1. Snakes, Shapes, and Gradient Vector Flow
Ayesha Mehboob
Memoona Iftikhar

2. Content What is Snake/Active Contour Model Applications
Working of traditional Snake Model
Improvements to Traditional Snake
GVF Snake Model
Examples of GVF

3. Given: initial contour (model) near desirable object
“Snakes”
Snakes, active contours [Kass, Witkin, Terzopoulos 1987]
In general, deformable models are widely used
Given: initial contour (model) near desirable object

4. “Snakes” Snakes, active contours [Kass, Witkin, Terzopoulos 1987]
Given: initial contour (model) near desirable object
Goal: evolve the contour to fit exact object boundary

5. gradient descent w.r.t. some function describing snake’s quality
A smooth 2D curve which matches to image data
Initialized near target, iteratively refined
Can restore missing data
intermediate
final
initial
Q: How does that work? ….

6. Applications

7. Applications

8. Working of Traditional Snake

9. Preview To find Gradient descent of 𝐹( 𝑥 𝑛 )
We have
Preview
To find Gradient descent of 𝐹( 𝑥 𝑛 )
𝑥 𝑛+1 = 𝑥 𝑛 -𝛻𝐹( 𝑥 𝑛 ), 𝑛≥0
We have,
𝐹( 𝑥 0 ) ≥𝐹 𝑥 1 ≥𝐹( 𝑥 2 )≥…

10. Preview f(x) Q: Is snake (contour) a point in some space? ... Yes
for simplicity, assume that
"snake” is a vector (or point) in R1
f(x)
assume some energy function f(x) describing snake’s “quality”
local minima
for f(x)
gradient descent for 1D functions
Q: Is snake (contour) a point in some space?
... Yes

11. Parametric Curve Representation (continuous case)
A curve can be represented by 2 functions
parameter
closed curve
open curve
Note: in computer vision and medical imaging the term “snake” is commonly associated with
such parametric representation of contours. (Other representations will be discussed later!)

12. Parametric Curve Representation (discrete case)
A curve can be represented by a set of 2D points
parameter

13. Measuring snake’s quality: Energy function
We can define some energy function E(C) that assigns some number (quality measure) to all possible snakes
WHY?: Somewhat philosophical question, but specifying a quality function E(C) is an objective way to define what “good” means for contours C.
Moreover, one can find “the best” contour (segmentation) by optimizing energy E(C).
E(C)
(contours C)
(scalars)

14. Internal energy encourages smoothness or any particular shape
Energy function
Usually, the total energy of snake is a combination of internal and external energies
Internal energy encourages smoothness or any particular shape
Internal energy incorporates prior knowledge about object boundary allowing to extract boundary even if some image data is missing
External energy encourages curve onto image structures (e.g. image edges)

15. Internal Energy (continuous case)
The smoothness energy at contour point v(s) could be evaluated as
Elasticity/stretching
Stiffness/bending
Then, the interior energy (smoothness) of the whole snake is

16. Internal Energy (discrete case)
elastic energy
(elasticity)
bending energy
(stiffness)

17. Internal Energy (discrete case)
Elasticity
Stiffness

18. External energy
The external energy describes how well the curve matches the image data locally
Numerous forms can be used, attracting the curve toward different image features

19. External energy Suppose we have an image I(x,y)
Can compute image gradient at any point
Edge strength at pixel (x,y) is
External energy of a contour point v=(x,y) could be
continuous case
External energy term for the whole snake is
discrete case

20. elastic smoothness term
Basic Elastic Snake
The total energy of a basic elastic snake is
continuous case
discrete case
elastic smoothness term
(interior energy)
image data term
(exterior energy)

21. Basic Elastic Snake (discrete case)
Li-1 Li
Li+1
This can make a curve shrink
(to a point)

22. Basic Elastic Snake (discrete case)
The problem is to find contour
that minimizes
Optimization problem for function
can compute local minima via gradient descent

23. Basic Elastic Snake Synthetic example
(1)
(2)
(3)
(4)

24. Basic Elastic Snake Dealing with missing data
The smoothness constraint can deal with missing data:

25. Basic Elastic Snake Relative weighting
Notice that the strength of the internal elastic component can be controlled by a parameter,
Increasing this increases stiffness of curve
large
medium
small

26. Discrete Snakes Optimization
At each iteration we compute a new snake position within proximity to the previous snake
New snake energy should be smaller than the previous one
Stop when the energy can not be decreased within local neighborhood of the snake (local energy minima)
Optimization Methods
Gradient Descent

27. Gradient Descent Example: minimization of functions of 2 variables
negative gradient at point (x,y) gives direction of the steepest descent towards lower values of function E

28. Stop at a local minima where
Gradient Descent
Example: minimization of functions of 2 variables
update equation for a point p=(x,y)
Stop at a local minima where

29. Gradient Descent for Snakes
here, energy is a function of 2n variables C
simple elastic snake energy
update equation for the whole snake C

30. Gradient Descent for Snakes
here, energy is a function of 2n variables C
simple elastic snake energy
update equation for each node C

31. Problems with snakes
External energy: may need to diffuse image gradients, otherwise the snake does not really “see” object boundaries in the image unless it gets very close to it.
image gradients
are large only directly on the boundary

32. Problems with snakes
(a) Convergence of a snake using (b) traditional potential forces, and (c) shown close-up within the boundary concavity.

33. Improvements in Traditional Snake

34. Alternative Way to Improve External Energy
Use instead of where D() is
Distance Transform (for detected binary image features, e.g. edges)
binary image features
(edges)
Distance Transform
Distance Transform can be visualized as a gray-scale image
Generalized Distance Transform (directly for image gradients)

35. Distance Transform (see p.20-21 of the text book)
Image features (2D) 3 4 2 5 1
Distance Transform
Distance Transform is a function that for each image pixel p assigns a non-negative number corresponding to distance from p to the nearest feature in the image I

36. Problem with distance
(a) Convergence of a snake using (b) distance potential forces, and (c) shown close-up within the boundary concavity.

37. Gradient Vector Flow (GVF) (A new external force for snakes)
Detects shapes with boundary concavities.
Large capture range.

38. Diffusing Image Gradients
image gradients diffused via Gradient Vector Flow (GVF)
Chenyang Xu and Jerry Prince, 98

39. Model for GVF snake
The GVF field is defined to be a vector field
V(x,y) =
Force equation of GVF snake
V(x,y) is defined such that it minimizes the energy functional
f(x,y) is the edge map of the image.

40. |f | is small, energy dominated by first term
( smoothing )
|f | is large, second term dominates
minimal when v= f
μ is tradeoff parameter, increase with noise

41. Diffusion Equations
GVF field can be obtained by solving following Euler equations
2 Is the Laplacian operator.
Reason for detecting boundary concavities.
The above equations are solved iteratively using time derivative of u and v.

42. Diffussion Equation
The steady-state solution of these linear parabolic equations is the desired solution

43. For convenience, we rewrite (14) as follows:

44. To set up the iterative solution, let the indices i, j, and n correspond to x,y , and t, respectively,

45. Substituting these approximations into (15) gives our iterative solution to GVF as follows:

46. Stability of GVF
GVF is stable whenever the step-size restriction is maintained. Since normally ∆𝑥, ∆𝑦 ,𝜇 and are fixed
we find that the restriction 𝑟≤1/4, must be maintained in order to guarantee convergence of GVF:

47. Results of GVF Field
(a) Convergence of a snake using (b) GVF external forces, and (c) shown close-up within the boundary concavity.

48. Traditional external force field v/s GVF field
Traditional force
GVF force
(Diagrams courtesy “Snakes, shapes, gradient vector flow”, Xu, Prince)

49. Difference in Results Image with initial contour Traditional snake
GVF snake

50. Snake Initialization and Convergence
(a) Initial curve and snake results from (b) a balloon with an outward pressure, (c) a distance potential force snake, and (d) a GVF snake.

51. Snake Initialization and Convergence
(a) Initial curve and snake results from (b) a traditional snake, (c) a distance potential force snake, and (d) a GVF snake.

52. References
M. Kass, A. Witkin, and D. Terzopoulos, "Snakes: Active contour models.“, International Journal of Computer Vision. v. 1, n. 4, pp , 1987.
Laurent D.Cohen , “Note On Active Contour Models and Balloons“, CVGIP: Image Understanding, Vol53, No.2, pp , Mar
C. Xu and J.L. Prince, “Gradient Vector Flow: A New External Force for Snakes”, Proc. IEEE Conf. on Comp. Vis. Patt. Recog. (CVPR), Los Alamitos: Comp. Soc. Press, pp , June 1997.