Deformable Models Highdimensional deformable models

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Deformable Models(High-dimensional deformable
models)
Petia Radeva(part III)

• Centre de Visió per ComputadorUniversitat
  Autonoma de Barcelona

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Deformable sheets

• Implicit Euler time integration scheme yields to
  the evolution equation
• Stabilized solution Q ( t ) corresponds to a
  deformed sheet with
• minimal total energy.
• FDM is applied to discretize the deformable
  sheet.
• Deformable meshes - constructed by triangular or
  quadrilateral
• finite elements.
• - simplex angle and discrete mean curvature are
  dened in order to determine the internal energy

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Balloons
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Energy-Minimization of Balloon

• Applying Euler-Lagrange equation leads to a
  linear system
• A.V G
• where A is the stiness matrix, V is the vector of
  coordinates in the chosen
• basis (the values of the surface and its
  derivatives) at the nodes of tesselation.
• Note A is symmetric, positive denite and
  heptadiagonal (tridiagonal per bloc in the 3D
  case).
• Considering the evolution equation
• If elastic parameters are constant, the
  factorization is done once
• at the beginning.

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Deformable B-Grids (A. Gueziec, A. Amini)
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Different topologies of B-grids
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Energy of the deformable surface (Gueziec et al.)
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Three-Dimensional Deformable Models Deformable
B-solid
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Energy of the Deformable B-Solid

• The B-solid is a deformable body with an
  associated energy com-
• posed of internal and external energies
• External forces the B-solid approaches image
  features by minimizing its
• external energy.
• Internal forces the B-solid tends to a smooth
  volumetric shape or
• an ideal solid by minimizing its internal energy .

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Internal Energy of a Smoothing Deformable B-Solid
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Energy-Minimization Procedure
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Properties of the Energy-Minimization Procedure

• Computational scheme
• 1. Three independent linear equations for ( X
  Y Z ).
• 2. Numerical properties of the snake
  computational system.
• 3. Banded and constant stiffness matrices.
• 4. One factorization of the 3 matrices in the
  beginning.
• 5. Allows for displacement of the B-solid
• 6. Does not change the numerical stability of
  the linear system of the snakes
• Tensorial representation of energy-minimization
  procedure
• Decoupled wrt the parameters smoothing operator.
• 1. Possible estimate of the eect of each
  smoothing operator.
• 2. Different elastic parameters to re ect data
  anisotropy.

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Advantages of the B-solid as a volumetric
deformable grid

• C 2 continuity throughout the volume.
• A compact B-spline representation.
• Flexible model due to local control of
  B-splines.
• No strong constraints imposed by a model.
• Few parameters of the B-solid (elastic
  parameters).

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Dynamic analysis by snakes of SPAMM-MRI
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Dynamic analysis by snakes of SPAMM-MRI
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Dynamic analysis by snakes of SPAMM-MRI
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