⇥ M. Curve in parameter domain: t ⇥ [0, 1] ⇤ (t) ⇥ D. Geometric realization: ¯(t) def. = ⇥( (t)) M. For an embedded manifold M Rn: First fundamental form: I = ⇥ ⇥ui , ⇥ ⇥uj ⇥ ⇥ i,j=1,2 . u1 u2 ⇥ ⇥u1 ⇥ ⇥u2 L( ) def. = 1 0 ||¯ (t)||dt = 1 0 ⇥ (t)I (t) (t)dt. Length of a curve ¯ ¯
+ 2 (x)e2 (x)e2 (x)T with 0 < 1 2, Tensor eigen-decomposition: Local anisotropy of the metric: 4 ECCV-08 submission ID 1057 Figure 2 shows examples of geodesic curves computed from a single starting point S = {x1 } in the center of the image = [0, 1]2 and a set of points on the boundary of . The geodesics are computed for a metric H(x) whose anisotropy ⇥(x) (deﬁned in equation (2)) is increasing, thus making the Riemannian space progressively closer to the Euclidean space. Image f = .1 = .2 = .5 = 1 Image f = .5 = 0 = .95 = .7 (x) = ⇥1 (x) ⇥2 (x) ⇥1 (x) + ⇥2 (x) [0, 1] x e1 (x) M e2 (x) 2 (x) 1 2 1 (x) 1 2 { \ H(x) 1} Geodesics tend to follow e1 (x).
and x0 solves (t) = ⇥t H( (t)) 1 Ux0 ( (t)) (0) = x1 t > 0 with U(x) = d(x0, x) Distance map: Theorem: U is the unique viscosity solution of || U(x)||H(x) 1 = 1 with U(x0 ) = 0 where ||v||A = v Av
1(x)⇤V (x), ⇤V (x)⇥ = 1, V (x0 ) = 0. U V , ⇤V ⇥ = H1/2 , H 1/2⇤V ⇥ ||H1/2 ||||H 1/2⇤V || C.S. = 1 If V is smooth on : Let : x0 x be any smooth curve. U(x) = min :x0 x L( ) = 1 0 H( (t)) (t), (t) dt
1(x)⇤V (x), ⇤V (x)⇥ = 1, V (x0 ) = 0. U V , ⇤V ⇥ = H1/2 , H 1/2⇤V ⇥ ||H1/2 ||||H 1/2⇤V || = U(x) = min L( ) V (x) C.S. = 1 = 0 L( ) = 1 0 ||H1/2 || 1 0 ⇥ , ⌅V ⇤ = V ( (1)) V ( (0)) = V (x) If V is smooth on : Let : x0 x be any smooth curve. U(x) = min :x0 x L( ) = 1 0 H( (t)) (t), (t) dt
= 0 (t) = H 1( (t)) V ( (t)) x0 H , = H 1 V, V = 1 x (0) = x Let x be arbitrary. dV ( (t)) dt = (t), V ( (t)) = 1 If V is smooth on ([0, tmax )), then = (tmax ) = x0 = tmax 0 , V = V ( (tmax )) + V ( (0)) = V (x) One has: U(x) L( ) = tmax 0 H , = tmax 0 H ,
(derivative-free) formulation: U(x) = d(x0, x) is the unique solution of Manifold discretization: triangular mesh. U discretization: linear ﬁnite elements. H discretization: constant on each triangle. y U(x) = (U)(x) = min y B(x) U(y) + d(x, y)
(derivative-free) formulation: U(x) = d(x0, x) is the unique solution of Manifold discretization: triangular mesh. U discretization: linear ﬁnite elements. H discretization: constant on each triangle. Ui = (U) i = min f=(i,j,k) Vi,j,k Vi,j,k = min 0 t 1 tUj + (1 t)Uk on regular grid: equivalent to upwind FD. explicit solution (solving quadratic equation). xj xk xi txj + (1 t)xk y +||txj + (1 t)xk xi ||Hijk U(x) = (U)(x) = min y B(x) U(y) + d(x, y)
t 1 tUj + (1 t)Uk Distance function in (i, j, k): (U) i = min f=(i,j,k) Vi,j,k xi xj xk +||txj + (1 t)xk xi ||Hijk g Unknowns: Discrete Eikonal equation: = Vi,j,k U(x) = x xi, g + d gradient
t 1 tUj + (1 t)Uk Distance function in (i, j, k): (U) i = min f=(i,j,k) Vi,j,k X = (xj xi, xk xi ) Rd 2 u = (Uj, Uk ) R2 S = (X X) 1 R2 2 I = (1, 1) R2 xi xj xk +||txj + (1 t)xk xi ||Hijk g Notations: Unknowns: Discrete Eikonal equation: = Vi,j,k U(x) = x xi, g + d gradient Hi,j,k = w2Id 3 (for simpliﬁty)
0 X g + dI = u = d2 2bd + c = 0 a = SI, I b = SI, u c = Su, u w2 = = S(u dI) Quadratic equation: Find g = X , R2 and d = Vi,j,k . || U(xi )|| = ||g|| = w Discrete Eikonal equation: ||XS(u dI)||2 = w2
d = b + a Admissible solution: dj = Uj + Wi ||xi xj || (ui ) = d if 0 min(dj, dk ) otherwise. xi xj xk 1 0 0 X g + dI = u = d2 2bd + c = 0 a = SI, I b = SI, u c = Su, u w2 = = S(u dI) Quadratic equation: Find g = X , R2 and d = Vi,j,k . || U(xi )|| = ||g|| = w Discrete Eikonal equation: ||XS(u dI)||2 = w2
RN solution of (U[N]) = U[N] Linear interpolation: ˜ U[N](x) = i U[N] i i (x) Uniform convergence: || ˜ U[N] U|| N + ⇥ 0 Continuous geodesic distance U(x).
For a mesh with N points: U[N] RN solution of (U[N]) = U[N] Linear interpolation: ˜ U[N](x) = i U[N] i i (x) Uniform convergence: || ˜ U[N] U|| N + ⇥ 0 Numerical evaluation: Continuous geodesic distance U(x).
The value of Ui depends on {Uj }j with Uj Ui . Compute (U) i using an optimal ordering. Front propagation, O(N log(N)) operations. u = (U) i is the solution of max(u Ui 1,j, u Ui+1,j, 0)2+ max(u Ui,j 1, u Ui,j+1, 0)2 = h2W2 i,j (upwind derivatives) Isotropic H(x) = W(x)2Id, square grid. xi+1,j xi,j+1 xi,j
The value of Ui depends on {Uj }j with Uj Ui . Compute (U) i using an optimal ordering. Front propagation, O(N log(N)) operations. triangulation with no obtuse angles. Bad Good u = (U) i is the solution of max(u Ui 1,j, u Ui+1,j, 0)2+ max(u Ui,j 1, u Ui,j+1, 0)2 = h2W2 i,j (upwind derivatives) Isotropic H(x) = W(x)2Id, square grid. Surface (ﬁrst fundamental form) xi xj xk Good Bad xi+1,j xi,j+1 xi,j
from Front to Computed . Iteration Front Ft , Ft = {i \ Ui t} Ft State Si {Computed, Front, Far} 3) Update Uj = (U) j for neighbors 1) Select front point with minimum Ui and
R2. Piecewise-smooth boundary S. Geodesic distance in S for uniform metric: dS (x, y) def. = min ⇥P(x,y) L( ) where L( ) def. = 1 0 | (t)|dt, Shape S Geodesics