toma multiforme, (b) Fibrillary Astrocytoma. IV. RESULTS AND DISCUSSION. Some of the results are shown in Figures 2 and 3. In Fig- ure 2a the low relaxation ...
A Multi-strategy Method for MRI Segmentation M. Martín-Landrove1,2, M. Paluszny3, G. Figueroa4, W. Torres5, and G. Padilla4 1
Centro de Física Molecular y Médica, Universidad Central de Venezuela, Caracas, Venezuela 2 Centro de Diagnóstico Docente Las Mercedes, Caracas, Venezuela 3 Escuela de Matemáticas, Universidad Nacional de Colombia, Medellín, Colombia 4 Centro de Geometría, Universidad Central de Venezuela, Caracas, Venezuela 5 Fundación Instituto de Ingeniería, Caracas, Venezuela
Abstract—An accurate method for T2-weighted MRI segmentation according to tissue transversal magnetization decay rates is presented. By means of a sequence of geometric image filters a classification of the pixels’ intensity decay curves is provided. This can be done through a double strategy: First a log-convexity filter is applied in order to regularize image intensity decay by adjusting its geometrical properties to those that are expected from noiseless data, i.e., monotonous and convex behavior. In doing so, image noise is somewhat filtered and controlled. Data points are fitted by an over determined interpolation procedure. Decay rate distributions are obtained and tissue classification is performed by means of the determination of principal decay rates or decay modes using a suitable mathematical morphology operator, i.e., watershed or similar. Image segmentation is performed by linear regression analysis on a pixel by pixel basis assuming that the pixel intensity decay is composed by a linear superposition of the decay modes previously obtained from the decay rate distribution function. The main advantage of the proposed multi-strategy approach rests in the accuracy and speed of calculation with respect to other methods such as Inverse Laplace Transform algorithms. The method could be easily extended to any exponentially decaying set of images such as diffusion-weighted MRI. Keywords—Image resonance imaging, morphology.
segmentation, filtering, magnetic pattern recognition, mathematical
I. INTRODUCTION The determination of relaxation rate distributions for T2weighted Carr-Purcell-Meiboom-Gill (CPMG) MRI has been previously used [1-3], for tissue classification and tumor segmentation, particularly for obtaining nosologic maps of tumoral lesions in brain [4]. Those efforts relay on the application of simulated annealing and Metropolis algorithm to perform an inverse Laplace transform on relaxation data and whose details are discussed elsewhere [1,3]. Even though these methods are extremely precise and robust for the determination of relaxation rate distributions, they are also extremely slow and require some adjustments and side processes to be applicable on a patient basis. The decay of
pixel intensity through the set of CPMG T2-weighted MR images can be modeled by a discrete sum of positive exponential functions [5]. The fact that pixel intensity sampling is made at equally spaced time intervals, transforms the initial fitting problem into a problem of finding the solutions of a set of non linear polynomial equations [6]. In the present work, transversal relaxation rate distribution functions obtained by solving analytically [7] these equations are used for segmentation of tumor lesions in MR brain images.
II. MATERIALS AND METHODS A. Image Measurement Multi-echo T2-weighted images were acquired using Carr-Purcell-Meiboom-Gill (CPMG) sequence with a total of 16 equally separated echoes, starting at TE = 22 ms. To cover the totality of the tumoral lesion, images for 8 axial slices were obtained, each one 5 mm thick. Pixel intensity is commonly assumed to behave as a single exponential given by
p n = pO exp(− nTER2 )
(1)
where R2 = 1/T2, n being the echo index and T2 the transversal relaxation time. B. The Partial Volume Problem A common situation in MR images is that even when they exhibit a very high spatial resolution axially, i.e., over the 2D image, the spatial resolution in the longitudinal direction, i.e., related to slice width, could be very low. As a consequence, it can be assumed that there could be a mixture of tissues within the image voxel, i.e., a partial volume problem [8,9], and in correspondence a mixture of relaxation rates R2. In that case, (1) can be replaced by a more refined equation to describe image intensity behavior within the voxel
O. Dössel and W.C. Schlegel ( Eds.): WC 2009, I FMBE Proceedings 25/IV, pp. 1222–1225, 2009. www.springerlink.com
A Multi-strategy Method for MRI Segmentation
1223
N
p n = b + ∑ C i exp(− n α λi )
(2)
i =1
where α stands for the image acquisition parameter, i.e., TE, Ci is the proportion of tissue i in the voxel, λi represents its characteristic decay parameter, b is a baseline correction to the pixel intensity and N is the maximum number of tissues that could be present in the voxel. C. Analytic Solution of Non Linear Polynomial Equations As shown in [7], equation (2) can be written as: N
p n = b + ∑ Ci X i =1
n i
(3)
where
X i = exp(− αλi )
⎡ q3 ~ ⎢ M = ⎢q 4 ⎢⎣ q5
(8)
− q2 − q3 − q4
q1 ⎤ q 2 ⎥⎥ q3 ⎥⎦
(9)
a modified Toeplitz matrix,
X1 + X 2 + X 3 ⎡ ⎤ G ⎢ Z = ⎢ X 1 X 2 + X 2 X 3 + X 1 X 3 ⎥⎥ X1X 2 X 3 ⎣⎢ ⎦⎥
(10)
a set of symmetric functions, and (4)
The equally spaced sampling of the magnetization decay allows for a polynomial representation of each data point according to (3). If it is assumed [3,4], that at most 3 tissues are present in each voxel; normal or unaffected tissue, lesion tissue and cerebrospinal fluid, CSF, a set of non linear polynomial equations [6] can be written:
p1 = b + C1 X 1 + C 2 X 2 + C 3 X 3 p 2 = b + C1 X 12 + C 2 X 22 + C 3 X 32
⎡q ⎤ G ⎢ 4⎥ Q = ⎢ q5 ⎥ ⎢⎣ q 6 ⎥⎦
pi − pi +1 ≡ qi
(11)
Solutions of (7) determine a set of non linear algebraic equations:
Z 1* = X 1 + X 2 + X 3 Z 2* = X 1 X 2 + X 2 X 3 + X 1 X 3
(12)
Z = X1X 2 X 3 * 3
(5)
# 7 p 7 = b + C1 X 1 + C 2 X 27 + C 3 X 37
Solutions of set (12) can be obtained as the roots of the cubic equation: X 3 − Z 1* X 2 + Z 2* X − Z 3* = 0
(13)
Solutions to the set (5) must fulfill the following conditions, imposed by the physical conditions of image acquisition:
The rest of the variables can be calculated by replacing the solutions of (13) into the initial set (5).
C i ≥ 0, b ≥ 0
D. Image Filtering
0 ≤ Xi
