Snakes in Images (Active contour tutorial)
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The document describes how snakes, or active contours, can be used to model shapes in images. It discusses how snakes work by defining an energy function along a curve and minimizing that energy to find the optimal curve. The energy includes an internal term based on curvature and an external term from image features. Level sets are used to propagate the curves towards the minimum energy configuration using gradient descent. Key steps include modeling the shape as a curve, defining the energy function, deriving the curve to minimize energy via calculus of variations, and propagating the curves using level sets.
![Snake -> Curves
Consider a closed planar curve C ( p) {x( p), y( p)}, p [0,1]
C(0.1)
S(p) is the arclength
C(0.2)
C(0.7)
C(0)
C(0.4)
C(0.95)
Cp =tangent
C(0.8)
Css
| Cp |
Css
N
N
1
C(0.9)
Cs
Cp
C
The geometric trace of the curve can be alternatively
represented implicitly as C {( x, y ) | ( x, y ) 0}
1
0
0
1](https://image.slidesharecdn.com/snakecurveslevelset-140129163014-phpapp01/85/Snakes-in-Images-Active-contour-tutorial-4-320.jpg)
![Properties of Level Sets
The level set curvature
Css =div
|
|
Proof. zero change along the level sets,
ss
So,
( x, y )
d
( x xs
ds
,
|
|
y
|
ys )
|
[
,T
d
ds
x
xx s
xy
ys ,
ss
,T
d
ds
x
xy s
yy
ys ],
|
0 , also
, N
|
Numerical Geometry of Images:
http://webcourse.cs.technion.ac.il/236861/Winter2012-2013/en/ho_Tutorials.html](https://image.slidesharecdn.com/snakecurveslevelset-140129163014-phpapp01/85/Snakes-in-Images-Active-contour-tutorial-6-320.jpg)
![Snake Model
1
arg min
C
Cp
0
2
C pp
tension
2
rigidity
2 g (C ) dp
external energy from the image
Energy
g (C )
|
[G (C ) * I (C )] |2
A snake that minimize E must satisfy the Euler equation
Cpp
Cpppp
g C
0
To find a solution
dC
dt
C pp
C pppp
g C
When C stabilizes and the term Ct vanishes, a solution is achieved.
Snakes: Active Contour Models, 1988](https://image.slidesharecdn.com/snakecurveslevelset-140129163014-phpapp01/85/Snakes-in-Images-Active-contour-tutorial-11-320.jpg)
![GVF Snakes
Replace the external force with gradient vector flow (GVF) force v.
dC
dt
C pp
C pppp
g C
V(x,y) = [u(x,y),v(x,y)]
Why?
f generally have large magnitudes only in the vicinity of the edges, the
capture range would be small.
It’s nearly zero in homogeneous regions.
V(x,y) = [u(x,y), v(x,y)] is defined as the vector field that minimizes
the energy functional:
http://www.iacl.ece.jhu.edu/static/gvf/](https://image.slidesharecdn.com/snakecurveslevelset-140129163014-phpapp01/85/Snakes-in-Images-Active-contour-tutorial-13-320.jpg)
Snakes in Images (Active contour tutorial)
- 1. Electrical &Computer Engineering Department University of Houston Snakes in Images Yan Xu Bio-Image Analytics Laboratory
- 2. How snake works in images http://www.iacl.ece.jhu.edu/~chenyang/research/levset/movie/index.html
- 3. How snake works in images Step 1: Model the shape by curve or surface Step 2: Define the energy function E along the curve E = internal energy (curvature) + external energy (images) Step 3: Derive the curve to minimize the energy (calculus of variations) Step 4: Propagate the curves (gradient descent) to reach the minimum using level set (easier to implement in reality). Keywords: Curve Energy Level set
- 4. Snake -> Curves Consider a closed planar curve C ( p) {x( p), y( p)}, p [0,1] C(0.1) S(p) is the arclength C(0.2) C(0.7) C(0) C(0.4) C(0.95) Cp =tangent C(0.8) Css | Cp | Css N N 1 C(0.9) Cs Cp C The geometric trace of the curve can be alternatively represented implicitly as C {( x, y ) | ( x, y ) 0} 1 0 0 1
- 5. Properties of level sets T The level set normal N | N | Proof. Along the level sets (equal 0) we have zero change, that is s 0 , but by the chain rule So, s | ( x, y ) | ,T x 0 xs | y | ,T ys T N | |
- 6. Properties of Level Sets The level set curvature Css =div | | Proof. zero change along the level sets, ss So, ( x, y ) d ( x xs ds , | | y | ys ) | [ ,T d ds x xx s xy ys , ss ,T d ds x xy s yy ys ], | 0 , also , N | Numerical Geometry of Images: http://webcourse.cs.technion.ac.il/236861/Winter2012-2013/en/ho_Tutorials.html
- 7. Constant Flow Ct N Propagation of Curves Change in topology Curvature Flow Ct Grayson N Vanish at a Circular point First becomes convex Gage-Hamilton
- 8. Curve propagation in Level Sets dC dt VN C(t) d dt y V x x, y, t ( x, y; t ) t 0 t N x xt y yt x xt | t C(t) level set , VN , Ct V | y y yt ,N ,N V V 0 , | x | V| |
- 9. Curve propagation in Level Sets Handles changes in topology Numeric grid points never collide or drift apart. Parametric Curve -> Level set in plane -> add time axis for propagation
- 10. Curve propagation in Level Sets Some flows Ct VN Ct N Ct g t t g, N N V 2 xx y div t 2 xy x y 2 2 x y 2 yy x div g g 2 xx y 2 xy x y 2 2 x y 2 yy x g,
- 11. Snake Model 1 arg min C Cp 0 2 C pp tension 2 rigidity 2 g (C ) dp external energy from the image Energy g (C ) | [G (C ) * I (C )] |2 A snake that minimize E must satisfy the Euler equation Cpp Cpppp g C 0 To find a solution dC dt C pp C pppp g C When C stabilizes and the term Ct vanishes, a solution is achieved. Snakes: Active Contour Models, 1988
- 12. Geodesic Active Contour arg min C L C 0 g C ds ds is the Euclidean arc-length. Trying to find minimum weighted distance I(x) Euler Lagrange gradient descent Ct g (C) n g (C), n n x g(x) d dt div g x, y Geodesic Active Contours, 1997 x
- 13. GVF Snakes Replace the external force with gradient vector flow (GVF) force v. dC dt C pp C pppp g C V(x,y) = [u(x,y),v(x,y)] Why? f generally have large magnitudes only in the vicinity of the edges, the capture range would be small. It’s nearly zero in homogeneous regions. V(x,y) = [u(x,y), v(x,y)] is defined as the vector field that minimizes the energy functional: http://www.iacl.ece.jhu.edu/static/gvf/
- 14. GVF Snake Results Original Snake Model 2d GVF vector field GVF Snake Model
- 15. itk Implementation for GVF Geodesic Active Contour Original Image Gradient Magnitude Gradient Initial Contour Sigmoid Rescaling Geodesic Active Contour Gradient Vector Flow Thresholding Insight Segmentation and Registration Toolkit (ITK)
- 16. itk Implementation for GVF Geodesic Active Contour gvfFilter->Update(); GeoAC->SetInput(initialContour->GetOutput()); GeoAC->SetAutoGenerateSpeedAdvection(false); GeoAC->SetFeatureImage(sigmoid->GetOutput()); GeoAC->GenerateSpeedImage(); GeoAC->SetAdvectionImage(gvfFilter->GetOutput());
- 17. Recommended: Numerical Geometry of Images: http://webcourse.cs.technion.ac.il/236861/Winter2012-2013/en/ho_Tutorials.html
- 18. Calculus of Variations Generalization of Calculus that seeks to find the path, curve, surface, etc., for which a given Functional has a minimum or maximum. Goal: find extrema values of integrals of the form F (u, u x )dx It has an extremum only if the Euler-Lagrange Differential Equation is satisfied, u d F ( u, u x ) dx u x 0
- 19. Calculus of Variations Example: Find the shape of the curve {x,y(x)} with shortest length: x1 y ( x1 ) 2 1 y x dx , y ( x0 ) x0 given y(x0),y(x1) x0 Solution: a differential equation that y(x) must satisfy, y xx 0 yx a y ( x ) ax b 3/ 2 2 1 yx x1