B-SPLINE SNAKE:
A FLEXIBLE TOOL FOR
PARAMETRIC CONTOUR DETECTION
P. Brigger, J. Hoeg, and M. Unser
Presented by Yu-Tseh Chi
OUTLINES
Snake revisited
 B-Spline curve
 B-Snake
 B-Spline Snake

 Cubic
B-Spline snake = Snake
 Implicit smoothness constraints
Results and Discussion
 Conclusion

By Yu-Tseh Chi
SNAKE

An energy minimizing parametric curve guided
by external and internal forces.
s*  min{Eint ( s )  Eext ( s )}
s

Internal energy enforce the continuity
constraint.
α |s’|2 + β |s’’|2
By Yu-Tseh Chi
SNAKE

External energy (data term) attracts the curve
to the image features such as edges and
corners.
By Yu-Tseh Chi
DISADVANTAGE OF SNAKE
Slow convergence due to large number of
coefficients
 Difficulty in determining weights associated
with smoothness constraints
Eint = α |s’|2 + β |s’’|2
 Description of the curve by a finite set of
disconnected points.
 High-order derivatives on the discrete curve
may not be accurate in noisy environments.

By Yu-Tseh Chi
OUTLINES
Snake revisited
 B-Spline curve
 B-Snake
 B-Spline Snake

 Cubic
B-Spline snake = Snake
 Implicit smoothness constraints
Results and Discussion
 Conclusion

By Yu-Tseh Chi
B-SPLINE CURVE

A piece-wise polynomial defined by
N 1
S (u )   Ci   id (u )
i 0
C1
C2
Ci : control points
d
 i : basis function of
degree d
C4
C0
C3
C5
By Yu-Tseh Chi
BASIS FUNCTION
u  ui
ui  d 1  u d 1
d 1
 (u ) 
βi (u)
βi 1 (u)
ui  d  ui
ui  d 1  ui 1
d
i
1 if ui  u  ui
{
 (u ) 
0
i
0
otherwise
ui are called knot values
 Number of knot values =
Number of ctrl pts + degree +1

C1
C2
C4
C0
C3
By Yu-Tseh Chi
C5
BASIS FUNCTION
Different ratios between knot intervals define
different curve
 For a degree 3 B-Spline curve with 5 control
points:
u1=[0 1 2 3 4 5 6 7 8] and
u2=[0 2 4 6 8 10 12 14 16] define the same
curve.

By Yu-Tseh Chi
BASIS FUNCTION
u  ui
ui  d 1  u d 1
d 1
 (u ) 
βi (u)
βi 1 (u)
ui  d  ui
ui  d 1  ui 1
d
i
1 if ui  u  ui
{
 (u ) 
0
i
0
otherwise
C1
C2
C4
C0
C3
C5
By Yu-Tseh Chi
B-SPLINE CURVE
N 1
S (u )   Ci   id (u )
i 0
C0

C1
C2
C3
C4
C5
Important properties
 Defined
by only few parameters
 Cd-1 continuity
A degree 3 B-Spline curve is C2
continuous
Duplicate knot values decrease
continuity by 1
By Yu-Tseh Chi
C1
C2
C4
C0
C3
C5
B-SPLINE CURVE
N 1
S (u )   Ci   id (u )
i 0
C0

C1
C2
C3
C4
Important properties
 Defined
by only few parameters
 Cd-1 continuity
A degree 3 B-Spline curve is C2
continuous
Duplicate knot values decrease
continuity by 1
By Yu-Tseh Chi
C5
B-SPLINE CURVE
N 1
S (u )   Ci   id (u )
i 0
C0

C1
C2
C3
C4
Important properties
 Defined
by only few parameters
 Cd-1 continuity
 Locality
By Yu-Tseh Chi
C5
CLOSED CURVE
For a degree d B-Spline curve, add d duplicate
control points in the end.
 Add knot values according.

C2
C8
C1 C7
C3
C4
C5
By Yu-Tseh Chi
C0
C6
OUTLINES
Snake revisited
 B-Spline curve
 B-Snake
 B-Spline Snake

 Cubic
B-Spline snake = Snake
 Implicit smoothness constraints
Results and Discussion
 Conclusion

By Yu-Tseh Chi
DISADVANTAGE OF SNAKE
Slow convergence due to large number of
coefficients
 Difficulty in determining weights associated
with smoothness constraints
Eint = α |s’|2 + β |s’’|2
 Description of the curve by a finite set of
disconnected points.
 High-order derivatives on the discrete curve
may not be accurate in noisy environments.

By Yu-Tseh Chi
B-SNAKE
Proposed by Medioni et. al.
 Curve is replaced by its B-Spline approximation

N 1
S (u )   Ci   id (u )
i 0

Advantages of this formulation
 Local
Control
 Continuity
 Less points to apply optimization
By Yu-Tseh Chi
ENERGY FUNCTION OF B-SNAKE
N 1
E (u )  {  (Ci  ( id (u ))')    (Ci  ( id (u ))")  F ( S (u ))}
i 0
d
(


i )' is the derivative of the basis function
d
d
d 1
d 1
i 
 i 1
=
ui  d  ui
ui  d 1  ui

F(S(u)) is the data term as defined in the orginal
Snake.
By Yu-Tseh Chi
ALGORITHM
Calculate the ctrl points based on some points
sampled from the user-defined curve.
 Update the ctrl points
Cik+1= Cik + η*∂E/ ∂ Cik

By Yu-Tseh Chi
DISADVANTAGE OF B-SNAKE
Does not take advantage of the implicit
smoothness constraint of B-Spline curve.
 Still have the regularization term explicitly

N 1
E (u )  {  (Ci  ( id (u ))')    (Ci  ( id (u ))")  F ( S (u ))}
i 0
By Yu-Tseh Chi
OUTLINES
Snake revisited
 B-Spline curve
 B-Snake
 B-Spline Snake

 Cubic
B-Spline snake = Snake
 Implicit smoothness constraints
Results and Discussion
 Conclusion

By Yu-Tseh Chi
B-SPLINE SNAKE
Proposed by Unser et. al.
 Same formulation as B-Snake
Using S (u)   C   (u) to outline contour in an
image.

N 1
i 0
By Yu-Tseh Chi
i
d
i
CONTRIBUTIONS
Optimal solution for Snake (a curvatureconstrained E ) is a cubic spline.
 Specify initial contour by node points of a Bspline curve instead of control points.
 Enforce smoothness constraint implicitly.
 Improve speed and robutsness by using a
multi-resolution scheme.

By Yu-Tseh Chi
SNAKE = CUBIC SPLINE

S * (u )  arg min  V (k , s(k ))    ( s" (u )) du
s
k

Similar formulation to the one of Snake
 S* is the optimal solution
 V is the data term.
 S(k) some points on the curve S(u)
 2nd term is the smoothness constraint.

By Yu-Tseh Chi
2
SNAKE = CUBIC SPLINE

S * (u )  arg min V (k , S (k ))    ( S " (u )) du
s
2

k
Define Sint(u) is the cubic spline interpolation of
the Snake S(u) and Sint(ui)=S(ui)
 Above equation can be rewritten as

S * (u )  arg min V (k , Sint (k ))
s
k

   S " (u )du

By Yu-Tseh Chi
SNAKE = CUBIC SPLINE

S * (u )  arg min  V (k , Sint (k ))    ( S " (u )) du
s

2

k
Using first integral equation, 2nd term can be
rewritten as




S " (u ) du   Sint " (u ) du
2
2


"
  ( S " (u )  Sint
(u ))2 du

By Yu-Tseh Chi
SNAKE = CUBIC SPLINE

S * (u )  arg min V (k , Sint (k ))    ( Sint " (u ))2 du
s
k


"
   ( S " (u )  Sint
(u ))2 du

S* is the optimal solution s.t. the energy
function is minimized.
 The energy function can be minimized if and
only if the 3rd term is minimized.

S * (u)  S * (u)
"
By Yu-Tseh Chi
"
int
SNAKE = CUBIC SPLINE
S * (u)  S * (u)
"
"
int
By intergrading twice, S(u) – Sint(u) = au+b
 Because of the interpolation condition
S(ui) = Sint(ui) , a = 0 and b=0
  S(u) = Sint(u)

By Yu-Tseh Chi
SNAKE = CUBIC SPLINE

Another way to prove it

S * (u )  arg min V (k , S (k ))    ( S " (u )) 2 du
s
k

Take the Euler-Lagrange of the 2nd term.
( 4)
(u )  0 S(u) is a cubic spline.
 S
 All Splines can be represented by a B-Spline.
 The optimal solution for Snake is a cubic BSpline

By Yu-Tseh Chi
CONTRIBUTIONS
Optimal solution for Snake (a curvatureconstrained E ) is a cubic spline.
 Specify initial contour by node points of a Bspline curve instead of control points.
 Enforce smoothness constraint implicitly.
 Improve speed and robutsness by using a
multi-resolution scheme.

By Yu-Tseh Chi
NODES POINTS
Provide more intuitive user interaction.
 Specify a B-spline curve from user defined
node points.

By Yu-Tseh Chi
NODE POINTS
Node points are points on the B-Spline S(u)
where u = ui ( are knot values of the B-Spline)
 To calculate ctrl points based on given node
points, we use S (u )   C   (u )

N 1
i 0
 S (u0 )    0
 S (u )   0
1 


 ...    .

 
...

  .
 S (un )  ...
N = B*C
-1
 C=B *N
By Yu-Tseh Chi
1
0
2
1
..
.
.
... 0  C0 
... 0   C1 
...  Points
.  Node
 u =ui
  when
.   ... 
. ... Cn 
i
3
i
Control points
CONTRIBUTIONS
Optimal solution for Snake (a curvatureconstrained E ) is a cubic spline.
 Specify initial contour by node points of a Bspline curve instead of control points.
 Impose smoothness constraint implicitly.
 Improve speed and robutsness by using a
multi-resolution scheme.

By Yu-Tseh Chi
IMPLICIT SMOOTHNESS CONSTRAINT

S * (u )  arg min V (k , S (k ))    ( S " (u )) du
s
k
2

2nd term is the smoothness constraint.
 Ignore 2nd term, by introducing smoothness
factor h.

By Yu-Tseh Chi
IMPLICIT SMOOTHNESS CONSTRAINT
Original parameterization
S (u )   C   (u ) with u =[0 1 2 3 …. N+3]
 We sample points S(ui) on the curve to do the
optimization.

N 1
i 0
By Yu-Tseh Chi
i
d
i
IMPLICIT SMOOTHNESS CONSTRAINT
New parameterization
S (u )   C   (u ) with u =[0 1h 2h …. (N+3)h]
 Sample points on S(u) where u is a integer
 h decides how dense we want to sample from
the curve

N 1
i 0
h=1
By Yu-Tseh Chi
i
d
i
h=2
h=4
h=8
IMPLICIT SMOOTHNESS CONSTRAINT
S * (u )  arg min  V ( k , S ( k ))   
s

k


( S " (u )) 2 du
h acts as the regularization factor in Snake.
By Yu-Tseh Chi
IMPLICIT SMOOTHNESS CONSTRAINT
S * (u )  arg min  V ( k , S ( k ))   
s

k


( S " (u )) 2 du
h acts as the regularization factor in Snake.
By Yu-Tseh Chi
CONTRIBUTIONS
Optimal solution for Snake (a curvatureconstrained E ) is a cubic spline.
 Specify initial contour by node points of a Bspline curve instead of control points.
 Impose smoothness constraint implicitly.
 Improve speed and robutsness by using a
multi-resolution scheme.

By Yu-Tseh Chi
MULTI-RESOLUTION SCHEME
Used in many Research topic.
 Image pyramid scheme.
 To increase speed and prevent local minimum.

By Yu-Tseh Chi
ENERGY FORMULATION

Energy function
g(S(i)) is the external potential function .
 Smoothed gradient of the input image.


Φ is a smoothing kernel (Guassian)
By Yu-Tseh Chi
OPTIMIZATION
N 1
S (u )   Ci   i3 (u )
i 0
By Yu-Tseh Chi

Use steepest descent algorithm to obtain new
control points.
k 1
i
C
By Yu-Tseh Chi

 C   (
)
Ci
k
i
OUTLINES
Snake revisited
 B-Spline curve
 B-Snake
 B-Spline Snake

 Cubic
B-Spline snake = Snake
 Implicit smoothness constraints
Results and Discussion
 Conclusion

By Yu-Tseh Chi
RESULTS


By Yu-Tseh Chi
RESULTS
By Yu-Tseh Chi
RESULTS
By Yu-Tseh Chi
RESULTS
By Yu-Tseh Chi
OUTLINES
Snake revisited
 B-Spline curve
 B-Snake
 B-Spline Snake

 Cubic
B-Spline snake = Snake
 Implicit smoothness constraints
Results and Discussion
 Conclusion

By Yu-Tseh Chi
CONCLUSIONS
Snake = cubic B-spline
 Improvement of optimization speed by
introducing # of free parameters of Snake
curve.
 Intuitive user-interaction.

By Yu-Tseh Chi
THANK YOU! QUESTIONS?

Have a nice summer break!
By Yu-Tseh Chi