FUZZY C-MEANS WITH VARIABLE COMPACTNESS

Abstract

Fuzzy c-means (FCM) clustering has been extensively studied and widely applied in the tissue classification of biomedical images. Previous enhancements to FCM have accounted for intensity shading, membership smoothness, and variable cluster sizes. In this paper, we introduce a new parameter called “compactness” which captures additional information of the underlying clusters. We then propose a new classification algorithm, FCM with variable compactness (FCMVC), to classify three major tissues in brain MRIs by incorporating the compactness terms into a previously reported improvement to FCM. Experiments on both simulated phantoms and real magnetic resonance brain images show that the new method improves the repeatability of the tissue classification for the same subject with different acquisition protocols.

1. INTRODUCTION

The fuzzy c-means (FCM) algorithm [1] has been extensively used in medical image segmentation. It has been especially successfully at classifying major tissues from magnetic resonance images of the human brain [2,3]. FCM converges readily, is scale and shift invariant, and allows for the straightforward incorporation of multichannel data. Furthermore, FCM directly yields soft segmentations (in the form of member-ship functions) that are typically desirable intermediate data structures (as opposed to hard classifications) for further analysis [4]. The original algorithm has also been adapted to account for image shading [5], membership smoothness [6], and variable cluster size [7, 8].

Recently, Clark et. al. [9] observed that there exists inconsistency in tissue segmentations arising from the choice of pulse sequence in data acquisition. We have had similar observations in our experiments. The inconsistency is caused by the fact that the relative size of clusters can be considerably different among multi-parametric scans of the same subject. This is an undesirable property as we prefer the performance of the segmentation algorithm to be independent of the data acquisition process. In order to achieve consistent segmentations, we introduce a new parameter, called compactness, into the FCM framework to obtain a novel variational formulation that has one compactness parameter per class predetermined for different acquisitions. We name the new algorithm FCM with variable compactness (FCMVC). The same idea is also incorporated into an enhanced variant of FCM — Fuzzy and Noise Tolerant Adaptive Segmentation Method (FANTASM) [6], which yields FANTASM with Variable Compactness (FVC).

In this paper, we describe an optimization process to solve the proposed FVC formulation. To validate the proposed method, we compare the performance of FVC against several existing methods, including FANTASM, FCM with a fuzzy covariance matrix [7] (FCMV), and Gaussian Mixture Model (GMM). Experiments show that FVC achieves improved consistency in segmentations when magnetic resonance brain images are acquired using two different pulse sequences (spoiled gradient (SPGR) and magnetization prepared rapid gradient echo (MP-RAGE)) from the same subject.

2. BACKGROUND

Mathematically, FCM is the solution of the following energy function:

$$J_{\text{FCM}}={\displaystyle \sum _{j\in \mathrm{\Omega }}{\displaystyle \sum _{k=1}^C{u}_{\mathit{\text{jk}}}^q{(y_i-{\upsilon }_k)}^2}}$$ (1)

where y_j is the observed image intensity at the j^th pixel, C is the number of classes, υ_k’s are class centroids, Ω is image domain, and u_jk is the membership function value of the j^th pixel for the k^th class. Membership functions must be non-negative, and satisfy the constraint ${\displaystyle {\sum }_{k=1}^C{u}_{\mathit{\text{jk}}}}=1,\forall j\in \mathrm{\Omega }$. The energy function is minimized if high membership values are assigned to observations close to centroids and low membership values to observations away from centroids. The parameter q is a weighting exponent and is constrained by q > 1. When q = 1, FCM reduces to the hard K-means algorithm with the membership functions taking binary values. As the value of q increases, the fuzziness of the membership functions also increase.

Since we are interested in segmenting three major tissues in the human brain – the cerebrospinal fluid (CSF), the gray matter (GM), and the white matter (WM), we consider a three class problem in the following manner. Let υ₁, υ₂, and υ₃ be the class centroids with υ₁ < υ₂ < υ₃. Let y_j vary over (−∞,+∞) and assume υ_k ’s are fixed. Recalling that $\forall j,{\displaystyle {\sum }_{k=1}^3{u}_{\mathit{\text{jk}}}}=1$, we can represent the membership functions [u_j1 u_j3 u_j2], for a particular y_j, as a point on the plane x + y + z = 1 (Fig. 1(a)) by taking u_j1 = x, u_j2 = z, u_j3 = y. We plot the membership points as a continuous function of y_j on this plane and show the 2D projection of the plane in Fig. 1(b) for the choices of q = 2 and q = 3. The three vertices in Fig. 1(b) starting from bottom left in clockwise direction are υ₁, υ₂, and υ₃, respectively. When y ≪ υ₁ or y ≫ υ₃, the curve tends to $\left[\frac{1}{3},\frac{1}{3},\frac{1}{3}\right]$, i.e, the center of the plane. In our case, for y_j ∈ [υ₁, υ₂], it is expected that y_j is a mixture of CSF and GM, with u_j3 being small. Similarly u_j1 is expected to be small for y_j ∈[υ₂, υ₃], as it is a mixture of GM and WM. Considering the effect of q on the memberships, we observe that, if q increases, WM membership u _j3 for y_j ∈ [υ₁, υ₂]increases, and so does the CSF membership u_j1 for y_j ∈[υ₂, υ₃] This demonstrates that the parameter q captures information about the variability of clusters. Based on this interpretation, we assume that if the WM cluster has large variance, then u _j3 for y_j ∈[υ₁, υ₂] should increase. And similarly, if CSF cluster has large variance, then u_j1 for y_j ∈[υ₂, υ₃] should increase.

Fig. 1.

(a)u_j1 + u_j2 + u_j3 = 1 plane. (b) Plot of u _jk(y_j ; q) on the plane. The left red line is u _j3 = 0 and the right red line is u_j1 = 0.

Several modifications of FCM have been proposed in the literature. A generalized L_p norm has been introduced in [10] to include variability of classes into energy function. Another way to take into account cluster variability is to incorporate a covariance matrix between classes, such as proposed with FCMV [7]. A simplified single-channel formulation is given as follows:

$$J_{\text{FCMV}}={\displaystyle \sum _{j\in \mathrm{\Omega }}{\displaystyle \sum _{k=1}^C{u}_{\mathit{\text{jk}}}^q}{\left(\frac{y_j-{\upsilon }_k}{{\sigma }_k}\right)}^2},$$ (2)

where σ_k is the variance of the k^th class. In the next section, we will describe a new approach to model the variability of clusters.

3. FCM WITH VARIABLE COMPACTNESS (FCMVC)

3.1. Problem Formulation

We introduce a fixed parameter p_k, k = 1, … ,C, in the FCM framework as minimization of the following function,

$$J_{\text{FCMVC}}={\displaystyle \sum _{j\in \mathrm{\Omega }}{\displaystyle \sum _{k=1}^C{\overline{u}}_{\mathit{\text{jk}}}^2{(y_j-{\upsilon }_k)}^{2p_k}.}}$$ (3)

Minimization of J_FCMVC gives the membership functions as,

$${\overline{u}}_{\mathit{\text{jk}}}=\frac{{(y_j-{\upsilon }_k)}^{-2p_k}}{{\displaystyle {\sum }_{l=1}^C{(y_j-{\upsilon }_l)}^{-2p_l}}}.$$

We note that p = [p₁ … p_C] = [1 … 1] is the same as FCM with q = 2. The plot of memberships ū_jk for a fixed set of υ_k and p_k are shown in Fig. 2. We observe that for ∀y_j ∈[υ₁, υ₂], ū_j3 with p₁ > p₂ > p₃ has increased from u _j3 with p = [1 1 1] (Fig. 2(a)). Similarly ∀_yj ∈[υ₂, υ₃], u¯ _j1 with p₁ < p₂ < p₃, has increased from u _j1 (Fig. 2(b)). It shows that with decreasing p_k, memberships increase for a fixed |y_j − υ _k|. So we infer that p_k is a measure of the compactness of cluster k. It is chosen to be large for small classes and small for large classes.

Fig. 2.

Plot of u¯_jk(y_j ; p), for different parameters p_k.

Based on the proposed method FCMVC, we now modify the FANTASM energy function [6] to FVC,

$$\begin{array}{c}{J}_{\text{FVC}}={\displaystyle \sum _{j\in \mathrm{\Omega }}{\displaystyle \sum _{k=1}^C{u}_{\mathit{\text{jk}}}^2{(y_j-g_j{\upsilon }_k)}^{2p_k}}}+\hfill \\ {\mathrm{\lambda }}_1{\displaystyle \sum _{j\in \mathrm{\Omega }}{\displaystyle \sum _{r=1}^2{{(D_r\star g)}_j}^2}}+{\mathrm{\lambda }}_2{\displaystyle \sum _{j\in \mathrm{\Omega }}{\displaystyle \sum _{r=1}^R{\displaystyle \sum _{s=1}^R{{(D_r\star D_s\star g)}_j}^2}}}\hfill \\ +\frac{\beta }{2}{\displaystyle \sum _{j\in \mathrm{\Omega }}{\displaystyle \sum _{k=1}^C{u}_{\mathit{\text{jk}}}^2}{\displaystyle \sum _{l\in N_j}{\displaystyle \sum _{m\ne k}{u}_{\mathit{\text{lm}}}^2}}}\hfill \end{array}$$

Ω, u_jk, p_k, y_j, υ_k have been described earlier. Here, g_j is a scalar gain field to account for image inhomogeneity, D_r and D_s are first order finite difference operators, R = 2 denotes vertical and horizontal directions, λ₁ and λ ₂ are regularizations on gain field, N_j is the set of first order neighbors of pixel j and β is a smoothing coefficient. λ₁, λ₂ and β are determined empirically.

Minimization of J_FVC can be performed using the following algorithm.

1. Choose suitable compactness parameters p_k (discussed in the following section).

2. Obtain initial estimate of υ_k.

3. Assume g_j = 1, ∀j ∈ Ω.

4. Compute membership functions,

   $$\begin{array}{c}{u}_{\mathit{\text{jk}}}=\hfill \\ \frac{{\left({(y_j-g_j{\upsilon }_k)}^{2p_k}+\beta {\displaystyle {\sum }_{l\in N_j}}{\displaystyle {\sum }_{m\ne k}{u}_{\mathit{\text{lm}}}^2}\right)}^{-1}}{{\displaystyle {\sum }_{k=1}^C{\left({(y_j-g_j{\upsilon }_k)}^{2p_k}+\beta {\displaystyle {\sum }_{l\in N_j}{\displaystyle {\sum }_{m\ne k}{u}_{\mathit{\text{lm}}}^2}}\right)}^{-1}}}.\hfill \end{array}$$
5. Update the centroids,

   $${\upsilon }_k=\frac{{\displaystyle {\sum }_{j\in \mathrm{\Omega }}{u}_{\mathit{\text{jk}}}^2{g}_j{(y_j-g_j{\upsilon }_k)}^{2p_k-2}{y}_j}}{{\displaystyle {\sum }_{j\in \mathrm{\Omega }}{u}_{\mathit{\text{jk}}}^2{g}_j^2{(y_j-g_j{\upsilon }_k)}^{2p_k-2}}}.$$
6. Update gain field coefficients by solving the spatially varying equation,

   $$\begin{array}{l}2y_j{\displaystyle \sum _{k=1}^C{u}_{\mathit{\text{jk}}}^2{p}_k{(y_j-g_j{\upsilon }_k)}^{2p_k-2}{\upsilon }_k=}\hfill \\ 2g_j{\displaystyle \sum _{k=1}^C{u}_{\mathit{\text{jk}}}^2{p}_k{(y_j-g_j{\upsilon }_k)}^{2p_k-2}{\upsilon }_k^2}\hfill \\ +{\mathrm{\lambda }}_1{(H_1\star g)}_j+{\mathrm{\lambda }}_2{(H_2\star g)}_j.\hfill \end{array}$$

7. If the algorithm converges, stop; otherwise go to step 4.

4. RESULTS

The compactness parameters are estimated based on a set of SPGR training data [4], for which we have manually selected landmarks on the cortical boundaries [11]. An exhaustive search was performed to estimate p, such that the segmentation isocontours generated by FVC were close to the landmarks. The compactness parameters estimated on the training data are p_S = [0.9, 1.00, 1.06]. Next we estimated the compactness parameters for MP-RAGE using a training set of SPGR and MP-RAGE data of the same subject. The parameters for MP-RAGE are estimated based on an exhaustive search so that the isocontours of MP-RAGE segmentation line up with the isocontours of SPGR using p_S. The estimated compactness for MP-RAGE was p_M = [1.25, 1.0, 0.95]. Variances ${\sigma }_k^2,k=1,2,3,$ from Eq. 2, were also estimated in exactly the same way. The estimated variances were found to be σ_S ² = [3.1, 1.44, 1] for SPGR and σ_M ² = [1, 1.52, 2.55] for MP-RAGE. We used these values of p_S, p_M, σ_S, σ_M on the test images.

We conducted two experiments to verify that FVC can improve the consistency of the segmentation of the same subject under the two acquisition protocols. Our first experiment involves two synthetic data sets which we used to simulate the same object imaged with MP-RAGE and SPGR imaging parameters. We generated the phantoms from a fuzzy classification truth model using the statistical model outlined in [12]. The phantoms possessed 5% noise and did not have any inhomogeneity. The misclassification rate, defined as the ratio of the # of correctly classified voxels against the total # of non-background voxels, is reported in Tab. 1 for classifying the MP-RAGE and SPGR phantoms using GMM (with variable priors and variances), FANTASM, FCM with Fuzzy Covariance matrix (FCMV) and FVC. Our second experiment involved a pair of real SPGR and MP-RAGE T1 images for five subjects. We computed the hard segmentations using each of the methods used in the first experiment and report the Jaccard coefficient between the hard segmentations, averaged over all three classes. We used smoothing and inhomogeneity correction while using GMM and FCMV. The results are in Tab. 2, while Fig. 3 shows one of the data sets.

Table 1.

Percent Misclassification rate for SPGR/MP-RAGE with Ground Truth for two simulated phantoms. M is MPRAGE and S is SPGR. FCMV and FVC represent FCM with Fuzzy Covariance matrix and FANTASM with Variable Compactness. The two rows correspond to two different phantoms.

GMM		FANTASM		FCMV		FVC
S	M	S	M	S	M	S	M
5.64	12.78	3.62	5.66	4.45	5.11	3.55	3.51
7.51	9.54	6.32	8.43	5.32	6.11	4.72	4.86

Table 2.

Jaccard Coefficients, averaged over all three classes, between SPGR and MP-RAGE hard segmentation of five subjects. See Tab. 1 for abbreviations.

#	GMM	FANTASM	FCMV	FVC
1	0.7383	0.5611	0.7781	0.7952
2	0.7047	0.4764	0.7501	0.7538
3	0.8134	0.6670	0.8391	0.8559
4	0.7707	0.5209	0.6363	0.7921
5	0.7377	0.4205	0.4173	0.7588

Fig. 3.

Comparison between hard segmentation of real MP-RAGE and SPGR images, subject #1. See Tab. 1 for abbreviations.

Tab. 1 shows that FANTASM has nearly the same misclassification rate as FVC on SPGR images, but fails to do as well on MP-RAGE images. From Fig. 3, we observe that GMM tends to over-estimate gray matter in SPGR, while FANTASM tends to over-estimate CSF in MP-RAGE. We also observe that FANTASM produces visually good results on SPGR data. It is also observed from Fig. 3 that FCMV performs better than FANTASM on MP-RAGE, but fails to do so on SPGR. Finally we can see that FVC performs better on both SPGR and MP-RAGE.

5. DISCUSSION AND CONCLUSION

A new compactness parameter in the FCM framework is introduced and we outlined an automatic fuzzy segmentation method based on the parameter. We have satisfied our primary goal of obtaining more similar segmentations from the same data acquired under differing protocols, SPGR and MP-RAGE. The proposed approach is currently designed for single channel data, it can be readily extended to a multichannel algorithm. In the future, we would like to develop a robust and fully automated framework for estimating the compactness parameters, as opposed to our current practice of estimation based on training.