1. Fast marching methods The continuous way 1
© Alexander & Michael Bronstein,
© Michael Bronstein, 2010
tosca.cs.technion.ac.il/book
Advanced topics in vision
Processing and Analysis of Geometric Shapes
EE Technion, Spring 2010 1

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

3. Fast marching algorithms

4. Forest fire 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
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 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 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
Intrinsic gradient = projection of
extrinsic gradient on tangent plane
In coordinates of a parametrization ,
is the Jacobian matrix
whose columns span

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

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

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

15. Fast marching methods (FMM)
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.

16. Dijkstra’s algorithm Initialize and for the rest of the graph;
Initialize queue of unprocessed vertices
While
Find vertex with smallest value of ,
For each unprocessed adjacent vertex ,
Remove from
Return distance map

17. Fast marching algorithm
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 Dijkstra’s update Fast marching 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
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
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
Apparently, we have two equations with three variables.
However, is a unit vector, hence
where
Substitute and obtain a quadratic equation

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

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

26. Monotonicity condition
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 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
Solve the quadratic equation and select the largest solution
Compute propagation direction
If monotonicity condition is violated,
Set

29. Causality and monotonicity encore
Acute triangle
All directions in the triangle
satisfy causality and
monotonicity conditions.
Obtuse triangle
Some directions in the triangle
violate causality condition!

30. Fast marching on obtuse meshes
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
Kimmel & Sethian, “Computing geodesic paths on manifolds”, 1998 30

31. Mesh “unfolding”
Virtual connection splits obtuse angle into two acute ones
Kimmel & Sethian, “Computing geodesic paths on manifolds”, 1998

32. MATLAB® intermezzo
Fast marching

34. Eikonal equation on parametric surfaces
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
Eikonal equation in parametrization coordinates

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

40. Heap-based grid update
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
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
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
4 iterations
2 iterations
5 iterations
3 iterations
6 iterations

43. Raster scan fast marching
MATLAB® intermezzo
Raster scan fast marching 43

44. Raster scan fast marching
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

45. Parallellization Rotate scan directions by 450.
All updates performed along a row or column can be parallelized.
Constant CPU load – suitable for SIMD architecture and GPUs.

46. Parallel marching Rotate scan directions by 450.
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!

47. Minimal geodesics
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

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

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

50. Voronoi tessellation & sampling
Uses of fast marching
Geodesic
distances
Minimal geodesics
Voronoi tessellation & sampling
Offset
curves

51. Implicit surfaces Shape represented as level set of some
Examples: medical images, shape-from-X reconstruction, etc.
Triangulation is costly and potentially inaccurate 51

52. Implicit surfaces Two-manifold Narrow band of radius Co-dimension 1
Three-manifold with boundary
smooth if 52

53. Distances on implicit surfaces
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

54. Distances on implicit surfaces
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 be a minimal
geodesic on Then
Memoli & Sapiro, “Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces”, 2001 54

55. Eikonal equation on implicit surfaces
Explicit
Intrinsic eikonal equation
Implicit
Extrinsic eikonal equation
VISCOSITY SOLUTIONS CONVERGE AS 55

56. Narrow band fast marching
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 56