H. Salehian, G. Cheng, J. Sun, B. C. Vemuri Department of CISE University of Florida

Transcription

1 Tractography in the CST using an Intrinsic Unscented Kalman Filter H. Salehian, G. Cheng, J. Sun, B. C. Vemuri Department of CISE University of Florida

2 Outline Introduction Method Pre-processing Fiber Tracking Post-processing Experimental Results Patient 1 Patient 2 2

3 Introduction Our tractography method is based on the Intrinsic Unscented Kalman Filer (IUKF) [Cheng et al ISBI 12]. An extension of UKF [Malcolm TMI 10] to the space of symmetric positive definite matrices (P n ). Operations intrinsic to P n are employed. No explicit constraints are needed to guarantee the positive definiteness of the estimated diffusion tensors. 3

4 Properties 1. Preserves the positive definiteness of the diffusion tensors. 2. Fiber tracking & tensor estimation are done simultaneously. 3. No need to estimate the tensors all over the field. 4. Follows the streamline tracking framework in [Malcolm TMI 10]. 4 MR Signal Estima tion Tracki ng MR Signal Tensor Estimation Tracking

5 Method Pre-processing Fiber Tracking Post-processing 5

6 Pre-processing Each volume corresponding to a given magnetic gradient direction from the DW MRI dataset is denoised using the unbiased non-local means algorithm for Rician noise [Wiest et al MICCAI 08]. Since our tracking method is based directly on the MR Signals, no multi-fiber reconstruction over the whole image lattice is needed. A DTI reconstruction method [Barmpoutis et al ISBI 10] is employed at the seed points as the initialization for the IUKF. 6

7 Fiber Tracking IUKF combined with a streamline algorithm is used. At iteration step k for a single fiber, the reconstruction is performed by the IUKF using a bi-tensor model. The direction d k is computed as the major eigenvector of one of the tensors that is closer to the direction from the last step. The streamline algorithm then updates the position by computing x k+1 = x k + t d k. t is the step size. 7

8 Fiber Tracking (cont.) Tracker stops if one of these conditions are satisfied: 1. The angle between d k and the fiber direction exceeds an angle threshold (θ M ). 2. The signal strength at the current voxel is less than a threshold (S 0 ). 3. Fiber length exceeds a given threshold (L f ), in order to discard too large fibers. 4. The boundary of the dataset is reached. 8

9 IUKF Overview The IUKF [Cheng et al ISBI 12] is similar to the standard UKF [Malcolm TMI 10] with some key differences. The key difference being, some of the vector operations, e.g. the update of the posterior are replaced by the general linear (GL) group operation on P n. IUKF has three main components: 1. Observation model 2. State transition model 3. Filter The first two models are described here. 9

10 IUKF - Observation Model Bi-tensor diffusion model The covariance matrix of the observation noise for all the magnetic gradients is R = ri, where I is the identity matrix. Var. (i) D k (n) S k g n b n Description i-th diffusion tensor at step k (filter state) MR signal at step k (observation), for n-th magnetic gradient n-th magnetic gradient b-value corresponds to g n 10

11 IUKF State Transition Model For the bi-tensor model, the state transition model at step k is given by, F is the state transition GL-based operation, set to identity in our experiments. v k (i) : Gaussian distributed state transition noise for D k (i), in the tangent space T Dk (i) P 3. The covariance matrices of the state transitions are Q (1) = q 1 I and Q (2) = q 2 I, respectively. 11

12 Post-processing To discard unwanted fibers, we applied the following fiber reduction schemes: 1. Each fiber should pass through both the ROIs, one at the top of the brain, and the other in the brainstem. 2. An angle threshold criteria based on the structure of corticospinal tracts, the resulting fibers have to be sufficiently vertical. each fiber that has an angle less than a threshold (θ c ) to the X-Y plane, is discarded. 12

13 Experimental Results Tracking corticospinal fiber bundles from patient 1&2. Image size: 256 x 256 x 52, Voxel size: 1 x 1 x 2.6mm, b=1000mm/s 2 The parameter values for each neurosurgical case are listed below: Sunject t (mm) θ M (d) q 1 q 2 r θ c (d) S 0 L f Patient Patient

14 14 Patient 1 - ROIs

15 15 Patient 1 Fiber Bundle

16 16 Patient 1 Posterior View

17 17 Patient 1 Right View

18 18 Patient 1 Superior View

19 19 Patient 2 - ROIs

20 20 Patient 2 Fiber Bundle

21 21 Patient 2 Posterior View

22 22 Patient 2 Right View

23 23 Patient 2 Superior View

24 Acknowledgement This research was funded by the NIH grant NS to BCV. 24

25 References 25 Cheng, G., Salehian, H., Hwang, M.S., Howland, D., Forder, J., Vemuri, B.: A novel intrinsic unscented Kalman filter for tractography from HARDI. In: ISBI. (2012) Malcolm, Shenton, Rathi,: Filtered multi-tensor tractography, IEEE Trans. in Medical Imaging (TMI). (2010) Wiest-Daessle, N., Prima, S., Coupe, P., Morrissey, S., Barillot, C.: Rician noise removal by non-local means filtering for low signal-to-noise ratio mri: applications to dtmri. In: MICCAI. (2008) Barmpoutis, A., Vemuri, B.C.: A unified framework for estimating diffusion tensors of any order with symmetric positive-definite constraints. In: ISBI. (2010)