Bias Field Estimation and Adaptive Segmentation of MRI ... - CiteSeerX

Bias Field Estimation and Adaptive Segmentation of MRI Data Using a Modi ed Fuzzy C-Means Algorithm M. N. Ahmed , S. M. Yamany, A. A. Farag, and T. Moriarty  Univ. of Louisville, E.E. Dept., Louisville, KY 40292 E-mail:fmohamed,[email protected] Department of Neurological Surgery, University of Louisville, Louisville, KY 40292. +

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Abstract



images [3].

In this paper, we present a novel algorithm for adaptive fuzzy segmentation of MRI data and estimation of intensity inhomogeneities using fuzzy logic. MRI intensity inhomogeneities can be attributed to imperfections in the RF coils or some problems associated with the acquisition sequences. The result is a slowly-varying shading artifact over the image that can produce errors with conventional intensity-based classi cation. Our algorithm is formulated by modifying the objective function of the standard fuzzy c-means (FCM) algorithm to compensate for such inhomogeneities and to allow the labeling of a pixel (voxel) to be in uenced by the labels in its immediate neighborhood. The neighborhood eect acts as a regularizer and biases the solution towards piecewise-homogeneous labelings. Such a regularization is useful in segmenting scans corrupted by salt and pepper noise. Experimental results on both synthetic images and MR data are given to demonstrate the eectiveness and eciency of the proposed algorithm.

The removal of the spatial intensity inhomogeneity from MR images is dicult because the inhomogeneities could change with dierent MRI acquisition parameters from patient to patient and from slice to slice. Therefore, the correction of intensity inhomogeneities is usually required for each new image. In the last decade, a number of algorithms have been proposed for the intensity inhomogeneity correction. Meyer et al.[5] presented an edge-based segmentation scheme to nd uniform regions in the image followed by a polynomial surface t to those regions. The result of their correction is, however, very dependant on the quality of the segmentation step. Several authors have reported methods based on the use of phantoms for intensity calibration. Wicks et al.[4] proposed methods based on the signal produced by a uniform phantom to correct for MRI images of any orientation. Similarly, Tincher et al.[6] modeled the inhomogeneity function by a second-order polynomial and tted it to a uniform phantom-scanned MR image. These phantom approaches, however, have the drawback that the geometry relationship of the coils and the image data is typically not available with the image data. They also require the same acquisition parameters for the phantom scan and the patient. In addition, these approaches assume the intensity corruption eects are the same for dierent patients, which is not valid in general [3].

1 Introduction Spatial intensity inhomogeneity induced by the radio frequency (RF) coil in magnetic resonance imaging (MRI) is a major problem in the computer analysis of MRI data. Such inhomogeneities have rendered conventional intensity-based classi cation of MR images very dicult, even with advanced techniques such as nonparametric, multichannel methods.[1]-[2]. This is due to the fact that the intensity inhomogeneities appearing in MR images produce spatial changes in tissue statistics, i.e. mean and variance. In addition, the degradation on the images obstructs the physician's diagnoses because the physician has to ignore the inhomogeneity artifact in the corrupted  This work was supported in part by grants from the Alliant Health Care System and the NSF (ESC9505674)

The homomorphic ltering approach to remove the multiplicative eect of the inhomogeneity has been commonly used due to its easy and ecient implementation [2]. This method, however, is eective only on images with relatively low contrast. Some researchers [6][7] reported undesirable artifacts with this approach. Dawant et al.[7] used operator-selected reference points in the image to guide the construction of a 1

2 Background

thin-plate spline correction surface. The performance of this method substantially depends on the labeling of the reference points. Considerable user interactions are usually required to obtain good correction results. More recently, Gilles et al.[8] proposed an automatic and iterative B-spline tting algorithm for the intensity inhomogeneity correction of breast MR images. The application of this algorithm is restricted to MR images with a single dominant tissue class, such as the breast MR images. Another polynomial surface tting method [9] was proposed based on the assumption that the number of tissue classes, the true means and standard deviations of all the tissue classes in the image are given. Unfortunately, the required statistical information is usually not available.

The observed MRI signal is modeled as a product of the true signal generated by the underlying anatomy, and a spatially-varying factor called the gain eld. Yk = Xk Gk

8k 2 f1; 2; ::; N g

(1)

where Xk and Yk are the corrected and observed intensities at the kth voxel, respectively, Gk is the gain eld at the kth voxel, and N is the total number of voxels in the MRI volume. The application of a logarithmic transformation to the intensities allows the artifact to be modeled as an additive bias eld [1] yk = xk + k

A dierent approach used to segment images with intensity inhomogeneities is to simultaneously compensate for the shading eect while segmentating the image. This approach has the advantage of being able to use intermediate information from the segmentation while performing the correction. Recently, Wells et al.[1] developed a new statistical approach based on the EM algorithm to solve the bias eld correction problem and the tissue classi cation problem altogether in an alternative fashion. Guillemaud and Brady [10] further re ned this technique by introducing an extra class \other". There are two main disadvantages of this EM approach. First, the the EM algorithm is extremely computationally intensive, especially for large problems. Second, the EM algorithm requires a good initial guess for either the bias eld or for the classi cation estimate. Otherwise, the EM algorithm could be easily trapped in a local minimum, resulting in an unsatisfactory solution [3]. Xu et al.[11] followed Wells' approach and proposed a new adaptive fuzzy c-means technique to produce fuzzy segmentation while compensating for intensity inhomogeneities. Their method, however, is computationally intensive and is also very sensitive to noise.

8k 2 f1; 2; ::; N g

(2)

where xk and yk are the corrected and observed logtranformed intensities at the kth voxel, respectively, and k is the bias eld at the kth voxel. If the gain eld is known, then it is relatively easy to estimate the tissue class by applying a conventional intensitybased segmenter to the corrected data. Similarly, if the tissue classes are known, then we can estimate the gain eld. It may be problematic to estimate either without the knowledge of the other. We will show that by using an iterative algorithm based on fuzzy logic, we can estimate both.

3 Modi ed Fuzzy C-means (MFCM) Objective Function The standard FCM objective function for partitioning

fxk gNk into c clusters is given by =1

J=

c X N X i=1 k=1

upik jjxk ? vi jj2

(3)

where fvi gci=1 are the prototypes of the clusters and the array [uik ]=U repsents a partition matrix, U 2 U , namely

In this paper, we present a dierent approach for c X adaptive fuzzy segmentation of MRI data. Our algoUf u 2 [0 ; 1] j uik = 1 8k and ik rithm is formulated by modifying the objective funci=1 tion of the standard fuzzy c-means (FCM) algorithm (4) N X to compensate for such inhomogeneities and to allow 0 < uik < N 8ig the labeling of a pixel (voxel) to be in uenced by the k=1 labels in its immediate neighborhood. Taking into account the neighborhood eect acts as a regularizer and The parameter p is a weighting exponent on each biases the solution towards piecewise-homogeneous la- fuzzy membership and determines the amount of belings. Such a regularization is useful in segmenting fuzziness of the resulting classi cation. The FCM scans corrupted by salt and pepper noise. objective function is minimized when high membership 2

values are assigned to voxels whose intensities are 4.1 Membership Evaluation close to the centroid of its particular class, and low The constrained optimization in equation (7) will be membership values are assigned when the voxel data solved using Lagrange multipliers is far from the centroid [12].

 c X  p p uik Dik + uik i + (1 ? uik ) NR i=1 (8)

c X N  X

We propose to modify Eq.(3) by introducing a term Jm = i=1 k=1 that allow the labeling of a pixel (voxel) to be in uenced by the labels in its immediate neighborhood. As where mentioned before, the neighborhood eect acts as a Dik = jjyk ? k ? vi jj2 regularizer and biases the solution towards piecewisehomogeneous labelings. Such a regularization is useful and 0 1 in segmenting scans corrupted by salt and pepper noise. X The modi ed objective function is given by

i = @ jjyr ? r ? vi jj2 A : Jm =

c X N X

+N

R i=1 k=1

upik

X xr 2Nk

jjxr ? vi jj

2

!

Taking the derivative of Jm w.r.t uik and setting the result to zero, we have

 Jm

(5)



= pupik?1 Dik + Np upik i ?  =0 uik R uik =uik (11) Solving for uik we have

where Nk stands for the set of neighbors of xk and Nk . The neighbors eect term is controlled by the parameter . The relative importance of the regularizing term is inversely proportional to the signal to noise ratio (SNR) of MRI signal. Lower SNR would require higher value of the parameter . NR is the cardinality of

Since

Substituting Eq.(2) into Eq.(5), we have Jm =

c X N X

or

Pc



uik =

! p? 1

1

(12)

p(Dik + NR i )

j =1 ujk = 1 8k , then ! p?1 1 c X  =1  j =1 p(Djk + NR j )

=

upik jjyk ? k ? vi jj2

i=1 k=1 c X N  X

(10)

yr 2Nk

upik jjxk ? vi jj2

i=1 k=1 c X N  X

(9)

Pc  j =1

0 1 X upik @ jjyr ? r ? vi jj A (6)

p

 p? !p? 1

1

Djk + NR j )

(

(13)

1

1

(14)

Substituting into equation (12), the zero-gradient condition for the membership estimator can be rewritten as, 1 Formally, the optimization problem comes in the form uik = (15)  1 +N

R i=1 k=1

min

U; fvi gci=1 ; f k gNk=1

2

yr 2Nk

Jm

subject to

U

2U

Pc

Dik + NR i j =1 Djk + NR j

(7)

p?1

4.2 Cluster Prototype Updating

In the following derivation we use the standard Eucledian distance. Taking the derivative of Jm w.r.t vi and setting the result to zero we have;

4 Parameter Estimation

3 2X N p 77 66 uik (yk ? k ? vi ) + 77 66k N 7 64 X X  (yr ? r ? vi )5 upik

The objective function Jm can be minimized in a fashion similar to the standard FCM algorithm. Taking the rst derivatives of Jm with respect to uik , vi , and k and setting them to zero results in three conditions for Jm to be at a minimum. In the following subsections, we will derive these three conditions.

=0 (16)

=1

k=1

3

NR yr 2Nk

vi =vi

Solving for vi we have; v = i

6 Results





p  P k=1 uik (yk ? k ) + NR yr 2Nk (yr ? r ) In this section we describe the application of the P N p adaptive segmentation on synthetic images corrupted (1 + ) k=1 uik (17) with multiplicative gain and on brain MR images.

PN

All of the MR images shown in this section were obtained using a General Electric Signa 1.5 Tesla clinical MR imager. Results show that intensity In a similar fashion, taking the derivative of Jm w.r.t variations across patients, scans, and equipment k and setting the result to zero we have changes have been accomodated in the estimated # "X N c @ X bias eld without the need for manual intervention. upik (yk ? k ? vi )2 = 0 We implemented the MFCM algorithm on a Silicon i=1 @ k k=1 (18) Graphics Onyx Supercomputer. In all the examples, k = k Since only the kth term in the second summation de- we set the parameter  (the neighbors eect) to be 0.7. pends on k , we have # "X Figure 1 shows a synthetic test image. This image c @ 2 p contains a two-class pattern corrupted by a sinusoidal =0 uik (yk ? k ? vi ) @ gain eld of higher spatial frequency. The test image k i=1 (19) k = k is intended to represent two tissue classes, while the Dierentiating the distance expression, we obtain sinusoid represents an intensity inhomogeneity. This # " X model was constructed so that it would be dicult c c c X X to correct using homomorphic ltering or traditional =0 yk upik ? k upik ? upik vi  FCM approaches. As Shown in Figures 1c and 1d, the i=1 i=1 i=1 (20) k = k MFCM has succeeded in correcting and classifying the Thus, the zero-gradient condition for the bias eld es- data. timator is expressed as Pc up v Figure 2 shows the resutls of applying the MFCM ali  k = yk ? Pi=1 (21) gorithm c uikp to segement an image of an axialsectioned T1 i=1 ik MR brain. Strong inhomogeneities are apparent in the image. The MFCM algorithm segemented the image into three classes corresponding to background, gray The MFCM algorithm for correcting the bias eld and matter and white matter. Figure 3 illustrates intensity segmenting the image into dierent clusters can be pro les in the image before and after correction. These pro les correspond to the 128th column extract summarized in the following steps: from the original bias eld corrupted image and the c Step 1 Select initial class prototypes fvi gi=1 . Set corrected one using the MFCM algorithm. The gure demonstrates the ability of the MFCM algorithm to f k gNk=1 to zero. correct for the bias eld. Figure 4 shows another examStep 2 Update the partition matrix ple of T2 MR image where the MFCM was successful 1 u = (22) in recovering the bias eld and segmentating the image.

4.3 Bias Field Estimation

5 MFCM Algorithm

ik

Pc  Dik j =1

+ N

R i Djk + NR j

 p? 1

1

Figure 5 shows the results of applying the MFCM for the segementation of a noisy brain image. The Step 3 The prototypes of the clusters are obtained in results of using the traditional FCM without considthe form of weighted averages of the patterns ering the neighborhood eld eect and the MFCM   PN up (y ? ) +  P (y ? ) are presented. Notice that the segmentation of the r k k=1 ik k NR yr 2Nk r MFCM, which uses the the neighborhood eld eect, vi = P N p (1 + ) k=1 uik (23) is much less fragmented compared to the traditional FCM approach. As mentioned before, the relative Step 4 Estimate the bias term as follows importance of the regularizing term is inversely Pc up v proportional to the signal to noise ratio (SNR) of MRI i k = yk ? Pi=1 8k 2 f1; 2; ::; N g: c uikp signal. i=1 ik (24) Repeat steps 2-4 till convergence. We compared our results with the Expectation4

Maximization (EM) algorithm developed by Wells et ical Image Analysis, CVPR98, pp. 56-63, Sanata al.[1]. The MFCM algorithm produces similar results Barbara, California, 1998. as the EM algorithm with faster convergence and the added advantage that no user interaction is needed. [4] D.A.G. Wicks, G.J. Barker and P.S. Tofts, \Correction of intensity nonuniformity in MR images Correct estimates of region statistics are crucial for the of any orientation," Magnetic Resonance Imaging, convergence of the EM algorithm while in the MFCM, Vol. 11, pp. 183-196, 1993. approximate values for class centers are sucient. In noisy images, the MFCM technique greatly outper- [5] C.R. Meyer, P.H. Bland and J. Pipe, \Retroforms the EM algorithm as it compensates for noise spective correction of intensity inhomogeneities in by including a regularization term. While only 2D reMRI," IEEE Trans. Medical Imaging, Vol. 14, No. sults were presented in this summary, the MFCM can 1, pp. 36-41, 1995. be easily extended to 3D segmentation and bias eld correction. [6] M. Tincher, C.R. Meyer, R. Gupta and D.M. Williams, \Polynomial modeling and reduction of RF body coil spatial inhomogeneity in MRI," IEEE Trans. Medical Imaging, Vol. 12, No. 2, pp. 361-365, 1993. We have demonstrated a new modi ed fuzzy c-means (MFCM) algorithm for adaptive segmentation and [7] B. Dawant, A. Zijidenbos, and R Margolin \Corintensity correction of MR images. The algorithm rection of intensity variations in MR images for was formulated by modifying the objective function computer-aided tissue classi cation," IEEE Trans. of the standard fuzzy c-means (FCM) algorithm to Med. Imag., Vol. 12, pp. 770-781 1993. compensate for such inhomogeneities and to allow the labeling of a pixel (voxel) to be in uenced by the labels [8] S. Gilles, M.Brady, J. Declerck, J.P. Thirion and N. Ayache, \Bias eld correction of breast MR imin its immediate neighborhood. Taking into account ages," Proceedings of the Fourth International Conthe neighborhood eect acts as a regularizer and biases ference on Visualization in Biomedical Computing, the solution towards piecewise-homogeneous labelings. pp. 153-158, Hamburg, Germany, Sep. 1996. Such a regularization is useful in segmenting scans corrupted by salt and pepper noise. [9] C. Brechbuhler, G. Gerig and G. Szekely, \Compensation of spatial inhomogeneity in MRI based The MFCM segmentation increases the robustness on a parametric bias estimate," Proceedings of the and level of automation available for the segmentation Fourth International Conference on Visualization of MR images into tissue classes by correcting interscan in Biomedical Computing, pp. 141,146, Hamburg, intensity inhomogeneities. Via improved segmentation, Germany, Sep. 1996. this algorithm leads to an improvement in the quality of 3D reconstruction of brain structures, for visualiza- [10] R. Guillemaud and M. Brady, \Estimating the tion, surgical planning, disease research, and for other bias eld of MR images," IEEE Trans. Medical purposes. Imaging, Vol. 16, No. 3, pp. 238-251, 1997. [11] C. Xu, D. Pham, and Jerry Prince \Finding the brain cortex using fuzzy segmentation, isosurfaces, and deformable surfaces," Pro. of the XVth Int. [1] W.M. Wells, III, W.E.L. Grimson, R. Kikinis and Conf. on Information Processing in Medical ImagF.A. Jolesz \Adaptive segmentation of MRI data," ing (IPMI 97)., pp. 399-404, 1997. IEEE Trans. Med. Imag., Vol. 15, pp. 429-442, [12] J. Bezdek, \A convergence theorem for the fuzzy 1996. ISODATA clustering algorithms," IEEE Trans. on [2] B. Johnston, M.S. Atkins, B. Mackiewich and M. Pattern Ana. Machine Intell. (PAMI), 1980. Anderson, \Segmentation of multiple sclerosis lesions in intensity corrected multispectral MRI," IEEE Trans. Medical Imaging, Vol. 15, No.2, pp. 154-169, 1996. [3] S. Lai and M. Fang, \A new variational shape-fromorientation approach to correcting intensity inhomogeneities in MR images," Workshop on Biomed-

7 Conclusions

References

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(b)

Figure 2: Brain MRI example: (a) the original MR image corrupted with intensity inhomogeneities (b) the bias- eld corrected image using the proposed algorithm. The segmented image is shown in (c) while the recovered bias eld is displayed in (d).

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150.0

60.0

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Figure 1: Results of the MFCM on a two-class syn-

(a)

thetic image corrupted by sinusoidal bias eld. The original image is shown in (a). The second row presents the normalized bias eld and segmentation results at iteration 10. Third row presents the nal results of the bias eld and segmentation at iteration 20.

0.0 0.0

100.0

200.0

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(b)

Figure 3: The cross section at the 128th column extract from (a) the original bias eld corrupted image and (b) the corrected image using the MFCM algorithm. Intensity uniformity can be clearly noticed in (b). 6

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(b)

(c)

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(f)

Figure 4: Brain Tumor MRI example: (a) the orig- Figure 5: Brain Tumor MRI examples corrupted

inal MR image corrupted with intensity inhomogeneities (b) the bias- eld corrected image using the proposed MFCM algorithm. (c) Gray matter membership using traditional FCM. (d) Gray matter membership using MFCM. The segmented image using MFCM is shown in (e) while the recovered bias eld is displayed in (f).

with noise: The original MR images corrupted with salt and pepper noise are shown in (a) and (b). The segmented images using MFCM without any neighborhood consideration, are presented in (c) and (d). The segmented images using MFCM and neighborhood eect are presented in (e) and (f). See the text for a discussion on the results.

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