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

AxonAxon

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.

kTk DqbqeE

D – Diffusion Tensor

q – Applied gradient direction

Basser et al. (Biophys. J., 66, 1994 )

E – Signal attenuation

MIA, September 06, Paris)λλ(λ

)D(λ)D(λ)D(λ

2

3FA

23

22

21

23

22

21

)/3λλ(λD 321

Diffusion Tensor Imaging (DTI)

Taa

a

a UUD )(3

1

MIA, September 06, Paris

Fiber Tracking

Uncinate Fasciculus

Corpus Callosum& Cingulum

Corona Radiata

Inferior LongitudinalFasciculus

Superior LongitudinalFasciculus

MIA, September 06, Paris

Front View Rear View

Top View Side 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. : n nI D R

• We treat tensors which belong to matrix Lie-groups. n nR G

• Examples of matrix Lie-groups: O(N),

GL(N), Sp(N), etc.

MIA, September 06, Paris

Principal Chiral Model

( )a b abTr T T k the metric over the Lie-group manifold (killing form)

iT generators of the Lie-group, span the Lie-algebra

[ , ] ,c ca b ab c abT T F T F structure constants

1 ,A g g elements of the Lie-algebra, .a

aA A T

21( )

2L d x Tr A A

MIA, September 06, Paris

The Abelian Case

2( ) ( )( ) ( ) || ||a ba bTr A A u u Tr T T u

then1 exp( )( ) exp( )a b c

a b c

aa

A g g u T u T u T

u T

( ) exp( ( ) )aag x u x T

•We use the exp map to write

and

MIA, September 06, Paris

Lie-group PDE flowsEquations of motion

1( ) 0

" ( ) 0"

g g

div g

Gradient descent equation1 1( )

" ( )"

gg g g

tdiv g

1( )g

g g gt

Isotropic Lie-group PDE flow

MIA, September 06, Paris

Anisotropic Lie-group PDE flow

1( ( , , ) )g

g c x y t g gt

1( , , ) (|| ||)c x y t f g g

Examples:

1(|| ||)f g g 21

1

|| ||1

g gk

1 2

2

|| ||exp

g g

k

MIA, September 06, Paris

Synthetic data experiments

Original O(3) tensor field Noisy tensor fieldDenoised tensor field - PCM

MIA, September 06, Paris

Synthetic data experimentsThe 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.

P

, .N NT

N N

O IP J P J J

I O

MIA, September 06, Paris

Synthetic data experimentsTwo parameters subgroup of Sp(4,R)

original field noisy field

restored field

Image=Trace

MIA, September 06, Paris

-function formulation

2 11

2L d x g g

Equations of motion

1

1

1

'0

g gdiv g g

g g

1

1 1

1

' g ggg div g g

t g g

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

2R G

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

2 1 2 2( ) , 0dG Tr g g dG

2 2 2 1 2( )ds dx dy Tr g g

1 1( )Tr g g g g

Principle bundles

2 2 2ds dx dy

MIA, September 06, Paris

Beltrami framework

2 1 11( )

2L d x Tr g g g g

Variation of this action yields the equations of motion

1

2 2

0

" ( ) 0", x

g g

div D g D R

Gradient descent equations

11

" ( )"

gg g g

t

gdiv D g

t

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 .1 2,g g G 1 2g g G

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.

1 ( ( , )),

, ,

: , ( ),

:

n n n ng g h a g t

g G a A h

A G expm a

logm G A

MIA, September 06, Paris

DT-MRI regularization via Lie-group flows

3P

• The DT-MRI data is represented in terms of a 3x3 positive-definite symmetric matrices which forms a symmetric space

3P

• Polar decomposition 3 3(3) (3)tP O D O

• Is the group of 3x3 diagonal positive-definite matrices3 ( , )D GL n R

separately• We may use our framework to regularize and 3D(3)O

MIA, September 06, Paris

Synthetic data

Original P3 fieldNoisy P3 field Denoised 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.