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/
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
