Transcript
“MRI IMAGE SEGMENTATION USING ACTIVE CONTOUR MODEL “
Shashank Janardan
090912022
Mtech (Biomedical engineering )
Under guidance of
Dr. N. PRADHAN Mrs. RAJITHA K V andProfessor and Head & Mrs. HILDA MAYROSEDepartment of Psychopharmacology, Assistant Professors,NIMHANS, Department of BiomedicalBangalore. Engineering, MIT, Manipal.
Overview
• Objective ?To perform
• Segmentation?Subdividing an image in to its constituent object.
• Application like -Boundary detection of Acoustic neuroma.
Acoustic Neuroma
• Benign tumor arises from the Schwann cells of vestibular nerve.
• Causes ? Electromagnetic radiation from cellular phones
Mutation in NF2 chromosome
Continued.
• Diagnosis – Audiometric, MRI imaging.• Treatment – observation, microsurgical
removal.
Acoustic neuroma in the
right cranial nerve
Courtesy JLN hospital & Research centre
Classical Edge based image segmentation method
Problems with classical methods
• Not effective in presence of noise and artifacts.• Not deform into many different shapes of object boundary.• Manual and global operators.
To overcome these problems
ACTICE CONTOUR MODEL
Curve
• What is curve ?• Vector valued function.• Representation in Parametric form• A curve• s is a parameter arc length.
]1,0[)),(),(()( ssysxsX
X(s=.7)=(x(.7),y(.7))
S=0S=1y
x3
3
Why parametric curve?
• Parametric curve are much easier to draw.• Easy to sample the free parameter.• Commonly used in computer graphics.
Local properties of curve - continuity – defined by derivatives.
Derivatives of vector valued function• 1st derivative of vector valued function defines the
tangent vector at that point .• 2nd derivative of vector valued function defines normal
to the point of tangent vector.
Normal vector
curve
Tangent vector
Derivatives of vector valued function defines the shape of curve.A curve properties can be expressed using points in curve and tangent to that point.
Gradient descent method• Gradient ?• Optimization method.• To find local minimum.• Steps proportional to the negative of the gradient
of function at that point. Initialization of curve C1
1st step finds local minimum curve C2
Global minimum i.e. is final curve C4
Application of Gradient
• Gradient application to find extreme points.
Active contour model
• Active contour models are the deformable model that uses discontinuity-based segmentation method.
• The model moves within the boundaries of the desired object due to two types of driving forces.
• Internal forces (elasticity and bending of curve) are responsible to keep model smooth and continuous.
• External forces (of image edge, line, termination) to move the model toward an boundary.
Active contour model
• Active contour model are called SNAKES ?• Active contour model are deformable model?• Snakes deform to nearest salient feature i.e.
local minimum.• Seeks local minimum rather than global
solution.
Types and Formulation of Active contour model
Active contour MODEL
Geometric Active contour
model
PARAMETRIC Active contour
model
Parametric active contour model represent curves
explicitly in their parametric form during deformation
Formulation for parametric model
Energy minimization
Dynamic force
Energy minimization parametric active contour (EMPAC) model
• The energy minimization formulation used to compute the equilibrium configuration.
• Starting with non equilibrium geometry and ends with equilibrium geometry.
• Using Iterative gradient descent method.
EMPAC model (cont.)
• The properties and behavior of the EMPAC model is specified through a function called ‘energy function’
X(s) is a vector-valued function representing a contour and s is the arc length parameter. In order to produce a bounded line segment, the arc length parameter s takes values in the range of 0 to 1.
Internal energy
Internal energy basically sum of elastic and bending energy
•1st derivative of vector valued function, gives information of longitudinal contraction.•2nd derivative of vector valued function, gives information about curvature.• α and β are elastic and bending coefficient ,determines the extent to which contour is allowed to stretch or bend at that point.
Effect of alpha value on curve
Keeping alpha value less Keeping alpha value high
As the alpha value increases, longitudinal contraction increases and length of the curve decreases.
Effect of beta value on curve
Curve with beta value 0 . Curve with beta value high
As the beta value increases the sharp bending decreases, and the contour becomes smooth.
External energy
• Eext(x,y) external function derived from image
• Line functional is simply image function,
• The edge functional is defined by
termedgelineext EEEE
External energy (cont)
• Terminations are the endpoints of an edge. Junctions and terminations represent robust information.
• gx and gy are the first order derivatives of the image in x and y directions respectively.
• gxx and gyy are the second order derivative of image in x and y directions respectively.
• gx y is the second order derivative of image in direction.
Results after applying external energy function
MRI image Edges of image
Courtesy Jawaharlal Nehru research centre
Termination
Algorithm of EMPAC model
Input MRI image
Filter the image using Gaussian filter
Compute the external forces of smooth image by using gradient operator
Select the contour points near the desired solution
Compute the internal forces by creating pentagonal diagonal matrix
Move the model in the direction of negative gradient, using iterative gradient descent method.
Implementation in MATLAB
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Applying Euler Lagrange Equation
Energy forces converted to discrete form using finite difference equation using Pascal's triangle
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Equation implemented in MATLAB
Array of points in new position
Pentagonal diagonal cyclic matrix
Array of points in current position
Array of External energy function at current position
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Pentagonal cyclic matrix where r= 2α+6β, q=-
(α+4β), p=β, where α, β are responsible for
generating internal force
New position
of contour points
Current position of
contour points
Weighted External energy force of contour points
Algorithm descriptionInput MRI image
Smooth the image using Gaussian filter
compute the external forces of smoothed image-
Edge ,line, termination
Edge energy is negative square root of summation of gradient in x direction
and gradient in y direction .
Line energy is image intensity
Gradient of summation of edge energy and line energy
Termination energy calculated using image
derivatives.
Initialize the contour
Hold on the smooth image
Contour points are picked manually using mouse button
The coordinate values of picked point are placed in the matrix
1st column value and last column value are same to make the curve closed one
Interpolate curve with finer spacing
Generating pentagonal cyclic matrix(Internal forces)
Generate a matrix of dimension equal to number of columns x and y values .
Row is generated of ((2 *alpha+6*beta) , -(alpha +4*beta),beta , no. of zeros equal to 5-dimenstion of
generated matrix, beta, (alpha+4*beta) )
Generated matrix is made pentagonal cyclic matrix by circular shift the values of 1st row in 2nd row
Pentagonal diagonal matrix is generated that represent the internal forces
Inverse of pentagonal matrix using Lu factorization
Moving contour in each iteration External forces of contour points are
interpolated from the image external forces
The coordinate values of every contour point are subtracted from their respective weighted external
forces.
The new coordinate values of contour points are multiplied with the inverse pentagonal diagonal cyclic matrix to make the contour smooth
The contour is plotted on the image
The above steps are repeated for each of the iterations
Results
Results (cont)
Results (cont)
Results cont
More Results
FUTURE SCOPE
Discussions
• Parametric model wide range of applications, but still facing problem-
• Difficulty to deform into boundary concavity.• Difficulty dealing with splitting or merging
model parts.• The 2D EMPAC algorithm can be extended to
segment 3D volume image.
References• M. Kass, A. Witkin, D. Terzopoulos, “Snakes—active contour models”,
International Journal of Computer Vision 1 (4) (1987) 321– 331.• C. Xu, J.L. Prince, “Snakes, shapes, and gradient vector flow”, IEEE
Transactions on Image Processing 7 (3) (1998) 359–369• T. McInerney, D. Terzopoulos, Deformable models in medical image
analysis: A survey”, Medical Image Analysis, 1(2), 1996, Pages 91–108.• R.C. Gonzalez, R.E.Woods and S.L Eddins, “Digital Image Processing Using
MATLAB”, 2nd Edition, Gatesmark Publication, Knoxville, TN. 2009.• Demetri Terzopoulos, John Platt, Alan Barr and Kurt Fleischer, “Elastically
Deformable Models”, Computer Graphics, Volume 21, Number 4, July 1987.
• D.Terzopoulos and K.Fleischer, “Deformable models”, The Visual Computer Volume 4, Number 6, (1988), Page 306–331.
Thank you
Minimizing the energy functional
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Applying euler lagrange condition.
It can be viewed as force balance equation
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DEFORMABLE MODEL IS MADE DYNAMIC
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The model is made dynamic by treating X(s) as a function of time t as well as s.
When the solution finds equilibrium the left side vanishes and we get the solution
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Simplification
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Flowchart Finding X(s) curve that minimizes
the energy functional
Must satisfy Euler Lagrange equation.
In sight of force balance equation.
Deformable model is made dynamic .
Solution stabilized and we achieve a solution .
Numerical implementation
• Finite difference method is used to implement deformable model.
• Finite difference method is efficient to compute.
.
Cont. numerical implementation
• By approximating the derivatives in
And converting into vector notation ),( tj
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Pascal's triangle
Derivative estimated using finite difference equation
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Cont. numerical implementation
Converted to matrix form
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Solved by LU decomposition
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Interpolation
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PENTAGONAL CYCLIC DIAGONAL MATRIX
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Flowchart Approximate the derivatives using
finite difference equation
By converting it in to vector notation
Applying finite difference equation for each derivative
Convert it in to matrix form
Discretizing
• The contour X(s) set of control points.
• The curve is obtained by joining control point.
• Force equations applied to each control point separately.
• Each control point allowed to move freely under the. influence of the forces.
• The energy and force terms are converted to discrete form.
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