Loading in 5 sec....

Geometric Flows over Lie Groups PowerPoint Presentation

Geometric Flows over Lie Groups

- By
**val** - Follow User

- 111 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Geometric Flows over Lie Groups' - val

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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.

MIA, September 06, Paris

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)

Myelin

Axon

Axon

White Matter

Anisotropy - The diffusion depends on the gradient direction

MIA, September 06, Paris

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

DiffusionTensor Imaging (DTI)

MIA, September 06, Paris

Fasciculus

Uncinate Fasciculus

Corpus Callosum &

Cingulum

Inferior Longitudinal

Fasciculus

Corona Radiata

Fiber TrackingMIA, September 06, Paris

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

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

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 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

Lie-group PDE flows assigned.

Equations of motion

Gradient descent equation

Isotropic Lie-group PDE flow

MIA, September 06, Paris

Original O(3) tensor field assigned.

Noisy tensor field

Denoised tensor field - PCM

Synthetic data experimentsMIA, September 06, Paris

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 experiments assigned.

Two parameters subgroup of Sp(4,R)

Image=Trace

original field

noisy field

restored field

MIA, September 06, Paris

- function formulation assigned.

Equations of motion

Gradient descent equations

MIA, September 06, Paris

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

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 assigned.

Variation of this action yields the equations of motion

Gradient descent equations

MIA, September 06, Paris

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 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

- 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 flowsMIA, September 06, Paris

Synthetic data positive-definite

Original P3 field

Noisy P3 field

Denoised directions and eigenvalues

Denoised directions

MIA, September 06, Paris

DTI demonstration positive-definite

MIA, September 06, Paris

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 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

Download Presentation

Connecting to Server..