The accuracy on intervertebral relation reported by the system were also tested. The angle between two vertebrae is usually used in most spinal motion analysis. Thus, the accuracy on the angle reported by the system were evaluated.
The measurement methodology is shown in figure 5(a). The testing result is shown in figure 5(b). It shows the relative root-mean-square error (the angular difference between the tracked result and the physician-reported result) is quite large during initial phase but getting smaller after 30 frames. The relative root- mean-square error is lower than 10% in average in later stage.
Finally, the number of vertebrae that can be tracked by the system is evalu- ated. The result is shown in figure 4(b). It shows that totally four of the vertebrae, namely L2, L3, L4 and L5 can be tracked, provided that the illumination and the contrast is not varied a lot. The relative root-mean-square error reported is less than 10% in these four vertebrae.
Fig. 4. (a) The first two graphs show the reported location of the four vertebral corners along the time domain and time series data of location of the correspond- ing vertebral corners marked by physician respectively. The third graph combines the above two time series into one graph, (b) The tracking result of L2 to L5 vertebrae.
Tracking Lumbar Vertebrae in Digital Videofluoroscopic Video 161
Fig. 5. (a) The angle difference between middle two vertebrae is recorded in the experiment. (b) The left two graphs show the time series data on angle difference reported by the system and those measured by physician respectively. The third graph shows the corresponding relative root-mean-square error in percentage.
5 Conclusions
In this paper, a system for automatic spinal motion analysis is proposed. The proposed system requires less human intervention than common approaches by automating the edge detection and snake fitting. Operators may need to setup initial snake position in the first frame only. The edge will then be detected automatically using pattern recognition and the snake will fit toward the edge accordingly. The initial snake position in the next frame will be predicted through the use of dynamic that learnt from previous observations. Experimental results show that the proposed system can segment vertebrae from videofluoroscopic images automatically and accurately.
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A New Algorithm Based on Fuzzy Gibbs Random Fields for Image Segmentation*
Gang Yan and Wufan Chen
Key Lab for Medical Image Processing, BME, First Military Medical University, Guangzhou, China
yangang@fimmu.edu.cn, chenwf@fimmu.edu.cn
Abstract. In this paper a new unsupervised segmentation algorithm based on Fuzzy Gibbs Random Field (FGRF) is proposed. This algorithm, named as FGS, can deal with fuzziness and randomness simultaneously. A Classical Gibbs Random Field (CGRF) servers as bridge between prior FGRF and origi- nal image. The FGRF is equivalent to CGRF when no fuzziness is considered;
therefore, the FGRF is obviously a generalization of the CGRF. The prior FGRF is described in the Potts model, whose parameter is estimated by the maximum pesudolikelihood (MPL) method. The segmentation results are ob- tained by fuzzifying the image, updating the membership of FGRF based on maximum a posteriori (MAP) criteria, and defuzzifying the image according to maximum membership principle (MMP). Specially, this algorithm can filter the noise effectively. The experiments show that this algorithm is obviously better than CGRF-based methods and conventional FCM methods as well.
1 Introduction
Image segmentation is a key technique in the pattern recognition, computer vision and image analysis. The accurately segmented medical images is very helpful for clinical diagnose and quantitative analysis. Automated segmentation is however very complicated, facing difficulties due to overlapping intensities, anatomical variability in shape, size, and orientation, partial volume effects, as well as noise perturbation, intensity inhomogeneities, and low contrast in images [1]. To overcome those diffi- culties, there has recently been growing interesting in soft segmentation methods [2]- [4]. The soft segmentation, where each pixel may be classified into classes with a varying degree of membership, is a more natural way. Introducing the fuzzy set the- ory into the segmentation is the outstanding contribution for soft segmentation algo- rithms. The algorithm is called soft image segmentation scheme if it is based on fuzzy set. When the fuzziness is eliminated according to some rules, the segmented image is exactly obtained.
Although the FCM algorithms are widely used in image segmentation, there are still some disadvantages. It is assumed that the data are of spatial independence or no
*This work was supported by 973 Program of China (No: 2003CB716101) and Key NNSF of China (No: 30130180).
G.-Z. Yang and T. Jiang (Eds.): MIAR 2004, LNCS 3150, pp. 163-170, 2004.
© Springer-Verlag Berlin Heidelberg 2004
164 G. Yan and W. Chen
context. Those assumptions are unreasonable. The FCM algorithm hardly deals with noised images. It is indispensable to incorporate the contextual constraint into the algorithm during segmentation. Another, the statistical approaches are increasingly used. Among them, Markov random fields (MRF)-based methods are of most impor- tance due to well modeling the prior information [5], [6], but they poorly deal with fuzziness. Furthermore, only hard segmentation was obtained with these methods.
The fuzzy-based methods and MRF-based methods have their respective advan- tages. It can be predicated that integrating the fuzzy set theory with MRF theory will create wonderful results. The mixing of Markov and fuzzy approaches is discussed in [7]-[9]. Only two-class segmentation is discussed by adding a fuzzy class [7], [8]. H.
Caillol and F. Salzenstein only had discussion about generalization to multi-class segmentation. S. Ruan et al used the fuzzy Markov Random Field (FMRF) as a prior to segmented MR medical image [9], which is a multi-class problem. However, only two-tissue mixtures are considered. Three-tissue or more-tissue mixtures are not con- cerned. The idea of merging the fuzziness and randomness is to be refreshed. The new concept of FMRF based on fuzzy random variable should be proposed. Every pixel is considered as fuzzy case, and is the mixture of all the classes.
The paper is organized as follows. In section 2 some preliminaries about our model are mentioned. The concept of FGRF is represented in Section 3. Our model based on FGRF is described in section 4. Section 5 gives the algorithm and some experiments.
The final section is concerning conclusions and discussion on this paper.
2 Preliminaries
Fuzzy set theory is the extension of conventional set theory. It was introduced by Prof. Lotfi A. Zadeh of UC/Berkeley in 1965 to model the vagueness and ambiguity.
Given a set U , a fuzzy set à in U is a mapping from U into interval [0,1], i.e., whereÃ(x) is called membership function. Given is the membership value for element All fuzzy sets in U are denoted by
Fuzzy set à in U is denoted by If the set U is finite, then fuzzy set à can be written as or as a fuzzy vector We always consider that the fuzzy set is the same nota- tion as its membership function.
Uncertainty of data comes from fuzziness and randomness modeled by the random variables and the fuzzy sets respectively. Fuzzy random variable can model the two kinds of uncertainty first proposed by Kwakernaak [10]. In the definition, fuzzy ran- dom variable is a fuzzy-valued mapping. It is necessary to define the probability of fuzzy event, thereby.
The probability of fuzzy events in the probability space is defined as
A New Algorithm Based on Fuzzy Gibbs Random Fields for Image Segmentation 165 Where E denotes the expectation. If the sample space is the discrete set, then
Generally is called primary probability.
3 Fuzzy Markov Random Fields
Traditional MRF-based segmentation algorithm requires modeling two random fields.
For S = {1,2,ããã,n}, the set of pixels, is unobservable MRF, also called the label field. The image to be segmented is a realization of the observed random field Random variable must be generalized to fuzzy random variables for treating the vagueness. In this case, each pixel corresponds to a fuzzy set of label field L = {1,2,ããã,k}. Soft segmentation can be realized, and the final result may be obtained flexibly by many defuzzification methods.
In detail, each pixel i is attached with a fuzzy set represented by a k-dimensional fuzzy vector where is the membership value of the ith pixel to
the sth class, and the constraint is introduced. is called fuzzy random field if each is fuzzy random variable. Fuzzy random field is just
FMRF if the family of fuzzy random variables is constrained by Markovianity.
When no fuzziness is considered, is composed of all the indi- cator functions of each label, i.e., includes no fuzzy cases. The fuzzy random variables will certainly degenerate into classic random variables without fuzziness.
It is essential to know the joint probability of fuzzy event
where each is fuzzy set. The probability P(X = x) is given in a Gibbs form [6]
where U(x) stands for the energy function and Z is the normalizing constant.
The family of fuzzy random variables is said to be a fuzzy Gibbs random field (FGRF) if and only if its joint probability obeys the Gibbs form.
4 FGRF Model
When the image segmentation is formulated in a Bayesian rule, the goal is to estimate prior X from a posteriori The prior and the likelihood distribution are presented respectively. We adopt the MAP estimation as the statistical criterion.
166 G. Yan and W. Chen
The FGRF is used as the prior to describe the context information. A fuzzy Potts model is to be established based on the classical Potts model. Usually the neighbor- hood system V is the spatial 8-connexity. In fuzzy Potts model, only pair-site clique potential is considered and defined as
where is parameter, and is the fuzzy set in its labels. The distance between the two fuzzy sets is measured by the hamming distance
The larger the hamming distance, the larger the difference between two neighboring sites is, and the larger the pair-site clique potential is too.
The subtle difference between two neighboring sites is taken into account in FGRF.
It is concluded that FGRF is more powerful and flexible than CGRF. When no fuzzi- ness is considered, we yield the classical Potts model denoted by the fuzzy Potts model and the classical Potts model are a pair of random field that hold the same parameter thereby.
It is necessary to define the membership function. The membership function is de- veloped using geometrical features [11]. Let be the number of neighborhood pixels of the candidate pixel belonging to class j . The membership function is defined as
where denotes the degree of this pixel belonging to class j. We assume that each class obeys normal form with the mean and variance
If the fuzzy random variable takes a value Then the dis- tribution of conditional on also obeys normal form with mean and variance To calculate the likelihood distribution we made the assumptions that the observation components are conditionally independent given X, and the distribu- tion of each conditional on X is equivalent to its distribution conditional on Thus
The parameter is estimated in MPL method. The concavity of pseudolikelihood function determines simple and fast computation. It is key to deduce a formula so that MPL estimation can be implemented.
A New Algorithm Based on Fuzzy Gibbs Random Fields for Image Segmentation 167 For Potts model, its distribution is
where
Let be the numbers of label different from the candidate pixel in its neighbor. The MPL estimation of parameter is obtained by solving the following equation
where is the expectation with respect to conditional distribution and On the other hand, it is the concavity of the pseudolikelihood function that MPL estimation is assured to be unique [12].
5 Algorithm and Experiments
When the MAP is adopted as the segmentation criteria, the goal is to find X that maximize the posteriori, i.e., minimizing the posterior energy
where and were discussed in section 4. The parameter was esti- mated in pseudolikelihood method using Eq.(3). The unknown parameters is denoted by
The FGRF is denoted by the whose no fuzzy case is CGRF It is easy to understand that the CGRF builds a bridge between the prior FGRF and the observed image. In detail, our algorithm is described in the follow steps:
1.
2.
3.
4.
Set the class number. The initial segmentation is obtained in the k-means clustering procedure, and then the parameters can also be initialized;
Estimate the parameter and update during each iteration.
Fuzzify the CGRF and obtain the initial value of FGRF
Update FGRF X using iterated conditional modes (ICM) by solving Eq. (5)
168 G. Yan and W. Chen
5.
6.
7.
Defuzzify the using MMP and yield an updated CGRF Update the parameter using the empirical means and variances.
Repeat the step 4)-6) until convergence.
Our algorithm is tested on both simulated MR images from the Brain Web Simu- lated Brain Database at the McConnell Brain Imaging Center of the Montreal Neuro- logical Institute (MNI), McGill University, and real MR images. Simulated brain images are corrupted with different noise level. The control algorithms are the classi- cal GRF (CGS), maximum likelihood (ML) and fuzzy c-mean (FCM) algorithm.
Fig. 1 presents a comparison results. For FGS algorithm, the original image has been divided into the distinct and exclusive regions. Moreover it has more advantages than the control algorithms, such as smooth and continuous boundary and no noise. Fig. 2 present a comparison of segmentation results for image corrupted with 9% noise level. It is nearly no ability for FCM or ML algorithms to filter the noise. It is doubt- less that FGS algorithm is much more powerful than CGS algorithm for filtering noise.
To further testify the powerful property in filtering the noise, our method is also realized to segment a simulated image corrupted greatly with unknown noise level.
Fig. 3 shows that only our algorithm can obtain the correct segmentation.
To measure the robustness of the algorithms, the overlap metric is utilized as the criteria. The overlap metric is a measure for comparing two segmentations that is defined for a given class assignment as the sum of the number of pixels that both have the class assignment in each segmentation divided by the sum of pixels where either segmentation has the class assignment [13]. Larger metric means more similar for results. The segmented images corrupted by different noise level are compared with the no noise image using the different algorithms. Experiments show that our algorithm is much more robust than the others. Table 1 gives the overlap metrics of white matter (WM), gray matter (GM) and cerebrospinal fluid (CSF). It is satisfied that the overlap metric of our algorithm varies slowly with the noise level increasing, i.e., our algorithm is insensitive to noise.
Fig. 1. Comparison of segmentation results on real clinical MR image. (a) Original image, (b) using FGS, (c) using CGS, (d) using FCM
A New Algorithm Based on Fuzzy Gibbs Random Fields for Image Segmentation 169
Fig. 2. Comparison of segmentation results on stimulated MR image, (a) Original images with 9% noise level, (b) using FGS, (c) using CGS, (d) using ML, (e) using FCM
Fig. 3. Comparison of segmentation results on general stimulated image. (a) Original image, (b) using FGS, (c) using CGS, (d) using FCM
6 Conclusion and Discussion
The proposed algorithm takes into account the fuzziness and the randomness simulta- neously. Each pixel is modeled by fuzzy random variable. The FGRF is used to ob- tain the contextual information. All the experiments show that our algorithm can obtain accurate segmentations. But the intensity inhomogeneity is not taken into ac- count. We will try to settle this problem in the following work.
170 G. Yan and W. Chen
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Improved Fiber Tracking for Diffusion Tensor MRI
Mei Bai, Shuqian Luo
Capital University of Medical Sciences, Beijing 100054,China baimei12@yahoo.com
Abstract. Diffusion tensor MRI is currently the only imaging method that can provide information about water molecule diffusion direction, which reflects the patterns of white matter connectivity in the human brain. This paper presents a fiber-tracking algorithm based on an improved streamline tracking technique (STT). Synthetic datasets were designed to test the stability of the fiber tracking method and its ability to handle areas with uncertainty or isotropic tensors. In vivo DT-MRI data of the human brain has also been used to evaluate the performance of the improved STT algorithm, demonstrating the strength of the proposed technique.
1 Introduction
Diffusion tensor MRI (DT-MRI) is an in vivo imaging modality with the potential of generating fiber trajectories of the human brain to reflect the anatomical connectivity.
Furthermore, the various water diffusion information provided by DT-MRI can reflect microstructure and texture characteristics of the brain tissues [1,2,3]. Thus far, there is no “gold standard” for in vivo fiber tractography [4]. In vitro validation of the fiber tract obtained by DT-MRI has been attempted histologically [5,6], but sample dissection, freezing, dehydration, and fixation can potentially change the microstructure of the tissue and distort the histological sample. Significant advances have been achieved in recent years by using the tract-tracing methods based on chemical tracers, in which chemicals agents are injected and their destinations are confirmed [7]. Due to the nature of the experiment design, these techniques are not suitable for in vivo studies. Quantitative validation of the virtual fiber bundles obtained by DT-MRI is an important field of research. The purpose of this paper is to describe an improved STT algorithm for fiber tracking, which is validated with both synthetic and in vivo data sets.
2 Material and Methods