1 jérôme pousin singular perturbations for an elastic model of the heart 24 juin 2008
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Jérôme Pousin
Singular perturbations for an elastic model of the heart
24 juin 2008
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Objectives
Extract the heart anatomy(3-D+t segmentation)
Combine complementary functional data
(Multimodal registration)
Difficulties
LVRV
- patient movement- moving deformable organ- acquisition geometry- the anatomy cannot be easily determined in all modalities
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Segmentation of the heart using an elastic deformable template
1. Elastic deformable template2. Boundary regularization3. Singular perturbation 4. Sketch of the proof
ICJ
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Deformed domain
Theoretical context
Continuum mechanics : equilibrium of an elastic body
x
Superficial forcesu(x)t(x)
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Initial domain
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Theoretical context
Equilibrium
Potential energy
Stress vector Strain vector
Elastic energy External energy
Stress tensor
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Incremental load
Elastic deformable template
Assumption : small displacements
[Vincent, 2001]
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Iterative local formulation
Elastic deformable template
Discretized expression
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Computing a 3-D force field
Force field deriving from a potential image
Gradient vector flow [Xu, 1998]
Imposed conditionForce field null on the border of the object to be segmented
- Potential minimum on the object border
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Computing a 3-D force field
Example : cube image
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Part 1.Segmentation of the heart using an
elastic deformable template
1. Elastic deformable template2. Improving the model’s convergence 3. Boundary regularization4. Model inititialization 5. Results of segmentation
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Boundary regularization
Large isotropic middle layer
Thin peripheral layers
Three-layer model
[Streeter, 1969]
- Ratio of circumferential fibers to longitudinal fibers 10:1
Longitudinal
Circumferential
Longitudinal
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fibers
[Ohayon, 1988] Constitutive law for a fiber-collagen model
pression
Boundary regularization
Asymptotic model ( 0)
t1, t2
[Destuynder, 1996]
Fiber direction
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Boundary regularization
Boundary constraint energy
Without boundary constraint With boundary constraint
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Singular perturbation
Elastic deformable template
Formaly take t=0
Vanishing elasticity allows larger motion
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Part 1.Segmentation of the heart using an
elastic deformable template
1. Elastic deformable template2. Improving the model’s convergence 3. Boundary regularization4. Model inititialization 5. Results of segmentation
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Numerical results