution with NCS
Fig. 3. GM CISD distributions with NCS= 0.02(a); 0.06 (b); 0.10 (c) and 0.18 (d) Therefore it is conflictive to choose a proper context size since assumption (1) asks the size as larger as possible, while the other two assumptions require the opposition.
As a function of NCS, misclassification rates (MCR) of MCFC on simulated data with 3% noise were plotted in Fig.2.in case of 0%, 20% and 40% bias field (INV) respectively. We can see that 0.06, a tradeoff between the two conflictive require- ments, yielded satisfying results.
Anatomy Dependent Multi-context Fuzzy Clustering 199 Given NCS, MCFC includes two stages: multi-context fuzzy clustering and infor- mation fusion. Firstly, multiple clustering contexts are generated for each voxel and fuzzy clustering is independently carried out in each context to calculate the member- ship of the voxel to each tissue class. The memberships can be regarded as soft deci- sions made upon the information from each information source, say the context. Then in stage 2, all the soft decisions are integrated as the final results. Implementation details of MCFC can be found in [11].
3 Anatomy-Dependent MCFC
Carefully studies on the MCR of each context resulted in an interesting finding that MCR in contexts varied at different position in the brain and most of the errors con- centrated in the area of brain stem and cerebella as shown in Fig. 4. The finding seemed similar for all the three data sets (3% noise and 0%, 20% and 40% bias fields).
Fig.4 Original simulated T1-weighted MR data and the misclassified pixels. Box indicates the area with concentrated misclassification.
Quantitatively analysis in FA suggested that higher FA made assumption (1) not well guaranteed, which was, at least, part of the reason to concentrated misclassifica- tion in such area. We enlarged contexts in these regions by an enlarging coefficient.
As a function of enlarging coefficient, averaged FA among the enlarged contexts were calculated and plotted in Fig. 5. We can see that FA in such area does decrease with the enlarging coefficient so that assumption (1) could be more correct. The im- plementation of adMCFC can be summarized as follows:
200 C.Z. Zhu et al.
Fig. 5. FA and enlarging coefficient Fig. 6 MCR and enlarging coefficient
Step 1 Find the enlarging anatomic area in target brain images
In practice, a binary mask could be rough created either by a manually drawing cov- ering brain stem and cerebella or by a rigid registration from the template as shown in Fig. 4 to the target.
Step 2 Perform modified MCFC with a given NCS.
During context window with NCS moving through the target image, context center is tested whether in the enlarging mask or not. If yes, enlarge the context by multiplying the NCS with a given enlarging coefficient; If not, keep original NCS unchanged. Do step 2 until all the contexts were processed.
In adMCFC, a proper enlarging coefficient is very important. Given NCS = 0.06 as in [11], three MCR curves were calculated and plotted as the function of the enlarging coefficient in Fig. 6 on the three sets of simulated T1-weighted MRI data respec- tively. When enlarging coefficient was set between 1 and 4, the MCR became smaller for all the intensity inhomogenieties conditions and the best classification results occurred at slightly different enlarging coefficients for the three conditions. In this work, we chose 3 as the enlarging coefficients in all experiments.
4 Experiments