Level Sets

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Level Sets
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Key idea of level sets.

• Not just a representational trick. Natural
  changes in the curve can be defined by natural
  changes to the level set function.

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• An equally important advantage is that it is easy
  to build accurate numerical schemes to
  approximate the equations of motion. That's
  because rather than track buoys around, which
  might end up colliding, one can instead stand on
  the xy plane and compute the answer. Using
  terminology from a completely different language,
  marker particle methods are man-to-man coverage,
  while level set methods are a zone defense (?!)

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Intuition on level sets.

• Given a level set function Phi,
• Which is perhaps a signed distance function.
• And whos zero level set represents a curve.
• How can we expand the curve?
• dPhi/dt ?
• How can we contract the curve?
• dPhi/dt ?
• How can we move the curve left?
• dPhi/dt ?

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The Mumford-Shah framework chan-vese99,
yezzi-tsai-willsky-99

• Mumford-Shah framework partitions the image into
  (multiple) classes according to a minimal length
• curve and reconstructing the noisy signal in
  each class
• Let us consider - a simplified version - the
  binary case and the fact that the reconstructed
  signal is piece-wise constant
• Where the objective is to reconstruct
• the image, using the mean values for the
• inner and the outer region
• Tractable problem, numerous solutions

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The Mumford-Shah framework chan-vese99,
yezzi-tsai-willsky-99

• Taking the derivatives with respect to piece-wise
  constants, it straightforward to show that their
  optimal value corresponds to the means within
  each region
• While taking the derivatives with respect to C
  and using the stokes theorem, the following flow
  is recovered for the evolution of the curve
• An adaptive (directional/magnitude)-wise balloon
  force
• A smoothness force aims at minimizing the length
  of the partition
• That can be implemented in a straightforward
  manner within the level set approach

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Relation with the Mumford-Shah functional

• The Chan and Vese model is a special case of the
  Mumford Shah model (minimal partition problem)
• ?0 and ?1?2?
• uaverage(u0 in/out)
• C is the CV active contour
• Cartoon model

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Level set formulation

• Considering the disadvantages of the active
  contour representation the model is solved using
  level set formulation
• level set form - no explicit contour

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Replacing C with F

• Introducing the Heaviside (sign) and Dirac (PSF)
  functions

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Replacing C with F

• The intensity terms

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Average intensities

• We can calculate the average intensities using
  the step function

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Level set formulation of the model
Combining the above presented energy terms we can
write the Chan and Vese functional as a function
of F. Minimization F wrt. F - gradient
descent The corresponding Euler-Lagrange
equation
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Approximation of the Curvature

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The algorithm

• Initialization n0
• repeat
• n
• Compute c1 and c2
• Evolve the level-set function
• until the solution is stationary, or nnmax

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The Mumford-Shah framework Criticism Results

• Account for multiple classes?
• Quite simplistic model, quite often the means are
  not a good indicator for the region statistics
• Absence of use on the edges, boundary information