Subvoxel Segmentation and Representation of Brain Cortex Using Fuzzy Clustering and Gradient Vector Diffusion

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1 Subvoxel Segmentation and Representation of Brain Cortex Using Fuzzy Clustering and Gradient Vector Diffusion Ming-Ching Chang Xiaodong Tao GE Global Research Center {changm, research.ge.com SPIE Medical Imaging, 2010

2 Background and Motivation Motivation Geometric representation of brain cortex is a critical step to visualization and study of sulcal and gyral patterns morphometric measurement, such as cortical thickness, area, etc functional brain mapping A Brain Cross-Section Previously Reported Methods Geometric deformable models, Han et al 04, Zeng et al 98 Parametric deformable models, Xu et al 99, Dale et al 99

3 Proposed Method No deformable surface models Using tissue classification and gradient vector diffusion

4 Fuzzy C-Means Classification Algorithm An iterative process consists of two steps Assign soft label (membership functions) to each voxel (y[j] v µ k [j] = k ) 2, k = 1...C. (y[j] v 1 ) 2 +(y[j] v 2 ) 2 +(y[j] v 3 ) 2 Estimate mean intensity for each class j v k = (µ k [j]) 2 y[j], k = 1...C. j (µ k [j]) 2 Bias Field Artifact MR images exhibit slowly varying bias field. Intensity distribution of a tissue class is location dependent.

5 Bias Field Correction Bias field g is smooth modeled as a low order polynomial MR intensity is only tissue dependent under bias-free condition Divide brain volume into regular grid, and used average intenisty of white matter in qualified grids to estimate polynomial parameters

6 Gray Matter Boundary Surfaces Inner surface (IS) and pial surface (PS) are generated as iso-surfaces of the following functions at level 0: φ IS = µ wm µ gm φ PS = µ wm + µ gm µ csf Additional smoothing step can be applied to the iso-surfaces.

7 Center Surface of GM Idealized Case Center surface locates at the center of the GM sheet µ GM is maximal at points on center surface in its normal direction Finding ridge points of µ GM will give us center surface Center surface as an iso-surface at value 0 of: φ CS = (µ csf µ wm )(1 µ gm ) The second term helps improve the accuracy of center surface.

8 Vector Field Diffusion GM membership function does not always have ridges at the center of the gray matter sheet. To enhance the ridge-ness, we apply an anisotropic gradient vector diffusion to the GM membership function.

9 Isotropic Diffusion of a Vector Field Given a vector field (u, v, w), the solution to the following PDE is an isotropic diffusion of (u, v, w) 1 : du dt = η 2 u dv dt = η 2 v dw dt = η 2 w. We dropped the edge strength term in the original formulation since we are only interested in enhancing the ridges. (1) 1 Xu and Prince, IEEE TMI 1998, with edge strength term dropped

10 Anisotropic Diffusion of a Vector Field Anisotropic diffusion of a vector field (u, v, w) 2 : du dt = η div(α( c, s) u) dv dt = η div(α( c, s)) v) dw dt = η div(α( c, s) w), (2) with α(θ) = α( c, s) = { ( ) e 2 c s c s 1 if c 0 and s 0 0 if c = 0 or s = 0. (3) 2 Yu and Bajaj, IEEE CVPR 2004

11 Ridge Enhancement for Gray Matter Initial vector field is defined as: V ( x) = ( u( x), v( x), w( x) ) = ( µ gm ( x) µ gm ( x ) ) x x x x, where x is the neighborhood of x that has the smallest µ gm value. V ( x) field points toward the center ( ridge ) of the cortex. In the valley, local minimal of µ gm, V ( x) is zero.

12 Center Surface Extraction Anisotropic vector diffusion is applied to V ( x) for N = 50 (emperically determined) iterations. A skeleton strength map is computed from the result of anisotropic vector diffusion. A non-maximal suppression step is applied to further enhance the ridge-ness. The result of this step is R( x), whose value is between 0 and 1, with value close to 1 indicating ridge. Center surface is extracted as an isosurface at value 0 of the following function: φ CS = (µ csf ( x) µ wm ( x)) (1 R(~x)) (1 µ gm ( x))

13 Segmentation Results Segmentation results on a sagittal slice (top) and a coronal slice (bottom). From left: original image, µ WM without bias field correction, µ WM with bias field correction, and bias field inside brain mask.

14 Surface Reconstruction Surface reconstruction results: (a) inner surface, (b) pial surface, (c) a renderring of both surfaces, surfaces superimposed on axial (d), coronal (e), and sagittal (f) cross sections.

15 Thickness Measurement As a demonstration, we computed the cortical thickness and mapped it to the center surface. Compute distance transform from WM hard segmentation, D WM ( x); Cortical thickness is appoxiamated by t C ( x) = 2D WM ( x), for every x on center surface. Cortical Thickness

16 Summary and Discussion Summary A direct method for cortical surface reconstruction No deformable models fast (under 10 minutes on a typical PC) Future work Validation of the algorithm Examine the topology the surfaces Compare with published algorithms

17 Acknowledgement NA-MIC National Alliance for Medical Image Computing, an NIH National Centers for Biomedical Computing, through NIH grant U54 EB UNC IDEA Lab Image Display, Enhancement, and Analysis Lab of University of North Carolina Chapel Hill, through NIH grant R01 EB

18 Thank you for your attention!