P. Brigger, J. Hoeg, and M. Unser Presented by Yu-Tseh Chi

By Yu-Tseh Chi  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  An energy minimizing parametric curve guided by external and internal forces.  Internal energy enforce the continuity constraint. α |s’| 2 + β |s’’| 2

By Yu-Tseh Chi  External energy (data term) attracts the curve to the image features such as edges and corners.

By Yu-Tseh Chi  Slow convergence due to large number of coefficients  Difficulty in determining weights associated with smoothness constraints E int = 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  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  A piece-wise polynomial defined by Ci : control points : basis function of degree d C0C0 C1C1 C2C2 C3C3 C4C4 C5C5

By Yu-Tseh Chi  u i are called knot values  Number of knot values = Number of ctrl pts + degree +1 C0C0 C1C1 C2C2 C3C3 C4C4 C5C5

By Yu-Tseh Chi  Different ratios between knot intervals define different curve  For a degree 3 B-Spline curve with 5 control points: u1=[ ] and u2=[ ] define the same curve.

By Yu-Tseh Chi C0C0 C1C1 C2C2 C3C3 C4C4 C5C5

 Important properties  Defined by only few parameters  C d-1 continuity A degree 3 B-Spline curve is C 2 continuous Duplicate knot values decrease continuity by 1 C0C0 C1C1 C2C2 C3C3 C4C4 C5C5 C 0 C 1 C 2 C 3 C 4 C 5

By Yu-Tseh Chi  Important properties  Defined by only few parameters  C d-1 continuity A degree 3 B-Spline curve is C 2 continuous Duplicate knot values decrease continuity by 1 C 0 C 1 C 2 C 3 C 4 C 5

By Yu-Tseh Chi  Important properties  Defined by only few parameters  C d-1 continuity  Locality C 0 C 1 C 2 C 3 C 4 C 5

By Yu-Tseh Chi  For a degree d B-Spline curve, add d duplicate control points in the end.  Add knot values according. C0C0 C1C1 C2C2 C3C3 C4C4 C5C5 C6C6 C7C7 C8C8

By Yu-Tseh Chi  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  Slow convergence due to large number of coefficients  Difficulty in determining weights associated with smoothness constraints E int = 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  Proposed by Medioni et. al.  Curve is replaced by its B-Spline approximation  Advantages of this formulation  Local Control  Continuity  Less points to apply optimization

By Yu-Tseh Chi  is the derivative of the basis function =  F(S(u)) is the data term as defined in the orginal Snake.

By Yu-Tseh Chi  Calculate the ctrl points based on some points sampled from the user-defined curve.  Update the ctrl points C i k+1 = C i k + η*∂E/ ∂ C i k

By Yu-Tseh Chi  Does not take advantage of the implicit smoothness constraint of B-Spline curve.  Still have the regularization term explicitly

By Yu-Tseh Chi  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  Proposed by Unser et. al.  Same formulation as B-Snake Using to outline contour in an image.

By Yu-Tseh Chi  Optimal solution for Snake (a curvature- constrained E ) is a cubic spline.  Specify initial contour by node points of a B- spline curve instead of control points.  Enforce smoothness constraint implicitly.  Improve speed and robutsness by using a multi-resolution scheme.

By Yu-Tseh Chi  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)  2 nd term is the smoothness constraint.

By Yu-Tseh Chi  Define S int (u) is the cubic spline interpolation of the Snake S(u) and S int (u i )=S(u i )  Above equation can be rewritten as

By Yu-Tseh Chi  Using first integral equation, 2 nd term can be rewritten as

By Yu-Tseh Chi  S* is the optimal solution s.t. the energy function is minimized.  The energy function can be minimized if and only if the 3 rd term is minimized.

By Yu-Tseh Chi  By intergrading twice, S(u) – S int (u) = au+b  Because of the interpolation condition S(u i ) = S int (u i ), a = 0 and b=0   S(u) = S int (u)

By Yu-Tseh Chi  Another way to prove it  Take the Euler-Lagrange of the 2 nd term.    S(u) is a cubic spline.  All Splines can be represented by a B-Spline.  The optimal solution for Snake is a cubic B- Spline

By Yu-Tseh Chi  Optimal solution for Snake (a curvature- constrained E ) is a cubic spline.  Specify initial contour by node points of a B- spline curve instead of control points.  Enforce smoothness constraint implicitly.  Improve speed and robutsness by using a multi-resolution scheme.

By Yu-Tseh Chi  Provide more intuitive user interaction.  Specify a B-spline curve from user defined node points.

By Yu-Tseh Chi  Node points are points on the B-Spline S(u) where u = u i ( are knot values of the B-Spline)  To calculate ctrl points based on given node points, we use Control points Node Points when u =u i N = B*C  C=B -1 *N

By Yu-Tseh Chi  Optimal solution for Snake (a curvature- constrained E ) is a cubic spline.  Specify initial contour by node points of a B- spline curve instead of control points.  Impose smoothness constraint implicitly.  Improve speed and robutsness by using a multi-resolution scheme.

By Yu-Tseh Chi  2 nd term is the smoothness constraint.  Ignore 2 nd term, by introducing smoothness factor h.

By Yu-Tseh Chi  Original parameterization with u =[ …. N+3]  We sample points S(u i ) on the curve to do the optimization.

By Yu-Tseh Chi  New parameterization 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 h=1 h=2 h=4 h=8

By Yu-Tseh Chi  h acts as the regularization factor in Snake.

By Yu-Tseh Chi  h acts as the regularization factor in Snake.

By Yu-Tseh Chi  Optimal solution for Snake (a curvature- constrained E ) is a cubic spline.  Specify initial contour by node points of a B- spline curve instead of control points.  Impose smoothness constraint implicitly.  Improve speed and robutsness by using a multi-resolution scheme.

By Yu-Tseh Chi  Used in many Research topic.  Image pyramid scheme.  To increase speed and prevent local minimum.

By Yu-Tseh Chi  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

 Use steepest descent algorithm to obtain new control points.

By Yu-Tseh Chi  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 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 = cubic B-spline  Improvement of optimization speed by introducing # of free parameters of Snake curve.  Intuitive user-interaction.

By Yu-Tseh Chi  Have a nice summer break!