Geometric Flows over Lie Groups
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Transcript of Geometric Flows over Lie Groups
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Geometric Flows over Lie Groups
Yaniv Gur and Nir Sochen
Department of Applied Mathematics
Tel-Aviv University, Israel
HASSIP, September 2006, Munich
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Motivation
• Diffusion Tensor MR imaging (DTI)
• Structure Tensor in imaging
• Continuous Mechanics: Stress, Strain, etc.
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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)
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White Matter
Neuron
AxonAxon
Myelin
•Anisotropy - The diffusion depends on the gradient direction
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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
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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
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Fiber Tracking
Uncinate Fasciculus
Corpus Callosum& Cingulum
Corona Radiata
Inferior LongitudinalFasciculus
Superior LongitudinalFasciculus
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Front View Rear View
Top View Side View
Courtesy of T. Schonberg and Y. Assaf
Pre operative planning
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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
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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.
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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
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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
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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
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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
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Synthetic data experiments
Original O(3) tensor field Noisy tensor fieldDenoised tensor field - PCM
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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
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Synthetic data experimentsTwo parameters subgroup of Sp(4,R)
original field noisy field
restored field
Image=Trace
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-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
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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
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•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
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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
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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
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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
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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
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Synthetic data
Original P3 fieldNoisy P3 field Denoised directionsDenoised directions and eigenvalues
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DTI demonstration
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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.
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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.