Segmentation Using Active Contour Model and Tomlab

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Segmentation Using Active Contour Model and
Tomlab

• By Dalei Wang
• 29/04/2003

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Content

• Introduction
• Mathematical definition of active contour, and
  its energies
• Program design and operation
• Performance and issues
• What to be done next

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The Project

• Started out as an investigation of how the
  interior point optimization algorithm applies to
  segmentation
• Popular segmentation algorithms Euler-Lagrange,
  Dynamic Program-ming, Greedy Algorithm

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The Project

• Objectives Design or find an interior point
  optimization algorithm. Design a self contained
  segmentation program based on interior point
  method.
• Later realized that interior point algorithms
  mostly apply to linear constrained problems,
  unsuitable for energy minimization

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The Project

• Interior point method for non-linear programming
  exists but resources are scarce and beyond my
  current level
• Finally decided to use an off the shelf
  optimization program Tomlab toolbox
• Tomlab will include an interior point solver soon

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Mathematical Definition of Snake Energies

• Active Contour Model, a.k.a. Snake, developed by
  Kass et al. in 1987
• An ordered set of snaxelsp1, p2, pn defined on
  a bounded two dimensional space 1
• A snake is defined so as to possess certain
  energy
• Energy conveniently defined so to reach minimum
  at boundary of objects 1,2

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Mathematical Definition of Snake Energies
Internal

• The internal energy controls the elasticity and
  stiffness of the snake
• Grants resistance to pulling and bending
• Smoothness constraints If no external force
  present the snake will be circular

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Mathematical definition of snake energies
Internal

• Mathematically, internal energy is defined as
  adv/ds2bd2v/ds22 v(s) is snake curve
  parametrically defined in terms of s, the curve
  length
• Must be discretized for the snake is not
  continuous, but defined as a set of points
• Discretized, the internal energy of ith snaxel
  is
• Eint(vi)avi-vi-12 bvi-1-2vi-vi12

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Mathematical definition of snake energies
Internal

• Total internal energy of a snake is the sum of
  all internal energies of individual snaxels.
• Can be expressed as x yAxt ytt, where xx1
  x2 x3..xn and yy1 y2 y3..yn and A is and n by
  n matrix whose elements are expressed in terms of
  a and b.

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Mathematical definition of snake energies
External

• The external energy depends only on the property
  of the image
• Let zI(x, y) be the image function, x, y are
  axes, z is 8 bit grey scale.
• Alternate definition depending on the type of
  image and the object one wants to extract

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Mathematical definition of snake energies
External

• For prostate ultra-sound images (noisy,
  non-homogeneous background,boundary not clearly
  defined) the external energy is

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Mathematical definition of snake energies
External
The gradient magnitude of prostate image

• This is the plot of the x components of image
  gradient
• The magnitude of gradient is strong near edge of
  prostate
• But also at false minima(noises)

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Mathematical definition of snake energies
External
This is the plot of Y gradient component
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Mathematical definition of snake energies

• Minimizing the total snake energy is therefore an
  unconstrained (but bounded) non-linear (due to
  the non-linearity of image function) programming
  problem
• There is an unconstrained non-linear routine in
  Tomlab ucSolve

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Program Design and Operation

• Snake initialization A function allows the user
  to select points on the prostate image
• For optimal results, one should select about 10
  points
• The program automatically interpolates along the
  curve and samples about 50 points

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Program Design and Operation

• Problem definition In order to define the
  problem in Tomlab format, the following data is
  stored in global variable image, gradient
  matrices of the image, matrix used in computing
  internal energy
• It is important for them to be calculated only
  once. They are needed for energy calculation,
  which is evaluated about 30005000 times per run

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Program Design and Operation

• Another design issue It is possible for some
  images to cause most or all snaxels to cluster to
  a region of the prostate edge, leaving a large
  gaps between some neighboring snaxels

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Program Design and Operation

• Solution Interpolating along the snake curve at
  each iteration and sample snaxels at equal
  intervals along the curve length

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A Typical Output
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A Typical Output

• The blue curve is initial snake, the red is the
  final
• Solution found in 32.35 seconds and 10 iterations
  (tested on Pentium III 1GHZ 384 MB Ram)
• This result does reflect typical time cost

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Issues

• Snakes as defined by Kass et al. has some
  intrinsic short comings
• Extreme sensitivity to parameters
• Extreme sensitivity to initialization
• Inability to displace far from original position
  due to weak external force fields away from edges
  
• Will NOT converge into concavities

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Example of Poor Initialization
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Sensitivity to Parameters
a0.3 b0.7 g0.1 -gt a0.3 b0.7 g0.5
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Whats Still to Be Done

• Performance analysis Compute how close the final
  snake is to real contour. Compare with other
  algorithms
• Solve the problem again with interior point
  solver when it comes out

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Bibliography

• 1 K. F. Lai, Deformable Contours Modeling,
  Extraction, Detection and Classiication,
  University of Wisconsin-Madison, 1994
• 2 A. Amini, T.Weymouth, and R. Jain, Using
  Dynamic Programming for Solving Variational
  Problems in Vision, in IEEE Transaction On
  Pattern Analysis And Machine Intelligence, Vol.12
  No.9, September 1990

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Bibliography

• 3 L. Zhang, Active Contour Model, Snake,
  Department of Computer Science, University of
  evade, Reno
• 4 K.Holmstrom.TOMLAB -An Environment for
  Solving Optimization Problems in Matlab. In
  M.Olsson,editor,Proceedings for the Nordic Matlab
  Conference 97,October 27-28 ,Stock-holm,
  Sweden,1997.Computer Solutions Europe AB.

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Thanks

• Prof. Salama
• Prof. Freeman
• Prof. Vanelli
• Jessie Shen
• Bernard Chiu