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

3. Introduction First, we review the Fast Marching Method for orthogonal grids. First, we review the Fast Marching Method for orthogonal grids. Then, for motivational reasons, we analyze the structure of this method on a triangulated planar grid constructed directly from an orthogonal grid. Then, for motivational reasons, we analyze the structure of this method on a 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. 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. As an application, we compute geodesic distance and minimal geodesic paths on manifolds. 1. Abstract 2. Introduction 3. The FM Method on Orthogonal Grid 4. FM on a Particular Triangulated Planar Domain 5. Construction of Minimal Geodesics

4. The Fast Marching Method on Orthogonal Grid We review the Fast Marching for computing the solution to the Eikonal equation. We review the Fast Marching for computing the solution to the Eikonal equation. non-linear equation formnon-linear equation form or equivalently,or equivalently, time T with speed F (x, y) in the normal direction at a point (x, y)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 when F=1.F (x, y) is typically supplied as known input, in the Fast Marching Method case when F=1. 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

5. The Fast Marching Method on Orthogonal Grid The Fast Marching Method process The Fast Marching Method process A.Given the initial curve (shown in red) B.stand on the lowest spot (which would be any point on the curve) C.build a little bit of the surface that corresponds to the front (shown in green) moving with the speed F Repeat B to CRepeat B to C When this process ends, the entire surface has been built.When this process ends, the entire surface has been built. 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

6. The Fast Marching Method on Orthogonal Grid 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 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). The Fast Marching Method considers the nature of upwind,entropy-satisfying approximations to the Eikonal equation (non- differentiable). 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

7. The Fast Marching Method on Orthogonal Grid Entropy-satisfying approximations to the Eikonal equation Entropy-satisfying approximations to the Eikonal equation Closed initial curve T (0) inClosed initial curve T (0) in Let T (t) be the one parameter family of curves, where t is timeLet 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 speed Fgenerated by moving in the initial curve along its normal vector field ( front )with speed F Let be the position vector which, at time t, parameterizes T (t) by s, 0 ≦ s ≦ S,Let be the position vector which, at time t, parameterizes T (t) by s, 0 ≦ s ≦ S, Travel along the curve in the direction of increasing sTravel along the curve in the direction of increasing s Written in term of the coordinatesWritten in term of the coordinates The equations of motionThe equations of motion 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

8. The Fast Marching Method on Orthogonal Grid First order finite difference schemes First order finite difference schemes For all approximation TFor all approximation T T : center point I, jT : center point I, j Upwind approximation t the gradient, given byUpwind approximation t the gradient, given by Eq.(1)means that information propagates “ one way ”,that is,from smaller values of T to larger values.Eq.(1)means that information propagates “ one way ”,that is,from smaller values of T to larger values. 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

9. The Fast Marching Method on Orthogonal Grid The algorithm rests on ” solving ” Eq.(1) by building the solution outward from the smallest T value. 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 algorithm is made fast by confining the “building zone” to a narrow band around the front. The Fast Marching Method algorithm is as follow : The Fast Marching Method algorithm is as follow : tag points in the initial conditions as Alivetag points in the initial conditions as Alive tag as Close all points one grid point awaytag as Close all points one grid point away Tag as Far all other grid pointsTag as Far all other grid points 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

10. The Fast Marching Method on Orthogonal Grid The center point i,j. If without loss of generality,that TA ≦ TC The center point i,j. If without loss of generality,that TA ≦ TC h is the uniform grid spacingh is the uniform grid spacing If T>TA and T>TC,update T If T>TA and T>TC,update T If T>TA and T ≦ TC,update T. If T>TA and T ≦ TC,update T. 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

11. The Fast Marching Method on Orthogonal Grid 1.Begin Loop : Let Trial be the point Close with the smallest T value 2.Add the point Trial to Alive ; remove it from Close 3.Tag as Close all neighbors of Trial that are not Alive : If the neighbor is in Far remove it from that list and add it to the set Close. 4.Recompute the values of T at all neighbors according to Eqn.(1) by solving the quadratic equation,using only values on points that are Alive. 5.Return to the top of Loop. 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

12. The Fast Marching Method on Orthogonal Grid This algorithm works because the process of recomputing the T values at upwind neighboring points can’t yields a value smaller than any of the Alive points. This algorithm works because the process of recomputing the T values at upwind neighboring points can’t yields a value smaller than any of the Alive points. M total points M total points The speed of the algorithm comes from a heapsort technique to efficiently locate the smallest element in the set Trial. The complexity : O(MlogM)The speed of the algorithm comes from a heapsort technique 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.Dijkstra’s method complexity : O(MlogM), but two points on the graph produces the network minimum length, which may not be optimal. 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

13. The center point i,j. If without loss of generality,that TA ≦ TC The center point i,j. If without loss of generality,that TA ≦ TC h is the uniform grid spacingh is the uniform grid spacing The equation of the plane determined by TA and TC and unknown T,namely 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, Computing the gradient we then want to update a value of T such that, 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 Fast Marching on Particular Triangulated Planar Domain

14. A Construction for Acute Triangulated Compute a possible value for T from each triangle that includes the center point as a vertex. 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 ). Update procedure for non obtuse triangle ABC in which the point to update is C ( T ). altitude h, t=ECaltitude h, t=EC u =T(B)-T(A)u =T(B)-T(A) t =T(C)-T(A)t =T(C)-T(A) Such thatSuch that 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

15. A Construction for Acute Triangulated a=BC, b=ACa=BC, b=AC Similarity t / b=DF/AD =u/ADSimilarity t / b=DF/AD =u/AD Thus CD= b-AD=b –bu/t= b(t-u) /tThus CD= b-AD=b –bu/t= b(t-u) /t By Law of Cosines:By Law of Cosines: And by the Sines :And by the Sines : 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

16. A Construction for Acute Triangulated Using right angle triangle CBGUsing right angle triangle CBG End up with the quadratic for t:End up with the quadratic for t: T() values that form a titled plane with a gradient magnitude equal to F.T() values that form a titled plane with a gradient magnitude equal to F. 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

17. A Construction for Acute Triangulated The solution t must satisfy u<t, and should be updated from within the triangle,namelyThe solution t must satisfy u<t, and should be updated from within the triangle,namely Update procedure :Update procedure : If u<t andIf u<t and then T(C)=min{ T(C), t+ T(A)}then T(C)=min{ T(C), t+ T(A)} else T(C)=min{ T(C), bF+ T(A), aF+ T(B)}else T(C)=min{ T(C), bF+ T(A), aF+ T(B)} 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

18. Extension to General Triangulations Vertex A( blue point) can be updated by its neighboring points only at a limited section of upcoming fronts.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.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.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.Then, the virtual directional edge from B to A. 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

19. Extension to General Triangulations Let hmax, hmin be the maximal and minimal altitudesLet hmax, hmin be the maximal and minimal altitudes Let θmax be the maximal obtuse angleLet θmax be the maximal obtuse angle α=Π-θmax the angle of the extended sectionα=Π-θmax the angle of the extended section θmin be the minimal (acute) angle for all triangles ( yellow 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 e max be the length of the longest edge (edge connected two black points) Let l be the virtual directionalLet l be the virtual directional edge (AB) edge (AB) 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

20. Extension to General Triangulations Assume α, θmin are samll enough angles so thatAssume α, θmin are samll enough angles so that Width of the narrow section is αWidth of the narrow section is α SoSo AB edge takes O(M), andAB edge takes O(M), and the running time the running time complexity is still optimal complexity is still optimal O(MlogM) O(MlogM) 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

21. Construction of Minimal Geodesics Algorithm to compute distance on triangulated manifolds, and construct minimal geodesicsAlgorithm 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 pointSolve 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 equationBacktrack along the gradient of the arrival time field by solving the ordinary difference equation Where X(s) traces out the geodesic path.Where X(s) traces out the geodesic path. 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

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