Summer Project Presentation

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Summer Project Presentation

• Presented byMehmet Eser
• Advisors
• Dr. Bahram Parvin
• Associate Prof. George Bebis

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Introduction

• What is morphing ?
• In what areas is morphing used ?
• What methods are used for morphing for solid
  shapes?

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What are Solid Shapes?
A slice from a brain MRI scan
Extracted Rendered Isosurface
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Problem Definition

• Interpolation of solid shapes
• Let S be a deformable closed surface such that
  a family of evolved surfaces with
  initial conditions at
• Construct intermediate solid shapes satisfying
  smoothness and continuity in time

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Approach to The Problem

• Defining the intermediate interpolated shapes
  implicitly
• such that
• The givens of the problem

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Regularization Method

• A numerical solution method
• Applied to the ill-posed problems
• The original problem is converted into a
  well-posed problem by satisfying some smoothness
  constraint.
• A smoothing parameter which controls the
  trade-off between an error term and the the
  amount of smoothing (regularization)

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Gradients can be helpful?
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Approach to The Problem

• Gradients can be used for finding a unique
  solution to the problem
• Disadvantages of this approach
• Global average may be small
• But locally gradient of f may change sharply (not
  good for a smooth interpolation of curves)

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Purposed Method

• Minimization of the supremum of the
• For minimization of the supremum of the gradients
  of the functions sup can be written
  as follows (in series)

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Purposed Method

• The minimization of this function can be achieved
  by using the Euler equation
• The result of the min of is the
  following

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Implementation

• Distance Field Transforms
• Finding an approximation to the problem with
  Distance Field Transform.
• Employing the regularization term
• Generation of the Morphing

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Distance Transformation

• Distance Transformations
• Obtained in time for 3D

D(x,y,z)
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An example to Distance Transform

• Original Image Distance Image

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DTs of a Cube and a Sphere
A slice of a distance transformed cube
A slice of a distance transformed sphere
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Signed Distance Transform

• Calculation of signed distance transform
• Take negative of the distance value if the pixel
  is inside the object
• Take positive of the distance value if the pixel
  is outside the object
• Morphing region is defined as

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Interpolation Region
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Interpolating Surfaces
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Why Distance Field ?

• A smooth and natural interpolation of surfaces
• Can be carried out at any desired resolution
• A good initial seed for the iteration with ILE
• PDE s can be calculated finite difference
  formulas

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Numerical Solution to ILE

• Get the interpolated surfaces
• Iterate using regularization term-ILE
• v ? iteration number
• ? step size
• F interpolated volume

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Iteration

• 1.Initialize F with boundary conditions
• 2.Initialize R with the approximated morphing
• 3.Update all points inside R with equation (1)
• 4.Compute
• 5.Repeat 3 4 till the local minimum of sup?F
  is reached.
• 6.Obtain morphed volumes
• S(t) (x,y,z,) F(x,y,z) t

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Results
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Special Thanks to
National Science Foundation (NFS)
UNR Computer Vision Laboratory (Assoc. Prof.
George Bebis lead)
LBL Vision Group
(Dr. Bahram Parvin lead)