Geometric Flows over Lie Groups

1. Yaniv Gur and Nir Sochen Geometric Flows over Lie Groups Department of Applied Mathematics Tel-Aviv University, Israel HASSIP, September 2006, Munich

2. Motivation • Diffusion Tensor MR imaging (DTI) • Structure Tensor in imaging • Continuous Mechanics: Stress, Strain, etc. MIA, September 06, Paris

3. Diffusion Imaging • Self Diffusion = Brownian Motion of water molecules. • Incellular tissuethe self diffusion is influencedby cellular compartments. • Water molecules aremagnetically labeledaccording to their position along an axis. • The signal is acquired after adiffusion timeperiod and depends on the displacement projection along this axis. MIA, September 06, Paris Stejskal and Tanner (J. chem. Phys, 1965)

4. Neuron Myelin Axon Axon White Matter Anisotropy - The diffusion depends on the gradient direction MIA, September 06, Paris

5. Diffusion Anisotropy • The diffusion profile is modeled asadiffusion tensor. • Measurements ofat least 6non-collineardirections are needed for unique solution. E –Signal attenuation D –Diffusion Tensor q–Applied gradient direction Basser et al. (Biophys. J., 66, 1994 ) MIA, September 06, Paris

6. DiffusionTensor Imaging (DTI) MIA, September 06, Paris

7. Superior Longitudinal Fasciculus Uncinate Fasciculus Corpus Callosum & Cingulum Inferior Longitudinal Fasciculus Corona Radiata Fiber Tracking MIA, September 06, Paris

8. Pre operative planning Front View Rear View Top View Side View Courtesy of T. Schonberg and Y. Assaf MIA, September 06, Paris

9. Denoising Tensors via Lie Group Flows Outline: • Tensor-valued images • Lie-group PDE flows - Principal Chiral Model - Beltrami framework • Lie-group numerical integrators • Synthetic data experiments • DTI demonstrations • Summary MIA, September 06, Paris

10. To each point of the image domain there is a tensor (matrix) assigned. • We treat tensors which belong to matrix Lie-groups. Tensor-valued images • Examples of matrix Lie-groups: O(N), GL(N), Sp(N), etc. MIA, September 06, Paris

11. Principal Chiral Model elements of the Lie-algebra, generators of the Lie-group, span the Lie-algebra structure constants the metric over the Lie-group manifold (killing form) MIA, September 06, Paris

12. The Abelian Case We use the exp map to write then and MIA, September 06, Paris

13. Lie-group PDE flows Equations of motion Gradient descent equation Isotropic Lie-group PDE flow MIA, September 06, Paris

14. Examples: Anisotropic Lie-group PDE flow MIA, September 06, Paris

15. Original O(3) tensor field Noisy tensor field Denoised tensor field - PCM Synthetic data experiments MIA, September 06, Paris

16. Synthetic data experiments The symplectic group: • The set of all (2N) X (2N) real matrices which obey the relation • The group is denoted Sp(2N,R). • We apply the PCM flow to a two-parameters subgroup of Sp(4,R). • Results are presented by taking the trace of the matrices. MIA, September 06, Paris

17. Synthetic data experiments Two parameters subgroup of Sp(4,R) Image=Trace original field noisy field restored field MIA, September 06, Paris

18. - function formulation Equations of motion Gradient descent equations MIA, September 06, Paris

19. Principle bundles • Matrix Lie-group valued images may be described as a principal bundle • A specific assignment of a Lie group element to a point on the base space (the image manifold) is called a section MIA, September 06, Paris

20. Principle bundles • The metric in the image domain is Euclidean • The metric over the fiber (killing form) is It is negative definite for compact groups (e.g, O(N)) • The metric over the principle bundle is • Calculation of the induced metric yields MIA, September 06, Paris

21. Beltrami framework Variation of this action yields the equations of motion Gradient descent equations MIA, September 06, Paris

22. Lie-group numerical integrators • The Beltrami flow may be implemented using directly the parameterization of the group. In this case we may use finite-difference methods. • It may also be implemented in a “coordinate free” manner. In this case we cannot use finite-difference methods. Let then . MIA, September 06, Paris

23. Lie-group numerical integrators • We may use Lie-group numerical integrators, e.g.: Euler Lie-group version time step operator. • Derivatives are calculated in the Lie-algebra (linear space) using e.g., finite difference schemes. MIA, September 06, Paris

24. The DT-MRI data is represented in terms of a 3x3 positive-definite symmetric matrices which forms a symmetric space Polar decomposition Is the group of 3x3 diagonal positive-definite matrices We may use our framework to regularize and separately DT-MRI regularization via Lie-group flows MIA, September 06, Paris

25. Synthetic data Original P3 field Noisy P3 field Denoised directions and eigenvalues Denoised directions MIA, September 06, Paris

26. DTI demonstration MIA, September 06, Paris

27. Summary • We propose a novel framework for regularization of Matrix Lie groups-valued images based on geometric integration of PDEs over Lie group manifolds. • This framework is general. • Using the polar decomposition it can be applied to DTI images. • An extension to coset spaces (e.g., symmetric spaces) is in progress. Acknowledgements We would like to thanks Ofer Pasternak (TAU) for useful discussions and for supporting the DTI data. MIA, September 06, Paris

28. Running times • The simulations were created on an IBM R52 laptop with 1.7 Ghz processor and 512 MB RAM. • Regularization of 39x45 grid using “coordinates Beltrami” takes 3 seconds for 150 iterations. • The same simulation using “non-coordinates Beltrami” takes 35 seconds for 150 iterations. MIA, September 06, Paris