Fast marching methods

Transcription

1 1 Fast marching methods Lecture 3 Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

2 Metric discretization 2 Approach I: discrete metric Discretized shape Discrete metric Metrication error Sampling theorem Approach II: consistently discretized metric Discretized metric

3 Fast marching algorithms 3 Imagine a forest fire

4 Forest fire 4 Fire starts at a source at. Propagates with constant velocity Arrives at time to a point. Fermat s (least action) principle: The fire chooses the quickest path to travel. Governs refraction laws in optics (Snell s law) and acoustics. Fire arrival time = distance map from source.

5 Distance maps on surfaces 5 Distance map on surface Mapped locally to the tangent space A small step in the direction changes the distance by is directional derivative in the direction.

6 Intrinsic gradient 6 For some direction, The perpendicular direction is the direction of steepest change of the distance map. is referred to as the intrinsic gradient. Formally, the intrinsic gradient of function at a point is a map satisfying for any

7 Extrinsic gradient 7 Consider the distance map as a function. The extrinsic gradient of at a point is a map satisfying for any direction In the standard Euclidean basis Usually called the gradient of. What is the connection between intrinsic and extrinsic gradients?

8 Intrinsic and extrinsic gradients 8 Intrinsic gradient = projection of extrinsic gradient on tangent plane In coordinates of a parametrization, is the Jacobian matrix whose columns span.

9 Eikonal equation 9 Let be a minimal geodesic between and. The derivative is the fire front propagation direction. In arclength parametrization. Fermat s principle: Propagation direction = direction of steepest increase of. Geodesic is perpendicular to the level sets of on.

10 Eikonal equation 10 Eikonal equation (from Greek εικων) Hyperbolic PDE with boundary condition Minimal geodesics are characteristics. Describes propagation of waves in medium.

11 Eikonal equation 11 Let be a minimal geodesic between and. The derivative is the fire front propagation direction. In arclength parametrization. Fermat s principle: Propagation direction = direction of steepest increase of. Geodesic is perpendicular to the level sets of on.

12 Uniqueness of solution 12 In classic PDE theory, a solution is a continuous differentiable function satisfying PDE theory guarantees existence and uniqueness of solution. Distance map is not everywhere differentiable. Solution is not unique! 1D example

13 13

14 Sub- and super-derivatives (1D case) 14 Superderivative: the set of all slopes above the graph Subderivative: the set of all slopes below the graph where is differentiable.

15 Viscosity solution 15 is a viscosity solution of the 1D eikonal equation if Monotonicity: viscosity solution Not a viscosity solution does not have local maxima. The largest among all Existence and uniqueness guaranteed. Viscosity solution

16 Fast marching methods (FMM) 16 A family of numerical methods for solving eikonal equation. Finds the viscosity solution = distance map. Simulates wavefront propagation from a source set. A continuous variant of Dijkstra s algorithm. Consistently approximate the intrinsic metric on the surface.

17 Fast marching algorithm 17 Initialize and mark it as black. Initialize for other vertices and mark them as green. Initialize queue of red vertices. Repeat Mark green neighbors of black vertices as red (add to ) For each red vertex For each triangle sharing the vertex Update from the triangle. Mark with minimum value of as black (remove from ) Until there are no more green vertices. Return distance map.

18 Update step 18 Dijkstra s update Vertex updated from adjacent vertex Distance computed from Path restricted to graph edges Fast marching update Vertex updated from triangle Distance computed from and Path can pass on mesh faces

19 Fast marching update step 19 Update from triangle Compute from and Model wave front propagating from planar source unit propagation direction source offset Front hits at time Hits at time When does the front arrive to? Planar source

20 Fast marching update step 20 Assume w.l.o.g. and. is given by the point-to-plane distance Solve for parameters and using the point-to-plane distance In vector notation where,, and. In a non-degenerate triangle matrix is full-rank

21 Fast marching update step 21 Apparently, we have two equations with three variables. However, is a unit vector, hence. where. Substitute and obtain a quadratic equation

22 Causality condition 22 Quadratic equation is satisfied by both and. Two solutions for Causality: front can propagate only forward in time. Causality condition

23 Causality condition 23 Causality condition In other words has to form obtuse angles with both triangle edges. Causality is required to obtain consistent approximation of the distance map. Smallest solution for is inconsistent and is discarded. If largest solution is consistent, live the largest solution!

24 Monotonicity condition 24 Viscosity solution has to be a monotonically increasing function. Monotonicity condition: increase when or increase. In other words: Differentiate w.r.t obtaining

25 Monotonicity condition 25 Substitute Monotonicity satisfied when both coordinates of have the same sign. is positive definite At least one coordinate Causality condition: of is negative Monotonicity condition:

26 Monotonicity condition 26 Since we have Rows of are orthogonal to triangle edges Monotonicity condition: Geometric interpretation: must form obtuse angles with normals to triangle edges. Said differently: must come from within the triangle.

27 One-sided update 27 Monotonicity condition: update direction must come from within the triangle. If it does not, project inside the triangle. will coincide with one of the edges. Update will reduce to Dijkstra s update or

28 Fast marching update 28 Solve for the quadratic equation Compute propagation direction If monotonicity condition is violated, Set

29 Consistency and monotonicity encore 29 Consistency Monotonicity Consistency Acute triangle All directions in the triangle satisfy consistency and monotonicity conditions. Obtuse triangle Some directions in the triangle violate consistency condition!

30 Fast marching on obtuse meshes 30 Inconsistent solution if the mesh contains obtuse triangles Remeshing is costly Solution: split obtuse triangles by adding virtual connections to non-adjacent vertices Done as a pre-processing step in

31 Mesh unfolding 31 Virtual connection splits obtuse angle into two acute ones

32 32 MATLAB intermezzo Fast marching

33 33

34 Eikonal equation on parametric surfaces 34 Parametrization of over. Compute distance map, from source. Chain rule Extrinsic gradient in parametrization coordinates Intrinsic gradient in parametrization coordinates

35 Eikonal equation on parametric surfaces 35 Eikonal equation in parametrization coordinates

36 Fast marching on parametric surfaces 36 Solve eikonal equation in parametrization domain March on discretized parametrization domain. We need to express update step in parametrization coordinates.

37 Fast marching on parametric surfaces 37 Cartesian sampling of with unit step. Some connectivity (e.g. 4- or 8-neighbor). Vertex updated from triangle Assuming w.l.o.g. or in matrix form

38 Fast marching on parametric surfaces 38 Inner product matrix Describes triangle geometry. lengths of the edges. cosine of the angle. Substitute into the update quadratic equation Only first fundamental form coefficients and grid connectivity are required for update. Can measure distances when only surface gradients are known.

39 Unfolding on parametric surfaces 39 Virtual connections can be made directly in parametrizatio domain Parametrization domain On the surface Kimmel & Spira, An efficient solution to the eikonal equation on parametric manifolds, 2004

40 Heap-based grid update 40 Fast marching and Dijkstra s algorithm use heap-based grid update. Next vertex to be updated is decided by extracting the smallest. Update order is unknown and data-dependent. Inefficient use of memory system and cache. Inherently sequential algorithm next update depends on previous one. Can we do better? Regular access to memory (known in advance). Vectorizable (parallelizable) algorithm.

41 Marching even faster 41 Danielsson s algorithm: update the grid in a raster scan order In Euclidean case, parametrization is trivial. Geodesics are straight lines in parametrization domain. Each raster scan covers ¼ of the possible directions of the geodesics. Euclidean distance map computed by four alternating raster scans.

42 Raster scan fast marching 42 Generally, geodesics are curved in parametrization domain. Raster scans have to be repeated to produce a convergent solution. Iterative algorithm. Number of iterations depends on geometry and parametrization. Practically, few iterations are required. 1 iteration 2 iterations 4 iterations 5 iterations 3 iterations 6 iterations

43 Raster scan fast marching 43 What we lost: No more a one-pass algorithm. Computational complexity is data-dependent. What we found: Coherent memory access, efficient use of cache. No heap, each iteration is. Raster scans can be parallelized. BBK, "Parallel algorithms for approximation of distance maps on parametric surfaces, 2007

44 Parallellization 44 Rotate scan directions by All updates performed along a row or column can be parallelized. Constant CPU load suitable for SIMD architecture and GPUs.

45 Parallel marching 45 Rotate scan directions by All updates performed along a row or column can be parallelized. Constant CPU load. Suitable for SIMD architecture and GPUs. GPU implementation computes geodesic on grid with 10,000,000 vertices in less than 50 msec. About 200 million distances per second!

46 Minimal geodesics 46 We have a numerical tool to compute geodesic distance. Sometimes, the shortest path itself is needed. Minimal geodesics are characteristics of the eikonal equation. In other words: Along geodesic, eikonal equation becomes an ODE with initial condition. Solve the ODE for.

47 Minimal geodesics 47 To find a minimal geodesic between two points Compute distance map from to all other points. Starting at, follow the direction of until Steepest descent on the distance map. is reached. In the parametrization coordinates Let be the preimage of in

48 Minimal geodesics 48 Substitute into characteristic equation Steepest descent on surface = scaled steepest descent in parametrization domain.

49 49 Numerical geometry of non-rigid shapes Fast Marching Methods Uses of fast marching Geodesic distances Minimal geodesics Voronoi tessellation & sampling Offset curves

50 Implicit surfaces 50 Shape represented as level set of some Examples: medical images, shape-from-x reconstruction, etc. Triangulation is costly and potentially inaccurate

51 Implicit surfaces 51 Two-manifold Co-dimension 1 Narrow band of radius Three-manifold with boundary smooth if

52 Distances on implicit surfaces 52 Since, for all Similarly, for, and hence The sequence is bounded and nondecreasing and hence converges to the supremum of its range For every and there exists such that

53 Distances on implicit surfaces 53 Uniform convergence of geodesic distances: For every there exists an such that for every, If then where is a constant dependent on the geometry of Convergence of minimal geodesics: Let be a unique minimal geodesic between and let geodesic on. Then be a minimal Memoli & Sapiro, Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces, 2001

54 Eikonal equation on implicit surfaces 54 Explicit Intrinsic eikonal equation Implicit Extrinsic eikonal equation VISCOSITY SOLUTIONS CONVERGE AS

55 Narrow band fast marching 55 Euclidean fast marching on Cartesian grid Only vertices inside narrow band do not participate in update Initial values of source set interpolated on the grid Heap or raster scan grid visiting