SNAKES: ACTIVE CONTOUR MODELS o Gradient flow at t,)=-c(t,9)+BC(t. ) +A VFl(c(t, q) 0,q)=c0(q), o Drawbacks of“s? Ahere F(n,e2)=g2(V/a、c) (,a)=c(t,b) o Curves' representation is not intrinsic. We could obtain different solutions by changing the parametrization while preserving the same initial curve Because of the regularity constraint, the model does not handle changes of topology We can reach only a local minimum, we have to choose initial curve close enough to the object to be detected The choice of a set of marker points for discretizing the parametrized evolving curve may need to be constantly updated
SNAKES: ACTIVE CONTOUR MODELS Gradient flow: Drawbacks of “snakes”: o Curves’ representation is not intrinsic. We could obtain different solutions by changing the parametrization while preserving the same initial curve. o Because of the regularity constraint, the model does not handle changes of topology. o We can reach only a local minimum, we have to choose initial curve close enough to the object to be detected. o The choice of a set of marker points for discretizing the parametrized evolving curve may need to be constantly updated. where
THE GEODESIC ACTIVE CONTOURS MODEL o Dropping the second order term of“ snakes” J1(c)=/d(a)l dq+x/9(vi(c(o))l ) do o Geodesic active contours model What's their relations? ()=2/Teo)(o) o Intrinsic! Let q=o(r),o: a,6-a,b1,o'>0 J2(c)=2vA g(VI(c(r)c(r)dr o Idea: weight g(vi(c(a)) defines a new Riemannian metric for which we search for geodesics
THE GEODESIC ACTIVE CONTOURS MODEL Dropping the second order term of “snakes” Geodesic active contours model Intrinsic! Let Idea: weight defines a new Riemannian metric for which we search for geodesics. What’s their relations?
CURVATURE. ELEMENTS OF DIFFERENTIAL GEOMETRY o Parametric curves (p)=(x1(p,x2(p)0≤p≤1 o Note T(p)= '(p)=(l(p), .?(p), the tangent vector at a(p), N(p=(.2(P),Ii(p)), the normal vector at .(p). s(p)=Va(r)2+(2(r))2dr, the curvilinear abscissa(or arc length) o Curvature x1(D)x2(D)-x2()x/1( +x℃ a(D)|p(| m/)=6my2 10/x′(p)
CURVATURE: ELEMENTS OF DIFFERENTIAL GEOMETRY Parametric curves: Note: Curvature: and
CURVATURE. ELEMENTS OF DIFFERENTIAL GEOMETRY o Parametric curves (p)=(x1(p,x2(p)0≤p≤1 o Note T(p)= '(p)=(l(p), .?(p), the tangent vector at a(p), N(p=(.2(P),Ii(p)), the normal vector at .(p). s(p)=Va(r)2+(2(r))2dr, the curvilinear abscissa(or arc length) 0 o Curvature(if the curve is parameterized by arc length) a2c N(S KS as 0s2
CURVATURE: ELEMENTS OF DIFFERENTIAL GEOMETRY Parametric curves: Note: Curvature (if the curve is parameterized by arc length):
CURVATURE. ELEMENTS OF DIFFERENTIAL GEOMETRY o Curves as level set of a function u: R2R x(s)={(x1(s),x2(8);u(1(s),2(S)=k} o Differentiating the equation u((a1(s), 2(s))= k x1(s)x1+x2(s)n2=0 入 2 x2(8)= o Suppose '(s)l= 1, then A2=t2 o Differentiating one more time A((ur1) 2un2+(2)a2-2un2 )+1(x1(8)2(s)-2(s)x1(s)=0 (un12u?+(rn)2u 2-2 ur, Ura ulr (vx1)2+(ux2)2)32 什=(F)
CURVATURE: ELEMENTS OF DIFFERENTIAL GEOMETRY Curves as level set of a function : Differentiating the equation Suppose , then Differentiating one more time