MUMFORD-SHAH FUNCTIONAL o Mumford-Shah(Ms) functional F(u,)=/(-20dx+a/ bp dr+b d6 MS 9-K Q-K K Observed image Set of discontinuities o Open questions. Conjecture 4.2.1 There exists a minimizer of F such that the edges(the discontinuity set K)are the union of a finite set of Cl, embedded curves Moreover, each curve may end either as a crack tip(a free extremity, i.e., K looks like a half-line) or in triple junction, that is, three curves meeting at their endpoints with 2T /3 angle between each pair. D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and applied mathematics. 42: 577-684. 1989
MUMFORD-SHAH FUNCTIONAL Mumford-Shah (MS) functional: Open questions: Observed Image Set of Discontinuities
FINITE DIFFERENCE APPROXIMATION OF MS MODEL: A CHAMBOLLE. 1995 o Finite difference approximation(ID) F(n)=b∑ h +h∑(nk-)2 k (k+1)h where Wn(t)=min(t2, 1/h)and gk kh F(u)=/(u-0)2dx+ u2d x+card(su Set of discontinuities of u o Finite difference approximation(2D) E(u)+2(座 +2∑(唉1一9)2 Hard to solve!
FINITE DIFFERENCE APPROXIMATION OF MS MODEL: A. CHAMBOLLE, 1995 Finite difference approximation (1D) Finite difference approximation (2D) where and Set of discontinuities of u Hard to solve!
CURVE EVOLUTION Snake Geodesic Active-Contour Chan-Vese Model Level set Method
CURVE EVOLUTION Snake, Geodesic Active-Contour, Chan-Vese Model, Level Set Method
OVERVIEW o Objective: automatically detect contours of objects plotted for t from 0 to 152m) o Questions: 0000 How contours(curves in 2D or surfaces in 3D)are represented? Explicit representation (parametric Implicit representation (level set How the locations of contours are determined? q(VI
OVERVIEW Objective: automatically detect contours of objects. Questions: o How contours (curves in 2D or surfaces in 3D) are represented? • Explicit representation (parametric) • Implicit representation (level set) o How the locations of contours are determined?
SNAKES: ACTIVE CONTOUR MODELS o(Kass, Witkin and Terzopoulos, 1987) J(c)/d(o dq id(o) dgha g(vi(c(o))dq ceC internal energy external energy where C=c: [ a, b]-02, c piecewise CI, c(a)=c(b) c(q)=(c1(q),c2(q),c(q) dci dc? do dq (s)=1/(1+ Edge Indicator Function Makes curves act membrane like Makes curves act thin plate like Makes curves be stuck at edges
SNAKES: ACTIVE CONTOUR MODELS (Kass, Witkin and Terzopoulos, 1987) where Makes curves act membrane like. Makes curves act thin plate like. Makes curves be stuck at edges. Edge Indicator Function