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Yaniv Gur and Nir Sochen. Geometric Flows over Lie Groups . Department of Applied Mathematics Tel-Aviv University, Israel. HASSIP, September 2006, Munich. Motivation. Diffusion Tensor MR imaging (DTI) Structure Tensor in imaging Continuous Mechanics: Stress, Strain, etc.

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Geometric flows over lie groups l.jpg

Yaniv Gur and Nir Sochen

Geometric Flows over Lie Groups

Department of Applied Mathematics

Tel-Aviv University, Israel

HASSIP, September 2006, Munich


Motivation l.jpg
Motivation

  • Diffusion Tensor MR imaging (DTI)

  • Structure Tensor in imaging

  • Continuous Mechanics: Stress, Strain, etc.

MIA, September 06, Paris


Diffusion imaging l.jpg
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)


Slide4 l.jpg

Neuron

Myelin

Axon

Axon

White Matter

Anisotropy - The diffusion depends on the gradient direction

MIA, September 06, Paris


Diffusion anisotropy l.jpg
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


Slide6 l.jpg

DiffusionTensor Imaging (DTI)

MIA, September 06, Paris


Fiber tracking l.jpg

Superior Longitudinal

Fasciculus

Uncinate Fasciculus

Corpus Callosum &

Cingulum

Inferior Longitudinal

Fasciculus

Corona Radiata

Fiber Tracking

MIA, September 06, Paris


Pre operative planning l.jpg
Pre operative planning

Front View

Rear View

Top View

Side View

Courtesy of T. Schonberg and Y. Assaf

MIA, September 06, Paris


Denoising tensors via lie group flows l.jpg
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 l.jpg

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


Principal chiral model l.jpg
Principal Chiral Model assigned.

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


The abelian case l.jpg
The Abelian Case assigned.

We use the exp map to write

then

and

MIA, September 06, Paris


Lie group pde flows l.jpg
Lie-group PDE flows assigned.

Equations of motion

Gradient descent equation

Isotropic Lie-group PDE flow

MIA, September 06, Paris


Anisotropic lie group pde flow l.jpg

Examples: assigned.

Anisotropic Lie-group PDE flow

MIA, September 06, Paris


Synthetic data experiments l.jpg

Original O(3) tensor field assigned.

Noisy tensor field

Denoised tensor field - PCM

Synthetic data experiments

MIA, September 06, Paris


Synthetic data experiments16 l.jpg
Synthetic data experiments assigned.

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 experiments17 l.jpg
Synthetic data experiments assigned.

Two parameters subgroup of Sp(4,R)

Image=Trace

original field

noisy field

restored field

MIA, September 06, Paris


Function formulation l.jpg
- function formulation assigned.

Equations of motion

Gradient descent equations

MIA, September 06, Paris


Principle bundles l.jpg
Principle bundles assigned.

  • 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


Slide20 l.jpg

Principle bundles assigned.

  • 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


Beltrami framework l.jpg
Beltrami framework assigned.

Variation of this action yields the equations of motion

Gradient descent equations

MIA, September 06, Paris


Lie group numerical integrators l.jpg
Lie-group numerical integrators assigned.

  • 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 integrators23 l.jpg
Lie-group numerical integrators assigned.

  • 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


Dt mri regularization via lie group flows l.jpg

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


Synthetic data l.jpg
Synthetic data positive-definite

Original P3 field

Noisy P3 field

Denoised directions and eigenvalues

Denoised directions

MIA, September 06, Paris


Dti demonstration l.jpg
DTI demonstration positive-definite

MIA, September 06, Paris


Summary l.jpg
Summary positive-definite

  • 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 l.jpg
Running times positive-definite

  • 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


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