Rician Noise Removal in Diffusion Tensor MRI

1. Rician Noise Removal in Diffusion Tensor MRI • Saurav Basu, Tom Fletcher, Ross Whitaker • School of Computing • University of Utah

2. Why DT-MRI filtering? • DT-MRI plagued by low SNR • Multiple scans needed to increase SNR • Issues: long acquisition time, patient comfort system throughput • Noise in DT-MRI adversely affects tensor measurements used in clinical studies

3. Rician noise in DT-MRI • DW images are magnitudes of complex valued signals. • If the real and imaginary components of the signal are assumed to have a Gaussian noise, the resulting magnitude image will have Rician distributed noise. gaussian magnitude where is zero mean, stationary Gaussian noise with standard deviation

4. Rician Noise A signal is said to be corrupted with Rician noise if the pdf of the noisy signal has a Rice distribution Unlike the normal distribution the pdf is not symmetric about the true signal value A

5. p(x) A Rice Distribution

6. How does Rician noise affect estimated tensors? We performed Monte Carlo simulations with two cases: Tensor splitting gradient direction Tensor aligned with gradient direction

9. Previous filtering approaches 2 categories Tensor Space DWI space Anisotropic Diffusion Parker(2000) Riemannian Space filtering Pennec (2004) Bayesian regularization using Gaussian markov random fields. Martin (2004) Constrained Variational approach Wang, Vemuri (2004) Very effective techniques, but do not explicitly handling Rician noise as part of the filtering process.

10. Rician Bias Correction Filter • Based on maximum a posteriori (MAP) approach to image reconstruction • In statistics MAP estimation is used to obtain a point estimate of an unobserved quantity based on empirical data

11. Given an initial noisy image u0 we want to estimate the clean image u. • We know that p(u0|u) has a Rician distribution. • To estimate the clean value we want to maximize p(u|u0) From Baye’s Rule constant for a given noisy image u0

12. maximize with gradient ascent noise model (likelihood ) prior posterior prior: what is pdf of the unobserved data (clean image) which we are trying to estimate? noise model (likelihood) : what is the conditional probability of the observed data( noisy image) , given a particular value of the unobserved data (clean image)? posterior: The probability of the unobserved data (clean image) given the observed data (noisy image)

13. Rician likelihood term Using the Rician pdf for the noise model we get rician likelihood term Taking derivative w.r.t. u, Rician attachment Term

14. Combining with the prior: The Rician attachment term can be combined with any image prior. We use a Gibbs prior with Perona Malik Energy functional. edge preserving smoothing prior Gibb’s prior Perona Malik energy weighing factor conductance

15. Combining the Rician correction term with prior we get the update equation for the filtered image

16. Preliminary Results • Parameters optimized for RMS error in tensor components. • For both synthetic and real data we used 7 images for each slice (6 gradient directions + 1 baseline) We compared 4 different filtering methods on both synthetic and real datasets DWI Space Tensor Space Anisotropic Diffusion without Rician attachment 2. Rician Bias Correction filter Anisotropic Diffusion in Euclidean space Anisotropic Diffusion on the Riemannian manifold Error Metrics: RMS error in tensor components Fractional Anisotropy Trace

17. Synthetic Data Results • 10x10x4 volume of tensors • 2 tensor orientations (along gradient and splitting the gradient directions) • Synthetic rician noise Noisy (SNR=15) Clean

18. DWI Space Filters Rician DWI Aniso DWI

19. Tensor Space Filters Riemannian Euclidean

23. Real Data Results Issue: No ground truth data available for DT-MRI ! • we developed a method to estimate a ground truth data from repeated scans of the same object • if {xi} is a set of intensities for the same voxel in N repeated scans we find the ML estimate of the true value A by maximizing the log likelihood function: How do we evaluate filtering performance quantitatively? Solution: p(x|A) is the Rician pdf

24. Generated ground truth from 5 scans • added known Rician noise (SNR=10,15,20) • compared errors as before Clean Coronal Slice Noisy Coronal Slice(SNR=15)

25. DWI Space Rician DWI Aniso DWI Both Aniso-DWI and Rician DWI gave very good results with Rician being marginally better

26. Tensor Space Riemannian Tensor Euclidean Tensor