C omputing G eodesic P aths on M anifolds

1. Computing Geodesic Paths on Manifolds R. Kimmel J.A. Sethian Department of Mathematics Lawrence Berkeley Laboratory University of California, Berkeley Proc. National. Academy of Sciences 1997

2. Abstract 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a Particular Triangulated Planar Domain 5. Construction of Minimal Geodesics • The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(MlogM) steps. M : total number of grid points • We extend the Fast Marching Method to triangular domains with the same computational complexity. • We provide an optimal time algorithm for computing geodesic distance and thereby extracting shortest paths on triangulated manifolds.

3. Introduction 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a Particular Triangulated Planar Domain 5. Construction of Minimal Geodesics • First, we review the Fast Marching Method for orthogonal grids. • Then, for motivational reasons, we analyze the structure of this method ona triangulated planar grid constructed directly from an orthogonal grid. • We then follow with a general procedure for computing the solution of the Eikonal equation on arbitrary acute triangulated domains, followed by an extension to general (non-acute) triangulates. • As an application, we compute geodesic distance and minimal geodesic paths on manifolds.

4. The Fast Marching Method on Orthogonal Grid 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM ona Particular Triangulated Domain 6. Construction of Minimal Geodesics • We review the Fast Marching for computing the solution to the Eikonal equation. • non-linear equation form • or equivalently, • time T with speed F (x , y)in the normal direction at a point (x , y) • F (x , y)is typically supplied as known input , in the Fast Marching Method case whenF=1.

5. The Fast Marching Method on Orthogonal Grid 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM ona Particular Triangulated Domain 6. Construction of Minimal Geodesics • The Fast Marching Method process • Given the initial curve (shown in red) • stand on the lowest spot (which would be any point on the curve) • build a little bit of the surface that corresponds to the front (shown in green) moving with the speed F • Repeat B to C • When this process ends, the entire surface has been built.

6. The Fast Marching Method on Orthogonal Grid 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM ona Particular Triangulated Domain 6. Construction of Minimal Geodesics • The key idea is to build an a approximation to the gradient term which correctly deals with the development of corners and cusps in the solution. • The Fast Marching Method considers the nature of upwind ,entropy-satisfying approximations to the Eikonal equation (non-differentiable).

7. The Fast Marching Method on Orthogonal Grid 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM ona Particular Triangulated Domain 6. Construction of Minimal Geodesics • Entropy-satisfying approximations to the Eikonal equation • Closed initial curveT (0) in • Let T (t) be the one parameter family of curves, where t is time • generated by moving in the initial curve along its normal vector field ( front )with speedF • Let be the position vector which, at time t , parameterizesT (t) by s,0≦s ≦S , • Travel along the curve in the direction of increasing s • Written in term of the coordinates • The equations of motion

8. The Fast Marching Method on Orthogonal Grid 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM ona Particular Triangulated Domain 6. Construction of Minimal Geodesics • First order finite difference schemes • For all approximation T • T : center point I, j • Upwind approximation t the gradient, given by • Eq.(1)means that information propagates “ one way ” ,that is ,from smaller values of T to larger values.

9. The Fast Marching Method on Orthogonal Grid 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM ona Particular Triangulated Domain 6. Construction of Minimal Geodesics • The algorithm rests on ” solving ” Eq.(1) by building the solution outward from the smallest T value. • The algorithm is made fast by confining the “building zone” to a narrow band around the front. • The Fast Marching Method algorithm is as follow : • tag points in the initial conditions as Alive • tag asCloseall points one grid point away • Tag as Far all other grid points

10. The Fast Marching Method on Orthogonal Grid 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM ona Particular Triangulated Domain 6. Construction of Minimal Geodesics • The center point i ,j. If without loss of generality ,that TA≦TC • h is the uniform grid spacing • If T>TA and T>TC ,update T • If T>TA and T ≦ TC ,update T.

11. The Fast Marching Method on Orthogonal Grid 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM ona Particular Triangulated Domain 6. Construction of Minimal Geodesics • Begin Loop : Let Trial be the pointClose with the smallest Tvalue • Add the point Trial to Alive ; remove it from Close • Tag as Close all neighbors ofTrialthat are not Alive : If the neighbor is in Farremove it from that list and add it to the set Close . • Recompute the values of Tat all neighbors according to Eqn.(1) by solving the quadratic equation ,using only values on points that are Alive . • Return to the top of Loop.

12. The Fast Marching Method on Orthogonal Grid 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM ona Particular Triangulated Domain 6. Construction of Minimal Geodesics • This algorithm works because the process of recomputing the Tvalues at upwind neighboring points can’t yields a value smaller than any of the Alive points. • M total points • The speed of the algorithm comes from a heapsorttechnique to efficiently locate the smallest element in the set Trial. The complexity : O(MlogM) • Dijkstra’s method complexity : O(MlogM) , but two points on the graph produces the network minimum length , which may not be optimal.

13. 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM ona Particular Triangulated Domain 6. Construction of Minimal Geodesics The Fast Marching on Particular Triangulated Planar Domain • The center point i ,j. If without loss of generality ,that TA≦TC • h is the uniform grid spacing • The equation of the plane determined by TA and TC and unknown T ,namely • Computing the gradient we then want to update a value of T such that,

14. A Construction for Acute Triangulated 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM on a Particular Triangulated Domain 6. Construction of Minimal Geodesics • Compute a possible value for T from each triangle that includes the center point as a vertex. • Update procedure for non obtuse triangle ABC in which the point to update is C (T ). • altitude h , t=EC • u=T(B)-T(A) • t=T(C)-T(A) • Such that

15. A Construction for Acute Triangulated 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM on a Particular Triangulated Domain 6. Construction of Minimal Geodesics • a=BC , b=AC • Similarity t / b=DF/AD =u/AD • Thus CD= b-AD=b –bu/t= b(t-u) /t • By Law of Cosines: • And by the Sines :

16. A Construction for Acute Triangulated 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM on a Particular Triangulated Domain 6. Construction of Minimal Geodesics • Using right angle triangle CBG • End up with the quadratic for t: • T() values that form a titled plane with a gradient magnitude equal to F.

17. A Construction for Acute Triangulated 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM on a Particular Triangulated Domain 6. Construction of Minimal Geodesics • The solution t must satisfy u<t , and should be updated from within the triangle ,namely • Update procedure : • If u<t and • then T(C)=min{ T(C), t+ T(A)} • else T(C)=min{ T(C), bF+ T(A), aF+ T(B)}

18. Extension to General Triangulations 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM on a Particular Triangulated Domain 6. Construction of Minimal Geodesics • Vertex A( blue point) can be updated by its neighboring points only at a limited section of upcoming fronts. • Connecting the vertex to any point in this section splits the obtuse angle into two acute ones. • Extend this section (gray section)by recursively unfolding the adjacent triangle (s), until a new vertex B( red point) is included in the extended section. • Then, the virtual directional edge from B to A.

19. Extension to General Triangulations 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM on a Particular Triangulated Domain 6. Construction of Minimal Geodesics • Let hmax , hmin be the maximal and minimal altitudes • Let θmaxbe the maximal obtuse angle • α=Π-θmax the angle of the extended section • θmin be the minimal (acute) angle for all triangles ( yellow section) • Let e max be the length of the longest edge (edge connected two black points) • Let l be the virtual directional edge (AB)

20. Extension to General Triangulations 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM on a Particular Triangulated Domain 6. Construction of Minimal Geodesics • Assume α, θmin are samll enough angles so that • Width of the narrow section is α • So • AB edge takes O(M), and the running time complexity is still optimal O(MlogM)

21. Construction of Minimal Geodesics 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM on a Particular Triangulated Domain 6. Construction of Minimal Geodesics • Algorithm to compute distance on triangulated manifolds , and construct minimal geodesics • Solve Eikonal equation with speed F=1 on the triangulated surface to compute the distance from source point • Backtrack along the gradient of the arrival time field by solving the ordinary difference equation • Where X(s) traces out the geodesic path.

22. Construction of Minimal Geodesics 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a • Particular • Triangulated • Planar Domain 5. FM on a Particular Triangulated Domain 6. Construction of Minimal Geodesics