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Masters Thesis Defense: University of Utah

Rician Noise Removal in Diffusion Tensor MRI

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- 1. Rician Noise Removal in Diffusion Tensor - MRI <ul><li>Thesis Defense </li></ul><ul><li>Saurav Basu </li></ul><ul><li>School of Computing </li></ul><ul><li>University of Utah </li></ul>
- 2. Organization <ul><li>Brief overview of DT MRI </li></ul><ul><li>Goals for this thesis </li></ul><ul><li>Motivation : </li></ul><ul><ul><li>why noise removal ? </li></ul></ul><ul><ul><li>why Rician noise ? </li></ul></ul><ul><ul><li>previous DT-MRI filtering methods </li></ul></ul><ul><li>Rician Bias Correction Filter </li></ul><ul><li>Results and discussion </li></ul><ul><li>Conclusion: summary, future work </li></ul><ul><li>Questions ? </li></ul>
- 3. March-April 2006 DT-MRI is the most recent in a series of astonishing breakthroughs in brain imaging
- 4. HyperStreamLines used to Visualize White Matter Fibres in the brain
- 5. Brief overview of DT-MRI <ul><li>Symmetric </li></ul><ul><li>Positive Definite </li></ul><ul><li>All eigenvalues are positive </li></ul>Diffusion Tensor Imaging technique to compute a 3x3 matrix (D) Characterizes diffusion of water across brain tissue Used to study structure of brain fibres Key: More diffusion along fibres than across fibres
- 6. <ul><li>Visualize the eigen values of D over the volume to infer connectivity and structure </li></ul>Tensor Orientation: Principle Eigen Vector Tensor Anisotropy: directional characteristics
- 7. How is the tensor computed ? A 0 A i g i Stejskal Tanner equation Known: b, g i Measured: A 0 , A i Find D Most Common: Linear Least Squares on
- 8. Goals for this Thesis: <ul><li>What is the best way to filter DT-MRI data? </li></ul><ul><li>How do current filtering methods compare ? </li></ul><ul><li>Is there a better way of doing filtering? </li></ul>Answer these questions.
- 9. Motivation <ul><li>Why is DT - MRI filtering important? </li></ul><ul><li>Why is it important to account for Rician noise in the filtering process? </li></ul>
- 10. <ul><li>DT MRI plagued by low SNR </li></ul><ul><ul><li>Multiple Scans needed to increase SNR </li></ul></ul><ul><ul><li>Issues: long acquisition time, patient comfort system throughput </li></ul></ul><ul><ul><li>Mis-registration issues (motion artifacts) </li></ul></ul><ul><ul><li>Partial voluming (volume averaging) : voxel covers a non homogeneous tissue region </li></ul></ul>Why DT - MRI filtering?
- 11. <ul><li>what is Rician noise? how does it arise in DT MRI? </li></ul><ul><li>how does it effect tensors? </li></ul><ul><li>previous filtering methods </li></ul>Why Rician noise removal?
- 12. <ul><li>DWI images are magnitudes of complex valued signals. </li></ul><ul><li>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. </li></ul>Rician noise in DT MRI ? gaussian magnitude where is zero mean , stationary Gaussian noise with standard deviation
- 13. Rician Noise A signal is said to be corrupted with Rician noise if the pdf of the noisy signal has a Rice distribution
- 14. p(x|A) A Rice Distribution
- 15. p(x) A Normal Distribution
- 16. 10000 samples , sigma=20
- 17. How does Rician noise affect estimated tensors? previous studies show noise trace and FA “ However, when we performed Monte Carlo simulations with Rician noise with diffusion tensors characteristic of those in the human brain we found FA and trace can be incorrectly estimated when tensors are aligned with gradient directions.” aligned tensors: noise FA trace
- 18. Tensor Splitting Gradient direction Tensor aligned with gradient direction
- 19. This tells us FA can be overestimated or underestimated depending on how a person sits inside the scanner ! Bottom Line! It is important to consider Rician noise in filtering process Can Seriously affect the validity of clinical studies using these FA estimates.
- 20. Previous filtering approaches 2 categories DWI space Tensor Space 1) Non linear smoothing for reduction of systematic errors. Parker(2000) 2) Constrained Variational approach Wang, Vemuri (2004) 2) Bayesian regularization using Gaussian markov random fields. Martin (2004) 1) Riemannian Space filtering Pennec (2004) Very effective techniques, but do not explicitly handling Rician noise as part of the filtering process. Others: Median filtering, K- Space(Fourier Domain) methods
- 21. <ul><li>DWI Space filter </li></ul><ul><li>Based on maximum a posteriori (MAP) approach to image reconstruction </li></ul><ul><li>( In statistics MAP estimation is used to obtain a point estimate of an unobserved quantity based on empirical data ) </li></ul>Rician Bias Correction Filter
- 22. MAP Image Reconstruction <ul><li>A Prior Model </li></ul><ul><li>A Likelihood or Noise Model </li></ul><ul><li>Optimization Scheme (maximize posterior) </li></ul>3 Key components
- 23. <ul><li>To estimate the clean value we want to maximize p(u|u 0 ) </li></ul>From Baye’s Rule: constant for a given noisy image u 0 MAP Formulation Given: Noisy Image u 0 estimate Output: Clean/Filtered Image u Known: p(u 0 |u) has a Rician distribution
- 24. posterior likelihood prior maximize with gradient ascent Capture some prior knowledge about the filtered image. Example: enforce smoothing criteria on the image Captures the noise model on the data Essentially says : What is the probability of the clean image given that i have a particular noisy image. For gradient ascent we need to take derivatives!
- 25. Likelihood Term Taking derivative w.r.t u , Rician attachment term or Bias correction term After Substituting for Rice pdf The Likelihood Term:
- 26. We use a Gibb’s prior with an energy functional which enforces a smoothness without blurring edges The Prior Term: Gibb’s prior Energy functional conductance weighing factor edge preserving smoothing prior
- 27. Combining the Rician correction term with the variational of the energy functional we get the update equation for the filtered image Derivative of likelihood term Variational of energy functional Implementation: Modify PDE Diffusion Filtering to use this modified update term 1. u = vector image of 7 or more DWIs 2 Bias correction term term computed independently for each component of the vector Note:
- 28. Results We compared 4 different filtering methods on both synthetic and real data sets DWI Space Tensor Space 1. Anisotropic Diffusion without Rician attachment 2. Rician Bias Correction filter 1. Anisotropic Diffusion in euclidean space. 2. Anisotropic Diffusion on the Riemannian manifold
- 29. Error Metrics <ul><li>Tensor Components </li></ul><ul><li>Fractional Anisotropy (FA) </li></ul><ul><li>Trace </li></ul>
- 30. <ul><li>10x10x4 volume of tensors </li></ul><ul><li>2 tensor orientations (along gradient and splitting the gradient directions) </li></ul><ul><li>Synthetic rician noise, baseline image intensity=250 </li></ul>Synthetic Data -1 Clean Noisy (SNR=15) Gradient Directions: ( 1 0 1 ) (-1 0 1 ) ( 0 1 1 ) ( 0 1 -1) ( 1 1 0 ) (-1 1 0 )
- 31. Aniso DWI Rician DWI DWI Space Filters
- 32. Euclidean Riemannian Tensor Space Filters
- 36. <ul><li>To check whether variability in directions affects results we generated a torus with tensors oriented in all possible directions. </li></ul><ul><li>Ran the filtering on the torus data set </li></ul>Synthetic Data Set -2 : Hollow Torus
- 37. Clean Tensor
- 38. Noisy Tensor (sigma=10)
- 39. Euclidean Filtering
- 40. Riemannian Filtering
- 41. Aniso DWI Filtering
- 42. Rician DWI Filtering
- 44. <ul><li>use repeated scans of the same subject. </li></ul>Real Data Results Issue: No ground truth data available for DT-MRI ! How do we evaluate filtering performance quantitatively? Solution: p(x/A) is the Rician pdf Maximize (Brent’s| Golden search method) ML Estimate: LIKELIHOOD FUNCTION
- 45. ML Estimator versus Averaging for generating Ground truth
- 51. <ul><li>5 scans of healthy volunteer </li></ul><ul><li>Resolution: 2 mm x 2 mm x 2 mm </li></ul><ul><li>3T scanner , scan time 12 mins. </li></ul>Real Data Filtering Results About the Real Data: Added Rician noise SNR levels of 10,15 and 20 with respect to white matter signal level. and ran our filtering methods.
- 52. Clean
- 53. Sigma=10 Euclidean Riemannian Aniso Rician Noisy Image
- 54. Noisy Image Euclidean Riemannian Aniso Rician
- 55. Noisy Image Euclidean Riemannian Aniso Rician
- 56. Noisy Image Euclidean Riemannian Aniso Rician
- 57. Noisy Image Euclidean Riemannian Aniso Rician
- 58.
- 59.
- 61. <ul><li>Rician Filter : best RMS error performance. </li></ul><ul><li>Real Data: Filtering DWIs better. </li></ul><ul><li>Riemannian filtering: overall performs poorly </li></ul>Discussion
- 62. Conclusions <ul><li>Bias effects of Rician noise </li></ul><ul><li>New Rician-bias correction filter </li></ul><ul><li>Systematic comparison </li></ul>Summary & Contributions
- 63. <ul><li>ML method: low noise DWIs </li></ul><ul><li>Filtering tools: </li></ul><ul><ul><li>Rician Bias Correction Filter </li></ul></ul><ul><ul><li>Riemannian Space Tensor Filter </li></ul></ul><ul><ul><li>Anisotropic Diffusion Filter on tensors </li></ul></ul><ul><ul><li>Anisotropic Diffusion Filter on DWIs </li></ul></ul>
- 64. <ul><li>DT-MRI Ground Truth: investigate ML methods </li></ul><ul><li>Noise effects: </li></ul><ul><ul><li>fiber-tractography, </li></ul></ul><ul><ul><li>diagnostic decisions. </li></ul></ul><ul><li>Rician Noise model : Tensor estimation </li></ul>Future Work:
- 65. Acknowledgments Advisor: Dr. P. T. Fletcher Committee: Dr. Ross T. Whitaker, Dr. Tolga Tasdizen Gordon for help with Deft and teem Josh for help with ITK Dr. Guido Gerig, Dr. Wei Lin from UNC for providing us the real DT-MRI data VIPER, NAMIC : Funding .
- 66. Thanks, Question?
- 67. Euclidean Space Gradient neighbors
- 68. Riemannian Space: Gradient neighbors

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