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

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

MIA, September 06, Paris Diffusion Imaging Self Diffusion = Brownian Motion of water molecules. In cellular tissue the self diffusion is influenced by cellular compartments. Water molecules are magnetically labeled according to their position along an axis. The signal is acquired after a diffusion time period and depends on the displacement projection along this axis. Stejskal and Tanner (J. chem. Phys, 1965)

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

MIA, September 06, Paris Diffusion Anisotropy The diffusion profile is modeled as a diffusion tensor. Measurements of at least 6 non-collinear directions are needed for unique solution. D – Diffusion Tensor q – Applied gradient direction Basser et al. (Biophys. J., 66, 1994 ) E – Signal attenuation

MIA, September 06, Paris Diffusion Tensor Imaging (DTI)

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

MIA, September 06, Paris Front ViewRear View Top ViewSide View Courtesy of T. Schonberg and Y. Assaf Pre operative planning

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

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

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

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

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

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

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

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

MIA, September 06, Paris 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 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 Principle bundles

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

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

MIA, September 06, Paris DT-MRI regularization via Lie-group flows 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 separately We may use our framework to regularize and

MIA, September 06, Paris Synthetic data Original P 3 fieldNoisy P 3 fieldDenoised directionsDenoised directions and eigenvalues

MIA, September 06, Paris DTI demonstration

MIA, September 06, Paris 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 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.